'203.252' ************** MAGMA ***************** Host 203.252.48.247 (203.252.48.247) Time: Sat Dec 31 14:53:23 2005 Input: nat:=5; pairs:=[{i,j}: i in [1..j-1], j in [1..nat]]; G:=PermutationGroup<#pairs | [[Index(pairs,p^g): p in pairs]: g in [(1,2),(1,2,3),(4,5)]]>; Rat:=Rationals(); R0:=InvariantRing(G,Rat); PI:=PrimaryInvariants(R0); print "*** Primary invariants"; PI; #PS:=SecondaryInvariants(R0); #print "*** Secondary invariants"; #PS; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 31 2005 14:53:03 on modular [Seed = 350336662] ------------------------------------- Errors: /bin/sh: line 1: 24073 Alarm clock nice -n 19 /usr/local/bin/magma '203.252' ************** MAGMA ***************** Host 203.252.48.247 (203.252.48.247) Time: Sat Dec 31 14:31:06 2005 Input: nat:=5; pairs:=[{i,j}: i in [1..j-1], j in [1..nat]]; G:=PermutationGroup<#pairs | [[Index(pairs,p^g): p in pairs]: g in [(1,2),(1,2,3),(4,5)]]>; Rat:=Rationals(); R0:=InvariantRing(G,Rat); PI:=PrimaryInvariants(R0); print "*** Primary invariants"; PI; PS:=SecondaryInvariants(R0); print "*** Secondary invariants"; PS; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 31 2005 14:30:45 on modular [Seed = 4161628035] ------------------------------------- Errors: /bin/sh: line 1: 24038 Alarm clock nice -n 19 /usr/local/bin/magma '203.252' ************** MAGMA ***************** Host 203.252.48.247 (203.252.48.247) Time: Sat Dec 31 14:13:07 2005 Input: nat:=4; pairs:=[{i,j}: i in [1..j-1], j in [1..nat]]; G:=PermutationGroup<#pairs | [[Index(pairs,p^g): p in pairs]: g in [(1,2),(3,4)]]>; Rat:=Rationals(); R0:=InvariantRing(G,Rat); PI:=PrimaryInvariants(R0); print "*** Primary invariants"; PI; PS:=SecondaryInvariants(R0); print "*** Secondary invariants"; PS; Output: Magma V2.11-10 Sat Dec 31 2005 14:13:07 on modular [Seed = 3409858710] ------------------------------------- *** Primary invariants [ x1, x2 + x3 + x4 + x5, x6, x2^2 + x3^2 + x4^2 + x5^2, x2*x3 + x4*x5, x2*x4 + x3*x5 ] *** Secondary invariants [ 1, x2^3 + x3^3 + x4^3 + x5^3 ] Total time: 0.210 seconds, Total memory usage: 3.43MB '203.252' ************** MAGMA ***************** Host 203.252.48.247 (203.252.48.247) Time: Sat Dec 31 14:03:34 2005 Input: nat:=3; pairs:=[{i,j}: i in [1..j-1], j in [1..nat]]; G:=PermutationGroup<#pairs | [[Index(pairs,p^g): p in pairs]: g in [(1,2),(1,2,3)]]>; Rat:=Rationals(); R0:=InvariantRing(G,Rat); PI:=PrimaryInvariants(R0); print "*** Primary invariants"; PI; PS:=SecondaryInvariants(R0); print "*** Secondary invariants"; PS; Output: Magma V2.11-10 Sat Dec 31 2005 14:03:33 on modular [Seed = 3744090820] ------------------------------------- *** Primary invariants [ x1 + x2 + x3, x1^2 + x2^2 + x3^2, x1^3 + x2^3 + x3^3 ] *** Secondary invariants [ 1 ] Total time: 0.200 seconds, Total memory usage: 3.43MB '212.138' ************** MAGMA ***************** Host 212.138.47.21 (212.138.47.21) Time: Tue Dec 20 09:25:33 2005 Input: P:=PolynomialRing(RationalField(),5); I:=ideal

; Radical(I); Output: Magma V2.11-10 Tue Dec 20 2005 09:25:32 on modular [Seed = 2940532906] ------------------------------------- Ideal of Polynomial ring of rank 5 over Rational Field Lexicographical Order Variables: x, y, z, u, v Dimension 0, Radical Groebner basis: [ x - v^16 - 8*v^14 - 32*v^12 - 80*v^10 - 138*v^8 - 168*v^6 - 144*v^4 - 80*v^2 - 26, y - v^8 - 4*v^6 - 8*v^4 - 8*v^2 - 5, z - v^4 - 2*v^2 - 2, u - v^2 - 1, v^32 + 16*v^30 + 128*v^28 + 672*v^26 + 2580*v^24 + 7664*v^22 + 18208*v^20 + 35296*v^18 + 56472*v^16 + 74944*v^14 + 82432*v^12 + 74624*v^10 + 54792*v^8 + 31776*v^6 + 13888*v^4 + 4160*v^2 - v + 677 ] Total time: 0.190 seconds, Total memory usage: 3.43MB '212.138' ************** MAGMA ***************** Host 212.138.113.12 (212.138.113.12) Time: Tue Dec 20 09:24:52 2005 Input: P:=PolynomialRing(RationalField(),5); I:=ideal

; Radical(I); Output: Magma V2.11-10 Tue Dec 20 2005 09:24:51 on modular [Seed = 2975524571] ------------------------------------- Ideal of Polynomial ring of rank 5 over Rational Field Lexicographical Order Variables: x, y, z, u, v Dimension 0, Radical Groebner basis: [ x - v^16 - 8*v^15 - 40*v^14 - 140*v^13 - 390*v^12 - 884*v^11 - 1702*v^10 - 2790*v^9 - 3980*v^8 - 4900*v^7 - 5282*v^6 - 4876*v^5 - 3910*v^4 - 2580*v^3 - 1440*v^2 - 567*v - 183, y - v^8 - 4*v^7 - 12*v^6 - 22*v^5 - 35*v^4 - 38*v^3 - 37*v^2 - 21*v - 13, z - v^4 - 2*v^3 - 4*v^2 - 3*v - 3, u - v^2 - v - 1, v^32 + 16*v^31 + 144*v^30 + 920*v^29 + 4620*v^28 + 19208*v^27 + 68348*v^26 + 212732*v^25 + 588380*v^24 + 1462760*v^23 + 3297580*v^22 + 6786000*v^21 + 12814320*v^20 + 22292560*v^19 + 35837420*v^18 + 53355230*v^17 + 73679935*v^16 + 94452240*v^15 + 112430520*v^14 + 124216240*v^13 + 127251670*v^12 + 120654560*v^11 + 105615510*v^10 + 85034690*v^9 + 62677680*v^8 + 42006568*v^7 + 25385078*v^6 + 13653832*v^5 + 6434290*v^4 + 2579820*v^3 + 849969*v^2 + 208088*v + 33673 ] Total time: 0.210 seconds, Total memory usage: 3.43MB '212.138' ************** MAGMA ***************** Host 212.138.47.24 (212.138.47.24) Time: Tue Dec 20 09:24:05 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Tue Dec 20 2005 09:24:05 on modular [Seed = 3176590049] ------------------------------------- >> P:=PolynomialRing(RationalField(),3); I:=ideal

> -x + (y^2+y+1), ^ User error: Identifier 'x' has not been declared or assigned >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.17 (212.138.47.17) Time: Tue Dec 20 09:19:52 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Tue Dec 20 2005 09:19:51 on modular [Seed = 2321668820] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x - z^4 - 2*z^3 - 4*z^2 - 3*z - 3, y - z^2 - z - 1, z^8 + 4*z^7 + 12*z^6 + 22*z^5 + 35*z^4 + 38*z^3 + 37*z^2 + 20*z + 13 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '212.138' ************** MAGMA ***************** Host 212.138.113.12 (212.138.113.12) Time: Tue Dec 20 09:18:42 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Tue Dec 20 2005 09:18:42 on modular [Seed = 2625096967] ------------------------------------- >> -x + y^2+y+1), ^ User error: bad syntax >> -y + z^2+z+1), ^ User error: bad syntax >> -z + (x^2+x+1) >; ^ User error: bad syntax >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '66.80.2' ************** MAGMA ***************** Host 66.80.213.2 (66.80.213.2) Time: Mon Dec 19 18:33:57 2005 Input: 123 Output: Magma V2.11-10 Mon Dec 19 2005 18:33:57 on modular [Seed = 3375221216] ------------------------------------- 123 Total time: 0.200 seconds, Total memory usage: 3.24MB '66.80.2' ************** MAGMA ***************** Host 66.80.213.2 (66.80.213.2) Time: Mon Dec 19 18:33:54 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Mon Dec 19 2005 18:33:54 on modular [Seed = 3425225954] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.200 seconds, Total memory usage: 3.24MB '130.83.' ************** MAGMA ***************** Host 130.83.2.27 (130.83.2.27) Time: Mon Dec 19 09:54:32 2005 Input: F2 := GF(2^163); a := 00000000 28194bf9 c17b1b81 9a71b20f 5ef0b51d 67cb6830; b := 00000009 8e59a5bf 2e0edbab 1487fcce ba428afa 855dfd12; E := EllipticCurve([1, 0, 0, a, b]); time #E; FactoredOrder(E); Output: Magma V2.11-10 Mon Dec 19 2005 09:54:32 on modular [Seed = 1653074050] ------------------------------------- >> a := 00000000 28194bf9 c17b1b81 9a71b20f 5ef0b51d 67cb6830; ^ User error: bad syntax >> b := 00000009 8e59a5bf 2e0edbab 1487fcce ba428afa 855dfd12; ^ User error: bad syntax >> E := EllipticCurve([1, 0, 0, a, b]); ^ User error: Identifier 'a' has not been declared or assigned >> time #E; ^ User error: Identifier 'E' has not been declared or assigned >> FactoredOrder(E);; ^ User error: Identifier 'E' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.83.' ************** MAGMA ***************** Host 130.83.2.27 (130.83.2.27) Time: Mon Dec 19 09:53:09 2005 Input: F2 := GF(2^163); a := 0000000028194bf9c17b1b819a71b20f5ef0b51d67cb6830; b := 000000098e59a5bf2e0edbab1487fcceba428afa855dfd12; E := EllipticCurve([1, 0, 0, a, b]); time #E; FactoredOrder(E); Output: Magma V2.11-10 Mon Dec 19 2005 09:53:09 on modular [Seed = 1820976753] ------------------------------------- >> a := 0000000028194bf9c17b1b819a71b20f5ef0b51d67cb6830; ^ User error: bad syntax >> b := 000000098e59a5bf2e0edbab1487fcceba428afa855dfd12; ^ User error: bad syntax >> E := EllipticCurve([1, 0, 0, a, b]); ^ User error: Identifier 'a' has not been declared or assigned >> time #E; ^ User error: Identifier 'E' has not been declared or assigned >> FactoredOrder(E);; ^ User error: Identifier 'E' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:46:38 2005 Input: 2^123497 Output: Magma V2.11-10 Mon Dec 19 2005 08:46:38 on modular [Seed = 98631666] ------------------------------------- 2001587198193684625170316366565213998770374765249539510968102874320528369280493\ 8986637440863966655910880436057015983629544805127690562348522105614355084980855\ 8100817350081063197408828461733641385699501604461211045438847140673688794520554\ 2154078678320893488245862834270199856499880144810356878526556017678836917907254\ 1046165766739171527647053135746574000765241140167521092967514930946037638159715\ 7324103041307853303707793081576249426290030160858776795634913065666764828870166\ 0817659761875008825402341766867988536473966628066571208495055021474203862053389\ 5619540415131014622090395555693412034725579822183946548196555711278388954151119\ 0335085512829608797342262217110627469824475552779402624494276981138185048459139\ 8524065478614504954527189393774015733718549520302500730232792551080567537098078\ 8504520403789845365307705116991604735772774428558035953536010649938224410067292\ 8726555743983504829486577211989987455967309277828078519623595338742342437569964\ 5480459542625040887895090424933868304713939583880575630273834790051610134516840\ 6227136764211875488062082160793844857874605530551098666226714806818667434144772\ 1063136347457836471378902998209687326426084007814921488032709718444818718797563\ 1141813287986845869664517980768746940850528726695163905943110109392108359161327\ 1435209214981086265090134155848586057194572973901021662468515412261193920494090\ 9274509060742943695708606635951092816028314742513588178522068898450751376914728\ 2105774082292189829524094815261817282380214547134492188670089396881892310558668\ 1075138822851095257150058382118481180777512842216174253916023460239766019544100\ 3440843242728325628214429228738744714732818782903251958599530525220184027857144\ 8494766691817545549179063010274278690882471994516169263645095088224017259735556\ 5460231370789994586701620182457752827395050864683700385365475052736126239627619\ 4565459534267572124445589815767503559146817323763563208395710939760067521823093\ 1039478071755774924299386829359477219344654557781020906230508052676142868199375\ 8820931570513731276070453033810404956787631704576192896431835499331291546070722\ 7190201851621353358192174806118358472970591935359590026199558874379595660861858\ 9696012980229179034987320784459133545806797826719157383137889513193005979305814\ 9517009315897690318613758217204056551173667110384474025519330720383291584977125\ 1501477939505655648279305065857100625251200428822652269455045440687939400145210\ 5944109328090321327466102115295188267167908443558655805705652964037232040454723\ 3214525907425420448703413811844845355431305416273212490421465116252485072159739\ 5832728124959630867540680772887561691838912831388869988133638291883062606868532\ 7010553445866529825347265106311408970375411578581162354756181776766855863863910\ 3666405469774824822749893808976093004702012502632596092309146114879185980357730\ 3861342700363386765252454917887675646729006062822382731919402870677014607812709\ 6742863782944785309137485591391473520882675261639169157028134563998259122606414\ 1214061808380859475240005407144224706478896212736978049847274888875101819616029\ 9966056275586323932325057051732694804116544104519152361253824143726585366108886\ 7824403554589285284935611921649878883484648078963332754980560806878465118995176\ 8636996763145767446119826236229282709824730386049555511195740119353423233856730\ 3003546375862251574339462096898282492902092773335958671071365209990805102574092\ 0636142630104451906526838644223997241041911264846371619293496929717047691412369\ 1587346265740663090731687558728136079480707562808197369049864379159635030104833\ 5462063207330037679739569439800054740440906773030574364446455020404814972154764\ 8465659046448645981163111625351420897515029106902997014825073863782367924689987\ 3076456293857228619376979994449524671977219099392394476306453997067225647326307\ 9651244784572003798516669285898768736293284876231569011061165312951792196750664\ 8004754964130050168046555775625521612295094815319045752147740812584490059471702\ 9110508498720322371497180824853974807955076774415446379857426101140477331235232\ 5282219526154347465870231715669282321911900759173373302053902742843137071740045\ 3621167367929877315269281203851500653681929449277541612917321658168275189795276\ 2412503712674455797837236292478679948347530840728542976552595796030279570515203\ 7940542968833411862734488251943840515303069923090930754867030519999763927008753\ 4109278412879725343113493554581197681691396093950913079134280924275077184989217\ 1982956692149258192788201462843811643821788250046018873375736899319270437924928\ 5540099696779485485716894358744992122125312773996048150216323732658153945679472\ 2320977755208176552178482693983039701785708172180465823200429017613355668581316\ 9197409885823806913479548221920578462725001628320343226209425531737323271123249\ 9727126380340937411653379777029732265276325290672904798107737192524653101768400\ 9963726349210748613853596548808286287341587314433611182352932800970657575890482\ 9602907447189590499440728343982860489151634746504481128663366540410107769278962\ 2181978575636973826070190685730400287078130504620330974191383418411304598280217\ 6289175130935471020191822476327340127497769105439927288109432302920619396686207\ 5563266955573972135825134090714013167104886818196119505018915027954165916798660\ 5640685233817787209455553922776298303907978125459096002683296118532244383049939\ 2102581707216392827636621552640316689749473277663821087276462030534324285817090\ 1574039834128452828945043423138163615088396652676945031102616772537236027362572\ 9093550129291730074222552412449785060754950844397709165842732972161473559312128\ 1648177665545877001929057577821788966777334571056138060542028896643985137026462\ 8061525933974099968310206635290264404637756991509321664429344972626116684300533\ 7861106196797563625824671581750709088611111005393211508348946836662320709164744\ 4854402831452730245851694998484779312170342406956127253906007800990005060969124\ 2395898623324386522359041849344412397966880334086445922750564087082180538353448\ 9634389254021811690315680739893729453852192126826862284872702040996640300451513\ 8200522354863683731681011043122873336107797215878814823980958952036686642194726\ 3706921381635816998209832545626576977827117750452679746326170192207125326473365\ 8497931921864125246498435153832638634428500668609805760957351666370173186152489\ 8549196644441257764954206799410775690220693547389493395063894074243026894236255\ 6121951407299610469418576123455522867323117795789526865250121923123020730121861\ 6624481869097437255443211332437606329843916191606975531157236542756283294761097\ 4454018830191740393255424759617907227542814955807285179342469298454284434333124\ 2188046685850140650249332877897138070156412249953068010125972434512590631995781\ 6632292281816295180543030856988598755698719014935611610497899707334483953087418\ 5979872236654381900555278741541091036775020178274565847573095539888643891558305\ 0175424376912130256217972415808265422685233550897840075159979797826139100476420\ 4214518939826524611784332641689623503135283070093056061865458136633828822285069\ 0004425896765346972392541315554962782742115932728784942482810982493570350990537\ 5108530943317744846866596119617581606715824753315301552311532186271382016305676\ 9063360495161630225801269986315318362731038779141167969552641858086429656721362\ 9599772990492455970027177634852018878282322897538817547405848057532086774798145\ 5364151801960828568783263237919368509672246292670138632820604703249416046299906\ 6309128599020726672129686873000548278929749049292091715882893019593031702086657\ 2174198190541394814812140410431243892023851682561822293927876173223740753378627\ 4994641265344197339284929991571856502679410265220369215765028835049768624721877\ 0121988303345502026704739350854447208331960688620845925110719853167768327430021\ 4142932824672056999929554415787065542872881312583864621789852055601802352448366\ 9108631443716501593765756505629326189956506734462415837555130395960452986739287\ 7660151949506573367420849284827494218187419458582547237607859031878846049644312\ 7920103300044594753464102678439477091221172525537648927003459027346572134757383\ 1030600313929525796904581198077505596731390132687144156670231248193846340305888\ 8787917882570263282367890190436422838181163776957771953426461883287295449828723\ 3488289675215560589564372515907540699914660691853939721373454211606692217598553\ 6584081788667858602130179979038955672511395920461593442738819291451583509219039\ 0618919475111553422287107453358753977381991894543719718939258589010970555112756\ 4846282563174724721875722821448159738351297436685228054903187961451796566642003\ 8877582017402536367794132418674778384181958606150038938978716935404757178232108\ 9544886249853827024538758904911767797137081414281149102511949914183355860551613\ 3966655353632315135931193933915921426295256616267943129810039331850010374647879\ 5019398749091731961928686112450404317715640599039825046649445559552364442288050\ 4878208618632646277400281860369252007926862480622634852930180193126283373580905\ 2188932240739236342043990025893410560025923506161687442534685002077189502865770\ 8628684577211773410041575281787067835156771184738929787375294329344608955759281\ 6377654751461037940825426469015541257395655125850006534845483689697461507641197\ 6033519972257906660048537471491539843946831599600103967246479411724790599911674\ 1836370606906770341983903573287845621138941743228357128906245733541012474702751\ 8453701557805174085939366323849083212393050852348898855243496187309225102337985\ 3152423482372965525069167089113730940204405410575250750673485929713480182810589\ 8682924721845452017908478942531854379243327576779228936408223207821032121141629\ 8026545748202577851026072354880919535700504372738944562621789406059507848845670\ 9314010730492365706988794320639036441537104845093729304487948798494150514394443\ 3321460335522184092101043554017852228116475405922692692379589150422884131353808\ 329 ** WARNING: Output too long, hence truncated. '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:46:24 2005 Input: 2^12349 Output: Magma V2.11-10 Mon Dec 19 2005 08:46:24 on modular [Seed = 250344695] ------------------------------------- 2626736171012131461696331866622504022562035768491774100353857649406801506742242\ 3084825445525830313237579267060342464266751281652973705149653235374277824190887\ 1189531386079630150950885566776783139898209075387224197253278610227046686860285\ 2146772936587820191698239338478093635758535173421076624226668386717908021430077\ 2575959758867004070058358245154844488213421278411858599405003098389395982774741\ 5207523985664026620091887697430049224315869743164135546469569593156053308579763\ 4263347484501060259378983778512308193286625242724234283776294831969816375131722\ 8936983092780315563728112542813599636673151675373431312055279054784023549919569\ 1219311126509583342457083167117071756897339079913318716330389289773213619097648\ 5462291545382776372231239190460511893638750161514624836332728442123550337942797\ 0107763595947903341784814752544645876919755965073714519271381674895702373091853\ 3215436670680437605890497876295398853415101607642404800691586567280726998586113\ 3155635070581153975745361022821609701475154342525458848449138355080543080276328\ 8562238102868156174523151272995385551848413365281813064332695534895707693916077\ 3662839142437371930375144772683967311099712589747556416445072489722773502081757\ 3932762811380394736879531383074542888986949490172385899865512107245330358348141\ 2002536006877334990455229759352997623012123459286967364906330023656745029008423\ 1318591798112344953149789799176371875210520880011310213974786327498190170775194\ 0246174540950463559914962203857965870347514848862996573094556026725122248029488\ 6275846517739215502335403747057022256585935251030078500156821873918117987513970\ 4426698969743209570301505361020908142913826815198427238143354973528877107200224\ 1362276706671793015984396682901357490556654981141630717756138713107208664121793\ 5964945552831824348164195179552274467535374454989803677496230655131121084486570\ 0261226878948470170102082465535705353261735614453198125197698538320433876443585\ 5548730497312029569853044174888346610679682782177024305897164452504267762524472\ 8942106845930309209375265735037991439661066606147717307209666084147940792076368\ 5347488239038281717789732344573987600622333170050711612768838066021919121172723\ 7211642627359164577023720988405242216072270697121621110267174598217721659879750\ 6359722942347060658884300635157725165894578694825025691943617203907024115252487\ 2604789418047694030384602063079611761683791030363420595196990670237588860702010\ 6169944921970214471348265581404547669212797420512416967703949848988112509424036\ 1134688460311879742839068107057958734150381319733809695520201181756924313844696\ 1918204395509649687931010893331484714802246461165949708658831409554486264656020\ 2841600039327715319442826732033633872044383688769435144495559623035113598100509\ 0075714762536621556465045676759336653654216176823133677119064993039589879343991\ 1804372050260066046422880519351824998305832815771105572582031735088076992900746\ 8766513825643038069839131037767895599272614757883512126120210656816117647905093\ 1080232106668263738912090292701279801420121666281544640796361809598122771169016\ 7769682855286385498577015573401574654445955805783975120408618104937011191205360\ 0034286567891044077209771318409439621839771975487282097247546270122766333673139\ 7818925249813795323067965555218133003053559443872659882156008658054660949373786\ 5912057785114351497720960768672901692108186937013204195723476497603434228407112\ 0818914716884554084155256682143282018872321808954078627677339016574811674828448\ 4545103570576976166155059054557530744414713978466406343423134640612927620214279\ 5177691324413778261863862669992600426858077025874394518306583020390601349897083\ 3017916526700157169696336864604125724185399555853360649287157768704465038886601\ 1190907156678096345969954990068832490888864044127228955581257071464844705313975\ 17312 Total time: 0.200 seconds, Total memory usage: 3.24MB '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:46:15 2005 Input: 2^1234 Output: Magma V2.11-10 Mon Dec 19 2005 08:46:14 on modular [Seed = 200208868] ------------------------------------- 2958112246080986290600446957161035907863396871353729922395562070506573507962389\ 2426105383724837805018644364775907095599312082089933038176093702721248284094494\ 1362110665443775183495726811929203861182015218323892077355983393191208928867652\ 6559936024879031137085494026686245211006117942703402327660993170980488874938090\ 23127398253860618772619035009883272941129544640111837184 Total time: 0.190 seconds, Total memory usage: 3.24MB '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:46:04 2005 Input: 2^1234324 Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 19 2005 08:45:44 on modular [Seed = 1955696309] ------------------------------------- Errors: /bin/sh: line 1: 2621 Alarm clock nice -n 19 /usr/local/bin/magma '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:45:33 2005 Input: 2^123 Output: Magma V2.11-10 Mon Dec 19 2005 08:45:33 on modular [Seed = 1905560389] ------------------------------------- 10633823966279326983230456482242756608 Total time: 0.190 seconds, Total memory usage: 3.24MB '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:45:26 2005 Input: eval("2^123") Output: Magma V2.11-10 Mon Dec 19 2005 08:45:26 on modular [Seed = 2057273413] ------------------------------------- >> eval("2^123"); ^ User error: Identifier 'eval' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:45:09 2005 Input: gap.eval("2^123") Output: Magma V2.11-10 Mon Dec 19 2005 08:45:08 on modular [Seed = 1736613194] ------------------------------------- >> gap.eval("2^123"); ^ User error: Identifier 'gap' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '82.58.6' ************** MAGMA ***************** Host 82.58.68.160 (82.58.68.160) Time: Mon Dec 19 08:11:01 2005 Input: 268165538371456392121266295429649186242633 Output: Magma V2.11-10 Mon Dec 19 2005 08:10:58 on modular [Seed = 2675600823] ------------------------------------- 268165538371456392121266295429649186242633 Total time: 0.210 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:10:23 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M2; Output: Magma V2.11-10 Mon Dec 19 2005 07:10:23 on modular [Seed = 3425250472] ------------------------------------- (0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0) [0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [1 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0] [1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1] [1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1] [1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0] [1 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0] [1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 1] [1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1] [1 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0] [1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1] [1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1] [1 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0] [1 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0] [1 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0] [1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1] [1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1] [1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1] [1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1] [1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0] [1 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0] Total time: 0.220 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:10:00 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M2; gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),M2); Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:09:51 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M2; gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),M2); C:=LinearCode(gen); gen; Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:09:32 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M2; gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),M2); C:=LinearCode(gen); IsSelfDual(C); MinimumWeight(C); Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:08:52 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M2; Output: Magma V2.11-10 Mon Dec 19 2005 07:08:52 on modular [Seed = 667076280] ------------------------------------- (0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0) [0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0] [1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1] [1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1] [1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1] [1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1] [1 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0] [1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0] [1 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0] [1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1] [1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1] [1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0] [1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1] [1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0] [1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0] [1 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0] [1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1] [1 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0] [1 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0] [1 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0] Total time: 0.200 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:08:36 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M1; Output: Magma V2.11-10 Mon Dec 19 2005 07:08:36 on modular [Seed = 751422883] ------------------------------------- (0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0) [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0] [0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1] [1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0] [0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0] [0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0] [0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1] [1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1] [1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1] [1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1] [1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0] [0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0] [0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0] [0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0] [0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0] [0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0] [0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1] [1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0] [0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0] Total time: 0.200 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:08:23 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); Output: Magma V2.11-10 Mon Dec 19 2005 07:08:23 on modular [Seed = 701286613] ------------------------------------- (1 0 1 1 1 1 0 0 0 1 1 1 1 1 0 1 1 0) Total time: 0.210 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:08:14 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),circulant(x)); Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:07:57 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),circulant(x)); C:=LinearCode(gen); IsSelfDual(C); MinimumWeight(C); Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:07:37 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; Output: Magma V2.11-10 Mon Dec 19 2005 07:07:37 on modular [Seed = 519831516] ------------------------------------- (1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1) Total time: 0.210 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:07:27 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); Output: Magma V2.11-10 Mon Dec 19 2005 07:07:26 on modular [Seed = 469695022] ------------------------------------- Total time: 0.230 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:07:12 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// Output: Magma V2.11-10 Mon Dec 19 2005 07:07:09 on modular [Seed = 15065382] ------------------------------------- Total time: 0.260 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:06:09 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),circulant(x)); C:=LinearCode(gen); IsSelfDual(C); MinimumWeight(C); Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '60.225.' ************** MAGMA ***************** Host 60.225.128.201 (60.225.128.201) Time: Mon Dec 19 00:36:02 2005 Input: Factorization(00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000090000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Mon Dec 19 2005 00:36:02 on modular [Seed = 3926912878] ------------------------------------- [ <2, 94>, <3, 2>, <5, 94> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.128.201 (60.225.128.201) Time: Mon Dec 19 00:34:19 2005 Input: Factorization(000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000007000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 19 2005 00:33:59 on modular [Seed = 3758482913] ------------------------------------- Errors: /bin/sh: line 1: 1446 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.128.201 (60.225.128.201) Time: Mon Dec 19 00:30:20 2005 Input: Factorization(0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000009000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000070000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 19 2005 00:29:59 on modular [Seed = 3661618379] ------------------------------------- Errors: /bin/sh: line 1: 1436 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.128.201 (60.225.128.201) Time: Mon Dec 19 00:28:12 2005 Input: Factorization(33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 19 2005 00:27:51 on modular [Seed = 3543585858] ------------------------------------- Errors: /bin/sh: line 1: 1429 Alarm clock nice -n 19 /usr/local/bin/magma '82.55.7' ************** MAGMA ***************** Host 82.55.71.167 (82.55.71.167) Time: Sun Dec 18 17:30:04 2005 Input: 10! Output: Magma V2.11-10 Sun Dec 18 2005 17:30:04 on modular [Seed = 3160022383] ------------------------------------- >> 10!; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '155.207' ************** MAGMA ***************** Host 155.207.209.176 (155.207.209.176) Time: Sun Dec 18 16:51:33 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-26*x^2+1; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); u1:=-9842-1526*a-16289/2*(a^2-1)-8036/4*(a^3 +a^2 -a -1); u2:=651+189*a+1113/2*(a^2-1)+532/4*(a^3 +a^2 -a -1); u3:=-42+0*a-63/2*(a^2-1)-42/4*(a^3 +a^2 -a -1); u4:=-63+21*a-161/2*(a^2-1)+56/4*(a^3 +a^2 -a -1); u5:=994-154*a+2457/2*(a^2-1)-812/4*(a^3 +a^2 -a -1); UnitEquation(a*u1,-a*u1,2*a*n); UnitEquation(a*u2,-a*u2,2*a*n); UnitEquation(a*u3,-a*u3,2*a*n); UnitEquation(a*u4,-a*u4,2*a*n); UnitEquation(a*u5,-a*u5,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 16:51:32 on modular [Seed = 266966483] ------------------------------------- [ 1, y, 1/2*(y^2 - 1), 1/4*(y^3 + y^2 - y - 1) ] true [ [-9842, -1526, -16289, -8036], [651, 189, 1113, 532], [-42, 0, -63, -42], [-63, 21, -161, 56], [994, -154, 2457, -812] ] [] [] [] [] [] Total time: 1.199 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.176 (155.207.209.176) Time: Sun Dec 18 16:46:21 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-26*x^2+1; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); u1:=-21-42*a-21/2*(a^2+1)+42/44*(a^3 + 11*a^2 + 43*a + 33); UnitEquation(a*u1,-a*u1,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 16:46:21 on modular [Seed = 2958437156] ------------------------------------- [ 1, y, 1/2*(y^2 - 1), 1/4*(y^3 + y^2 - y - 1) ] true [ [-9842, -1526, -16289, -8036], [651, 189, 1113, 532], [-42, 0, -63, -42], [-63, 21, -161, 56], [994, -154, 2457, -812] ] >> UnitEquation(a*u1,-a*u1,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral Total time: 0.440 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.124 (155.207.209.124) Time: Sun Dec 18 16:32:15 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-34*x^2+121; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); u1:=-21-42*a-21/2*(a^2+1)+42/44*(a^3 + 11*a^2 + 43*a + 33); UnitEquation(a*u1,-a*u1,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 16:32:13 on modular [Seed = 1269685551] ------------------------------------- [ 1, y, 1/2*(y^2 + 1), 1/44*(y^3 + 11*y^2 + 43*y + 33) ] true [ [-21, -42, -21, 42] ] [ [[84, 73, 18, -44] [81, 63, 15, -38]], [[-40, 73, 26, -44] [-43, 63, 23, -38]], [[0, 7, 2, -4] [-3, -3, -1, 2]], [[4, 7, 2, -4] [1, -3, -1, 2]], [[9, 5, 1, -3] [6, -5, -2, 3]], [[-6, 5, 2, -3] [-9, -5, -1, 3]], [[-1, 3, 1, -2] [-4, -7, -2, 4]], [[3, 3, 1, -2] [0, -7, -2, 4]], [[43, -63, -23, 38] [40, -73, -26, 44]], [[-81, -63, -15, 38] [-84, -73, -18, 44]] ] Total time: 1.229 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.124 (155.207.209.124) Time: Sun Dec 18 16:31:23 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-34*x^2+121; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); u1:=-21-42*a-21/2*(a^2+1)+42/44*(y^3 + 11*y^2 + 43*y + 33); UnitEquation(a*u1,-a*u1,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 16:31:23 on modular [Seed = 1421001286] ------------------------------------- [ 1, y, 1/2*(y^2 + 1), 1/44*(y^3 + 11*y^2 + 43*y + 33) ] true [ [-21, -42, -21, 42] ] >> u1:=-21-42*a-21/2*(a^2+1)+42/44*(y^3 + 11*y^2 + 43*y + 33); ^ Runtime error in '+': Arguments are not compatible Argument types given: FldNumElt, FldNumElt >> UnitEquation(a*u1,-a*u1,2*a*n); ^ User error: Identifier 'u1' has not been declared or assigned Total time: 0.350 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.124 (155.207.209.124) Time: Sun Dec 18 16:28:33 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-34*x^2+121; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u2:=14-35*z+7*z^2-7*z^3; u3:=-21*z+21*z^2+21*z^3; u4:=14-7*z-77*z^2-35*z^3; u5:=-56-49*z+217*z^2+105*z^3; UnitEquation(a*u1,-a*u1,2*a*n); UnitEquation(a*u2,-a*u2,2*a*n); UnitEquation(a*u3,-a*u3,2*a*n); UnitEquation(a*u4,-a*u4,2*a*n); UnitEquation(a*u5,-a*u5,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 16:28:33 on modular [Seed = 1589168614] ------------------------------------- [ 1, y, 1/2*(y^2 + 1), 1/44*(y^3 + 11*y^2 + 43*y + 33) ] true [ [-21, -42, -21, 42] ] >> UnitEquation(a*u1,-a*u1,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral >> UnitEquation(a*u2,-a*u2,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral >> UnitEquation(a*u3,-a*u3,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral >> UnitEquation(a*u4,-a*u4,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral >> UnitEquation(a*u5,-a*u5,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral Total time: 0.350 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:30:21 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u2:=14-35*z+7*z^2-7*z^3; u3:=-21*z+21*z^2+21*z^3; u4:=14-7*z-77*z^2-35*z^3; u5:=-56-49*z+217*z^2+105*z^3; UnitEquation(a*u1,-a*u1,2*a*n); UnitEquation(a*u2,-a*u2,2*a*n); UnitEquation(a*u3,-a*u3,2*a*n); UnitEquation(a*u4,-a*u4,2*a*n); UnitEquation(a*u5,-a*u5,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 14:30:20 on modular [Seed = 266990835] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] [] [] [ [[62, 72, -13, -15] [67, 91, -14, -19]], [[5, 0, -1, 0] [10, 19, -2, -4]], [[2, -4, -1, 1] [7, 15, -2, -3]], [[-7, -15, 2, 3] [-2, 4, 1, -1]], [[-10, -19, 2, 4] [-5, 0, 1, 0]], [[-67, -91, 14, 19] [-62, -72, 13, 15]] ] [] [] Total time: 1.250 seconds, Total memory usage: 3.70MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:29:01 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u2:=14-35*z+7*z^2-7*z^3; u3:=-21*z+21*z^2+21*z^3; UnitEquation(a*u1,-a*u1,2*a*n); UnitEquation(a*u2,-a*u2,2*a*n); UnitEquation(a*u3,-a*u3,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 14:29:00 on modular [Seed = 166720069] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] [] [] [ [[62, 72, -13, -15] [67, 91, -14, -19]], [[5, 0, -1, 0] [10, 19, -2, -4]], [[2, -4, -1, 1] [7, 15, -2, -3]], [[-7, -15, 2, 3] [-2, 4, 1, -1]], [[-10, -19, 2, 4] [-5, 0, 1, 0]], [[-67, -91, 14, 19] [-62, -72, 13, 15]] ] Total time: 1.260 seconds, Total memory usage: 3.70MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:27:53 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u2:=14-35*z+7*z^2-7*z^3; UnitEquation(a*u1,-a*u1,2*a*n); UnitEquation(a*u2,-a*u2,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 14:27:53 on modular [Seed = 81985293] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] [] [] Total time: 0.420 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:27:32 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u2:=14-35*z+7*z^2-7*z^3; UnitEquation(a*u1,a*u1,2*a*n); UnitEquation(a*u2,a*u2,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 14:27:31 on modular [Seed = 536470540] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] [] [] Total time: 0.430 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:26:43 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u1:=56-105*z-63*z^2+49*z^3; UnitEquation(a*u1,a*u1,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 14:26:42 on modular [Seed = 452783832] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] [] Total time: 0.430 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:26:27 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u1:=56-105*z-63*z^2+49*z^3; UnitEquation(a*u1,a*u1,2*a*n)); Output: Magma V2.11-10 Sun Dec 18 2005 14:26:27 on modular [Seed = 368304582] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] >> UnitEquation(a*u1,a*u1,2*a*n));; ^ User error: bad syntax Total time: 0.370 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:25:59 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63z^2+49*z^3; u1:=56-105*z-63z^2+49*z^3; UnitEquation(a*u1,a*u1,2*a*n)); Output: Magma V2.11-10 Sun Dec 18 2005 14:25:58 on modular [Seed = 284615910] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] >> u1:=56-105*z-63z^2+49*z^3; ^ User error: bad syntax >> u1:=56-105*z-63z^2+49*z^3; ^ User error: bad syntax >> UnitEquation(a*u1,a*u1,2*a*n));; ^ User error: bad syntax Total time: 0.420 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.183 (155.207.209.183) Time: Sun Dec 18 13:14:25 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; > >NormEquation(O, 4*(p*q)^4); Output: Magma V2.11-10 Sun Dec 18 2005 13:14:24 on modular [Seed = 1118086763] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] Total time: 0.380 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.81 (155.207.209.81) Time: Sun Dec 18 13:09:52 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=1; q:=5; > >NormEquation(O, 4*(p*q)^4); Output: Magma V2.11-10 Sun Dec 18 2005 13:09:51 on modular [Seed = 1286253194] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [0, 0, -5, 0] ] Total time: 0.330 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.81 (155.207.209.81) Time: Sun Dec 18 13:09:40 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p=1; q=5; > >NormEquation(O, 4*(p*q)^4); Output: Magma V2.11-10 Sun Dec 18 2005 13:09:39 on modular [Seed = 1236250499] ------------------------------------- [ 1, y, y^2, y^3 ] >> p=1; ^ User error: Identifier 'p' has not been declared or assigned >> q=5; ^ User error: Identifier 'q' has not been declared or assigned >> NormEquation(O, 4*(p*q)^4);; ^ User error: Identifier 'p' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.15 (212.138.47.15) Time: Sun Dec 18 10:08:42 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 10:08:41 on modular [Seed = 1387555329] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x + 216*z^4 - 504*z^3 - 24*z^2 + 371*z + 110, y + 6*z^2 - 7*z - 5, z^9 + 279936*z^8 - 1306368*z^7 + 1461888*z^6 + 1106784*z^5 - 1953720*z^4 - 775656*z^3 + 793998*z^2 + 492317*z + 73365 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '212.138' ************** MAGMA ***************** Host 212.138.47.23 (212.138.47.23) Time: Sun Dec 18 10:08:04 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 10:08:04 on modular [Seed = 1589143072] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x + 216*z^4 - 504*z^3 - 24*z^2 + 371*z + 110, y + 6*z^2 - 7*z - 5, z^8 - 14/3*z^7 + 47/9*z^6 + 427/108*z^5 - 335/48*z^4 - 775655/279936*z^3 + 44111/15552*z^2 + 492317/279936*z + 24455/93312 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '212.138' ************** MAGMA ***************** Host 212.138.47.29 (212.138.47.29) Time: Sun Dec 18 10:07:22 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 10:07:21 on modular [Seed = 1151499335] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x - 20670318404676804121590341679907342944438721410236055716516252786259218\ 420711433810584139369555228744398094045155409199104/1393743412806712329\ 99759010297844161860871496969373562963049485474223528276610984064461508\ 093880757021543538634046896781115514291717975*z^26 + 52817691742356886370701872211589394258641546772944677761355469155105730\ 384820419602679329873674311460101684589053757063168/1393743412806712329\ 99759010297844161860871496969373562963049485474223528276610984064461508\ 093880757021543538634046896781115514291717975*z^25 + 44084189102656559761036896914171515230703918700362814196736477697748482\ 349329165388194181519205694144483399773379046731904/1393743412806712329\ 99759010297844161860871496969373562963049485474223528276610984064461508\ 093880757021543538634046896781115514291717975*z^24 + 10282605380629114589617264989256557708549756853994112065698656467656094\ 04406261142299579793912180745541265316913236203077632/13937434128067123\ 29997590102978441618608714969693735629630494854742235282766109840644615\ 08093880757021543538634046896781115514291717975*z^23 - 22228881978492193293559859093936405405939009088196569488635554540757211\ 37017756817169737946803181622067694970688957503668224/13937434128067123\ 29997590102978441618608714969693735629630494854742235282766109840644615\ 08093880757021543538634046896781115514291717975*z^22 - 17181692219196114429115811806207940681105897576614572215614886254607317\ 41173496080747558758974447442649000940198623520859392/13937434128067123\ 29997590102978441618608714969693735629630494854742235282766109840644615\ 08093880757021543538634046896781115514291717975*z^21 - 61640289475596731450679340165759399618431062957953433768199378491567206\ 565511264862842657794696865313209618370001141849358336/1393743412806712\ 32999759010297844161860871496969373562963049485474223528276610984064461\ 508093880757021543538634046896781115514291717975*z^20 - 66095752571214459624518529004011368779461729417153411440699393722538310\ 17246088969923789514916084342288456155181781001857089536/13937434128067\ 12329997590102978441618608714969693735629630494854742235282766109840644\ 61508093880757021543538634046896781115514291717975*z^19 + 25238028072639823696679503339288656807992425850973474585354246877644706\ 187685801574587471601563331251578311852754405108094443328/1393743412806\ 71232999759010297844161860871496969373562963049485474223528276610984064\ 461508093880757021543538634046896781115514291717975*z^18 + 15100774264212872360384730726830227678249159853860787662574702196479871\ 49631337678553773700882486776776263487029256690143371264/13937434128067\ 12329997590102978441618608714969693735629630494854742235282766109840644\ 61508093880757021543538634046896781115514291717975*z^17 + 44841353739392875941013024676041792726615548527879166766758623056926826\ 3013268190022444874591316028252564801406550386388655271936/139374341280\ 67123299975901029784416186087149696937356296304948547422352827661098406\ 4461508093880757021543538634046896781115514291717975*z^16 - 14801931190140762886015675654974862129502652192610372634978790059234754\ 947512399230178102320439394185071889427571268295241900385632/1393743412\ 80671232999759010297844161860871496969373562963049485474223528276610984\ 064461508093880757021543538634046896781115514291717975*z^15 + 64118150688406168899806736311366044438266976175150900955009941229687123\ 359613473660565416868586665235987375007859422298128871370752/1393743412\ 80671232999759010297844161860871496969373562963049485474223528276610984\ 064461508093880757021543538634046896781115514291717975*z^14 - 59766235433460852251769712174872729369152399253394915317285117833345339\ 512065855473683081139767348030618064160139820193734351419392/1393743412\ 80671232999759010297844161860871496969373562963049485474223528276610984\ 064461508093880757021543538634046896781115514291717975*z^13 - 28951070792768735987119578603040775907790667817768845226622240582260527\ 922476126045937075626141100552572245765618249291256309909688/2787486825\ 61342465999518020595688323721742993938747125926098970948447056553221968\ 12892301618776151404308707726809379356223102858343595*z^12 + 76499372079855456323013037441293431543773048046253148902645677881534577\ 4660856190940754360644032963410442949681497376069584715073536/139374341\ 28067123299975901029784416186087149696937356296304948547422352827661098\ 4064461508093880757021543538634046896781115514291717975*z^11 - 18265271834715021921507366802108499920741026425191081094680295339668321\ 8637033273500632516798684292743702866140155367781483358671872/139374341\ 28067123299975901029784416186087149696937356296304948547422352827661098\ 4064461508093880757021543538634046896781115514291717975*z^10 + 81234711366392755150707194044826385840652588795161583303875060921131616\ 16613798195316541777619380893565306500576380034572591855769816/13937434\ 12806712329997590102978441618608714969693735629630494854742235282766109\ 84064461508093880757021543538634046896781115514291717975*z^9 - 76301446847565195312748120513029598866156889065065860133024518164894305\ 3105777355729600924867262813718143714436913502675351522258163712/139374\ 34128067123299975901029784416186087149696937356296304948547422352827661\ 0984064461508093880757021543538634046896781115514291717975*z^8 - 82348805467619018020877218869999199242072232530941549101334278979740312\ 08331342268339335248205917407685848265330600950653722177270555648/13937\ 43412806712329997590102978441618608714969693735629630494854742235282766\ 10984064461508093880757021543538634046896781115514291717975*z^7 + 84412521960472857160911216066242401851090200344030094679739981190800010\ 375965825632433557714526225507464746855468694951312740791718236002/1393\ 74341280671232999759010297844161860871496969373562963049485474223528276\ 610984064461508093880757021543538634046896781115514291717975*z^6 - 19683600898511070798075127987078432887095168620485541000219071579591808\ 6564629037654229865133236791460268215512602009300843346282091915264/139\ 37434128067123299975901029784416186087149696937356296304948547422352827\ 6610984064461508093880757021543538634046896781115514291717975*z^5 + 10666103245020463025460572567380480364748019759231156004389783614618856\ 4278509574124073487675029961578117423850555852445478175878833243136/139\ 37434128067123299975901029784416186087149696937356296304948547422352827\ 6610984064461508093880757021543538634046896781115514291717975*z^4 + 96184597403301187882098000111345017149388757160701053527995880628846239\ 321327676865418804844922873757179730046636417630135402601301240819/1393\ 74341280671232999759010297844161860871496969373562963049485474223528276\ 610984064461508093880757021543538634046896781115514291717975*z^3 - 23484072682953374360602270413978122643907931573358680446186791183604617\ 35064254014182229707643805925968450130018541132374369207378926592/55749\ 73651226849319990360411913766474434859878774942518521979418968941131064\ 439362578460323755230280861741545361875871244620571668719*z^2 - 26306871523575483715460693403165967104582376597576450916239167773516248\ 016865114672591341164780430740062095698813097584417659321598050816/1393\ 74341280671232999759010297844161860871496969373562963049485474223528276\ 610984064461508093880757021543538634046896781115514291717975*z - 17685951105934495101334685727679688500362705972794161671957590971299254\ 4928960696571368311776218030826201622816796879764966588533652253/278748\ 68256134246599951802059568832372174299393874712592609897094844705655322\ 196812892301618776151404308707726809379356223102858343595, y - 52327418222015905902570988536026443305400447176512318167424696241950974\ 64867103815670348284947982736369877307627698015104/13937434128067123299\ 97590102978441618608714969693735629630494854742235282766109840644615080\ 93880757021543538634046896781115514291717975*z^26 + 53322713168254937049374753704315866133748322761477018891704908145697222\ 36175698315958165401476066818724050651331474596288/13937434128067123299\ 97590102978441618608714969693735629630494854742235282766109840644615080\ 93880757021543538634046896781115514291717975*z^25 + 16200344963969739629738687977305337274986451247440630132666003014205689\ 06154724466997188357122175984508399767599792176664/13937434128067123299\ 97590102978441618608714969693735629630494854742235282766109840644615080\ 93880757021543538634046896781115514291717975*z^24 + 24267663491107314919267151607004192719477817031022986301215056432744760\ 4502999413958546800557457229730073541614908067430272/139374341280671232\ 99975901029784416186087149696937356296304948547422352827661098406446150\ 8093880757021543538634046896781115514291717975*z^23 - 22941289203469812166392436718685805905743949242949535266547366634353399\ ** WARNING: Output too long, hence truncated. '212.138' ************** MAGMA ***************** Host 212.138.47.18 (212.138.47.18) Time: Sun Dec 18 10:06:52 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sun Dec 18 2005 10:06:32 on modular [Seed = 1084651922] ------------------------------------- Errors: /bin/sh: line 1: 26986 Alarm clock nice -n 19 /usr/local/bin/magma '212.138' ************** MAGMA ***************** Host 212.138.47.17 (212.138.47.17) Time: Sun Dec 18 10:05:35 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sun Dec 18 2005 10:05:15 on modular [Seed = 1219391945] ------------------------------------- Errors: /bin/sh: line 1: 26978 Alarm clock nice -n 19 /usr/local/bin/magma '212.138' ************** MAGMA ***************** Host 212.138.47.24 (212.138.47.24) Time: Sun Dec 18 10:04:18 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 10:04:17 on modular [Seed = 1905739818] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x - 607747819788532155708373696835866320181177034137286756794368/1534183140\ 25015422015563440134202740864381613046837395925*z^26 - 1644194756182732188695998750143313136448830566796183794089984/767091570\ 125077110077817200671013704321908065234186979625*z^25 + 1519370344618022681485678701207219405338713755668400813637632/697355972\ 84097919097983381879183064029264369566744270875*z^24 + 5657683761495134333516311523334543610877895279826294418178048/153418314\ 025015422015563440134202740864381613046837395925*z^23 - 54919850897379097829474394140593057612213076270900171882102784/76709157\ 0125077110077817200671013704321908065234186979625*z^22 - 23749346132955892738971647263493830752398879458430423918444544/15341831\ 4025015422015563440134202740864381613046837395925*z^21 - 20700681958857365834122812251195046038933464222145607741997056/76709157\ 0125077110077817200671013704321908065234186979625*z^20 + 329797041412983216677707986578040306567545389152753038938669056/7670915\ 70125077110077817200671013704321908065234186979625*z^19 + 5743093752282330081008122287919690804163733840955113232662528/153418314\ 025015422015563440134202740864381613046837395925*z^18 - 223440592050718037577946396909580575534889719118640118530048/1450078582\ 467064480298331192194732900419485945622281625*z^17 + 105410529530167997137721778655638855754188729832104137565487104/3068366\ 2805003084403112688026840548172876322609367479185*z^16 + 885277150983208290850600622134945917710089103724880750058029056/7670915\ 70125077110077817200671013704321908065234186979625*z^15 - 12228856033159216097966918255958697335709910281377823340045697024/76709\ 1570125077110077817200671013704321908065234186979625*z^14 - 2428919581438507999580094225861501397229411784479549789109817344/153418\ 314025015422015563440134202740864381613046837395925*z^13 + 19686636770492932197315778634758016279089740344967393996509011968/76709\ 1570125077110077817200671013704321908065234186979625*z^12 + 37439000525180257727455377920560461618208300249854315692199601152/76709\ 1570125077110077817200671013704321908065234186979625*z^11 - 242567485796071338802342670543220142016253736503063297377912832/1534183\ 14025015422015563440134202740864381613046837395925*z^10 - 1851283411800758364534308335130407783249267591238748229467858432/306836\ 62805003084403112688026840548172876322609367479185*z^9 - 32281291693399996932626040560041370875138715107061454029379634176/76709\ 1570125077110077817200671013704321908065234186979625*z^8 + 15187976308103132481649219442943042891239263040607663202857682752/76709\ 1570125077110077817200671013704321908065234186979625*z^7 + 32541311772827410438962148277497379673569947455678018687923473506/76709\ 1570125077110077817200671013704321908065234186979625*z^6 + 13913681728440239381658207089017885127837169844456124071384306152/76709\ 1570125077110077817200671013704321908065234186979625*z^5 - 5201437434931523590484162612412046914431897286598813426355149518/767091\ 570125077110077817200671013704321908065234186979625*z^4 - 8369631173418546797554673043809526207557975516145664666830070596/767091\ 570125077110077817200671013704321908065234186979625*z^3 - 367049861123048438708477131899166019080891143075089642030257163/6973559\ 7284097919097983381879183064029264369566744270875*z^2 - 952768348549232830699641549698525626444725223809958208866600116/7670915\ 70125077110077817200671013704321908065234186979625*z - 18715587866479416200274107374916451473634359600279292058148243/15341831\ 4025015422015563440134202740864381613046837395925, y + 8588783796645609270041309550413627734427158955259021244760064/767091570\ 125077110077817200671013704321908065234186979625*z^26 - 637476922202747849091768753421039549247886904737543430864896/7670915701\ 25077110077817200671013704321908065234186979625*z^25 - 2991662828937587719651268641902300229276057309871396005292802048/767091\ 570125077110077817200671013704321908065234186979625*z^24 - 3905324324791053203620496116512887656336880862025760883867648/697355972\ 84097919097983381879183064029264369566744270875*z^23 + 27504966074299938066699391828241118157638650281304603241981935616/76709\ 1570125077110077817200671013704321908065234186979625*z^22 + 16954503077996297980763933818251459139545426272372724218478723072/76709\ 1570125077110077817200671013704321908065234186979625*z^21 - 10148400141775193758023270288734527185830514208009811283475562496/69735\ 597284097919097983381879183064029264369566744270875*z^20 - 136748565042632957498398260907316114042218819459638218594974040064/7670\ 91570125077110077817200671013704321908065234186979625*z^19 + 218073368412696644786151217621427176878167387891939188253975248896/7670\ 91570125077110077817200671013704321908065234186979625*z^18 + 476420567037107074864416424418893515283464984044841541861899829248/7670\ 91570125077110077817200671013704321908065234186979625*z^17 - 87911783741774450433038258838115842490336235736309248584654553088/76709\ 1570125077110077817200671013704321908065234186979625*z^16 - 865393107630329396744064599190108622066096619174761478073516490752/7670\ 91570125077110077817200671013704321908065234186979625*z^15 - 491732366580711670721311407822861406520038434484615375661704249344/7670\ 91570125077110077817200671013704321908065234186979625*z^14 + 734818272058152712148400269178951592247749820810396870290521808896/7670\ 91570125077110077817200671013704321908065234186979625*z^13 + 1053964300460031535169533803184852097290543328887561757855461595136/767\ 091570125077110077817200671013704321908065234186979625*z^12 + 8634030541199246468005771144699878375129632040218392070043614208/153418\ 314025015422015563440134202740864381613046837395925*z^11 - 76063092256498361515666815473156851226821587577053074339657521152/69735\ 597284097919097983381879183064029264369566744270875*z^10 - 605969915846052348237289545733521947789758114815909673885915143168/7670\ 91570125077110077817200671013704321908065234186979625*z^9 + 103462238301384026891084346638351778208358212884489589520021951616/7670\ 91570125077110077817200671013704321908065234186979625*z^8 + 206006578234046558080680416999501373559639471746243171719113212/4047976\ 62335133039618900897451722271409977870835982575*z^7 + 213087953931560709344296477834302537326554381040028500222583661956/7670\ 91570125077110077817200671013704321908065234186979625*z^6 - 1551500011152425928327059620628005368730724758729742956668117102/153418\ 314025015422015563440134202740864381613046837395925*z^5 - 14331700909217848590119018556215107062277900112783826180653965517/15341\ 8314025015422015563440134202740864381613046837395925*z^4 - 43953099780249835711295595369687919026992869322936517188129514756/76709\ 1570125077110077817200671013704321908065234186979625*z^3 - 13801578868727311991345220593097358202836764355928486394382755206/76709\ 1570125077110077817200671013704321908065234186979625*z^2 - 2357219767421710505084804774312577449297980257986286253398662112/767091\ 570125077110077817200671013704321908065234186979625*z - 35062495528309440485031693067359094546187759778808876771992076/15341831\ 4025015422015563440134202740864381613046837395925, z^27 - 21/2*z^25 - 19/3*z^24 + 49*z^23 + 532/9*z^22 - 49939/432*z^21 - 6517/27*z^20 + 228487/2592*z^19 + 74759341/139968*z^18 + 1224167/5184*z^17 - 86467843/139968*z^16 - 3761045885/5038848*z^15 + 290329361/1679616*z^14 + 1419975504253/1632586752*z^13 + 13760646589/30233088*z^12 - 131206787713/362797056*z^11 - 4113563481113/7346640384*z^10 - 12868110131287/78364164096*z^9 + 110838315489477121/609359740010496*z^8 + 827650776663089/4231664861184*z^7 + 8940368413909147/152339935002624*z^\ 6 - 652513311785843/25389989167104*z^5 - 4957091804726761/152339935002624*z^4 - 511975131257783/33853318889472*z^3 - 151452934564537/38084983750656*z^2 - 358514838196123/609359740010496*z - 23491990397845/609359740010496 ] Total time: 0.210 seconds, Total memory usage: 3.43MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sun Dec 18 09:57:59 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode(M); aut3 := AutomorphismGroup(C3); Order(aut3); Generators(aut3); aut2 := AutomorphismGroup(C2); Order(aut2); Generators(aut2); Output: Magma V2.11-10 Sun Dec 18 2005 09:57:58 on modular [Seed = 954074084] ------------------------------------- true [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] 7 { (1, 5, 2, 6, 3, 7, 4)(8, 12, 9, 13, 10, 22, 11)(14, 23, 15, 24, 16, 25, 17)(18, 26, 19, 27, 20, 28, 21)(29, 33, 30, 34, 31, 35, 32)(36, 40, 37, 41, 38, 42, 39)(43, 47, 44, 48, 45, 49, 46)(50, 54, 51, 55, 52, 56, 53) } 1 {} Total time: 1.330 seconds, Total memory usage: 3.93MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sun Dec 18 09:56:58 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode(M); aut3 := AutomorphismGroup(C3); Order(aut3); Generators(aut3); aut2 := AutomorphismGroup(C2); Order(aut2); Generators(aut3); Output: Magma V2.11-10 Sun Dec 18 2005 09:56:57 on modular [Seed = 666967220] ------------------------------------- true [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] 7 { (1, 5, 2, 6, 3, 7, 4)(8, 12, 9, 13, 10, 22, 11)(14, 23, 15, 24, 16, 25, 17)(18, 26, 19, 27, 20, 28, 21)(29, 33, 30, 34, 31, 35, 32)(36, 40, 37, 41, 38, 42, 39)(43, 47, 44, 48, 45, 49, 46)(50, 54, 51, 55, 52, 56, 53) } 1 { (1, 5, 2, 6, 3, 7, 4)(8, 12, 9, 13, 10, 22, 11)(14, 23, 15, 24, 16, 25, 17)(18, 26, 19, 27, 20, 28, 21)(29, 33, 30, 34, 31, 35, 32)(36, 40, 37, 41, 38, 42, 39)(43, 47, 44, 48, 45, 49, 46)(50, 54, 51, 55, 52, 56, 53) } Total time: 1.320 seconds, Total memory usage: 3.93MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sun Dec 18 09:51:22 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode(M); aut3 := AutomorphismGroup(C3); Order(aut3); Generators(aut3); Output: Magma V2.11-10 Sun Dec 18 2005 09:51:21 on modular [Seed = 801708908] ------------------------------------- true [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] 7 { (1, 5, 2, 6, 3, 7, 4)(8, 12, 9, 13, 10, 22, 11)(14, 23, 15, 24, 16, 25, 17)(18, 26, 19, 27, 20, 28, 21)(29, 33, 30, 34, 31, 35, 32)(36, 40, 37, 41, 38, 42, 39)(43, 47, 44, 48, 45, 49, 46)(50, 54, 51, 55, 52, 56, 53) } Total time: 1.320 seconds, Total memory usage: 3.93MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sun Dec 18 09:50:08 2005 Input: K := FiniteField(2); > C := LinearCode; D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode(M); aut3 := AutomorphismGroup(C3); Order(aut3); Generators(aut3); Output: Magma V2.11-10 Sun Dec 18 2005 09:50:08 on modular [Seed = 3560104159] ------------------------------------- >> D := Dual(S); ^ User error: Identifier 'S' has not been declared or assigned >> (D meet S) eq S; ^ User error: Identifier 'D' has not been declared or assigned >> M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); ^ User error: Identifier 'S' has not been declared or assigned >> M1:=EchelonForm(M); ^ User error: Identifier 'M' has not been declared or assigned >> M2:=Submatrix(M1,22,22,14,35); ^ User error: Identifier 'M1' has not been declared or assigned >> C2:=LinearCode(M2); ^ User error: Identifier 'M2' has not been declared or assigned >> WeightDistribution(C2); ^ User error: Identifier 'C2' has not been declared or assigned >> L:=MinimumWords(C2); ^ User error: Identifier 'C2' has not been declared or assigned >> C3:=LinearCode(M); ^ User error: Identifier 'M' has not been declared or assigned >> aut3 := AutomorphismGroup(C3); ^ User error: Identifier 'C3' has not been declared or assigned >> Order(aut3); ^ User error: Identifier 'aut3' has not been declared or assigned >> Generators(aut3); ; ^ User error: Identifier 'aut3' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sun Dec 18 09:37:24 2005 Input: K := FiniteField(2); > C := LinearCode; Output: Magma V2.11-10 Sun Dec 18 2005 09:37:24 on modular [Seed = 3274044620] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.14 (212.138.47.14) Time: Sun Dec 18 09:16:17 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 09:16:17 on modular [Seed = 4061454056] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x - 1296*z^9 + 4536*z^7 + 3240*z^6 - 5292*z^5 - 7560*z^4 - 600*z^3 + 4410*z^2 + 3101*z + 710, y + 6*z^3 - 7*z - 5, z^27 - 21/2*z^25 - 15/2*z^24 + 49*z^23 + 70*z^22 - 7811/72*z^21 - 1715/6*z^20 + 12985/432*z^19 + 133685/216*z^18 + 14063/32*z^17 - 274925/432*z^16 - 8654347/7776*z^15 - 62965/864*z^14 + 54259387/46656*z^13 + 88401125/93312*z^12 - 81855529/279936*z^11 - 260804915/279936*z^10 - 402754685/839808*z^9 + 224599585/1119744*z^8 + 1933044449/5038848*z^7 + 1197004745/6718464*z^6 - 823553731/40310784*z^5 - 8293758725/120932352*z^4 - 29056568867/725594112*z^3 - 3016860665/241864704*z^2 - 781606447/362797056*z - 715820345/4353564672 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '212.138' ************** MAGMA ***************** Host 212.138.47.14 (212.138.47.14) Time: Sun Dec 18 09:13:54 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 09:13:52 on modular [Seed = 4279882356] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x + 216*z^4 + 288*z^3 - 288*z^2 - 256*z + 165, y + 6*z^2 + 4*z - 5, z^8 + 8/3*z^7 - 8/9*z^6 - 160/27*z^5 + 23/162*z^4 + 1262/243*z^3 - 458/729*z^2 - 505855/279936*z + 162685/279936 ] Total time: 0.220 seconds, Total memory usage: 3.34MB '144.137' ************** MAGMA ***************** Host 144.137.185.243 (144.137.185.243) Time: Sun Dec 18 04:05:26 2005 Input: 2+1 divided by x = 4x Output: Magma V2.11-10 Sun Dec 18 2005 04:05:26 on modular [Seed = 3340870312] ------------------------------------- >> 2+1 divided by x = 4x; ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '216.239' ************** MAGMA ***************** Host 216.239.167.217 (216.239.167.217) Time: Sat Dec 17 21:24:05 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Sat Dec 17 2005 21:24:05 on modular [Seed = 3910034073] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.15 (212.138.47.15) Time: Sat Dec 17 13:59:08 2005 Input: Table[Timing[FullSimplify[{i, Cos[Pi/i], Sin[Pi/i]}]], {i, 1, 22}] Output: Magma V2.11-10 Sat Dec 17 2005 13:59:07 on modular [Seed = 1686337705] ------------------------------------- >> Table[Timing[FullSimplify[{i, Cos[Pi/i], Sin[Pi/i]}]], {i, 1, 22}] ; ^ User error: Identifier 'Table' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '85.89.1' ************** MAGMA ***************** Host 85.89.168.182 (85.89.168.182) Time: Sat Dec 17 13:57:50 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Sat Dec 17 2005 13:57:50 on modular [Seed = 2005953512] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.15 (212.138.47.15) Time: Sat Dec 17 13:52:57 2005 Input: P:=PolynomialRing(RationalField(),4); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:52:56 on modular [Seed = 250155959] ------------------------------------- >> P:=PolynomialRing(RationalField(),4); I:=ideal

: Rhs argument 4 is invalid for this constructor >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.34MB '212.138' ************** MAGMA ***************** Host 212.138.47.24 (212.138.47.24) Time: Sat Dec 17 13:50:38 2005 Input: P:=PolynomialRing(RationalField(),4); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:50:38 on modular [Seed = 183309509] ------------------------------------- >> -1 + m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2), ^ Runtime error in '^': Bad argument types Argument types given: RngMPolElt, RngMPolElt >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.23 (212.138.47.23) Time: Sat Dec 17 13:50:07 2005 Input: P:=PolynomialRing(RationalField(),4); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:50:06 on modular [Seed = 132127208] ------------------------------------- >> -t1 + m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2), ^ User error: Identifier 't1' has not been declared or assigned >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.15 (212.138.47.15) Time: Sat Dec 17 13:48:52 2005 Input: P:=PolynomialRing(RationalField(),4); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:48:52 on modular [Seed = 15146661] ------------------------------------- >> 1 = m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2), ^ Runtime error in '^': Bad argument types Argument types given: RngMPolElt, RngMPolElt >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.17 (212.138.47.17) Time: Sat Dec 17 13:48:05 2005 Input: P:=PolynomialRing(RationalField(),4); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:48:05 on modular [Seed = 486212168] ------------------------------------- >> t1 = m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2), ^ User error: Identifier 't1' has not been declared or assigned >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.24 (212.138.47.24) Time: Sat Dec 17 13:45:43 2005 Input: P:=PolynomialRing(RationalField(),1); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:45:42 on modular [Seed = 351472627] ------------------------------------- Ideal of Polynomial ring of rank 1 over Rational Field Lexicographical Order Variables: y Dimension 0, Radical Groebner basis: [ y^21 - 12*y^20 + 55*y^19 - 107*y^18 + 17*y^17 + 245*y^16 - 175*y^15 - 535*y^14 + 715*y^13 + 844*y^12 - 2407*y^11 + 1075*y^10 + 1459*y^9 - 503*y^8 - 2999*y^7 + 4018*y^6 - 2002*y^5 + 37*y^4 + 489*y^3 - 311*y^2 + 86*y - 13 ] Total time: 0.200 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.113.12 (212.138.113.12) Time: Sat Dec 17 13:44:41 2005 Input: P:=PolynomialRing(RationalField(),1); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:44:41 on modular [Seed = 301338297] ------------------------------------- Ideal of Polynomial ring of rank 1 over Rational Field Lexicographical Order Variables: x Dimension 0, Radical Groebner basis: [ x^7 - 2*x^6 - x^5 + x^4 + x^3 + x^2 - x - 1 ] Total time: 0.200 seconds, Total memory usage: 3.24MB '83.26.1' ************** MAGMA ***************** Host 83.26.157.172 (83.26.157.172) Time: Sat Dec 17 10:35:39 2005 Input: pi(51) Output: Magma V2.11-10 Sat Dec 17 2005 10:35:39 on modular [Seed = 301359720] ------------------------------------- >> pi(51) ^ User error: Identifier 'pi' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '83.26.1' ************** MAGMA ***************** Host 83.26.157.172 (83.26.157.172) Time: Sat Dec 17 10:35:28 2005 Input: piprime(51) Output: Magma V2.11-10 Sat Dec 17 2005 10:35:28 on modular [Seed = 751525726] ------------------------------------- >> piprime(51) ^ User error: Identifier 'piprime' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '83.26.1' ************** MAGMA ***************** Host 83.26.157.172 (83.26.157.172) Time: Sat Dec 17 10:29:43 2005 Input: prime(51) Output: Magma V2.11-10 Sat Dec 17 2005 10:29:43 on modular [Seed = 3442122777] ------------------------------------- >> prime(51) ^ User error: Identifier 'prime' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '83.26.1' ************** MAGMA ***************** Host 83.26.157.172 (83.26.157.172) Time: Sat Dec 17 10:28:07 2005 Input: pi(100) Output: Magma V2.11-10 Sat Dec 17 2005 10:28:07 on modular [Seed = 3257249654] ------------------------------------- >> pi(100) ^ User error: Identifier 'pi' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:18:15 2005 Input: G:=DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:18:13 on modular [Seed = 684237507] ------------------------------------- Group of Dirichlet characters of modulus 432 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 [ Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field ] q - 5*q^7 - 7*q^13 + q^19 - 5*q^25 + 4*q^31 - q^37 - 8*q^43 + 18*q^49 - 13*q^61 - 11*q^67 + 17*q^73 + 13*q^79 + 35*q^91 + 5*q^97 + O(q^99) Total time: 1.399 seconds, Total memory usage: 6.08MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:15:53 2005 Input: G: =DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:15:52 on modular [Seed = 3509672902] ------------------------------------- >> G: =DirichletGroup(432);G; ^ User error: bad syntax >> X :=Elements(G);X; ^ User error: Identifier 'G' has not been declared or assigned >> X :=Elements(G);X; ^ User error: Identifier 'X' has not been declared or assigned >> X; ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 2, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],99); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:15:32 2005 Input: G: = DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:15:32 on modular [Seed = 3593886941] ------------------------------------- >> G: = DirichletGroup(432);G; ^ User error: bad syntax >> X :=Elements(G);X; ^ User error: Identifier 'G' has not been declared or assigned >> X :=Elements(G);X; ^ User error: Identifier 'X' has not been declared or assigned >> X; ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 2, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],99); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:15:27 2005 Input: G: =DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:15:27 on modular [Seed = 3678097899] ------------------------------------- >> G: =DirichletGroup(432);G; ^ User error: bad syntax >> X :=Elements(G);X; ^ User error: Identifier 'G' has not been declared or assigned >> X :=Elements(G);X; ^ User error: Identifier 'X' has not been declared or assigned >> X; ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 2, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],99); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:15:16 2005 Input: G:=DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:15:14 on modular [Seed = 3223341821] ------------------------------------- Group of Dirichlet characters of modulus 432 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 [ Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field ] q + 4*q^5 + 3*q^7 - 4*q^11 + q^13 - 4*q^17 + q^19 - 4*q^23 + 11*q^25 + 4*q^31 + 12*q^35 - 9*q^37 + 8*q^43 + 12*q^47 + 2*q^49 - 8*q^53 - 16*q^55 - 4*q^59 - 5*q^61 + 4*q^65 - 11*q^67 - 8*q^71 + q^73 - 12*q^77 + 5*q^79 - 8*q^83 - 16*q^85 + 12*q^89 + 3*q^91 + 4*q^95 + 5*q^97 + O(q^99) Total time: 1.409 seconds, Total memory usage: 6.10MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:15:02 2005 Input: G: =DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:15:02 on modular [Seed = 3307556844] ------------------------------------- >> G: =DirichletGroup(432);G; ^ User error: bad syntax >> X :=Elements(G);X; ^ User error: Identifier 'G' has not been declared or assigned >> X :=Elements(G);X; ^ User error: Identifier 'X' has not been declared or assigned >> X; ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 2, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],99); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:14:49 2005 Input: G:=DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:14:48 on modular [Seed = 3391766813] ------------------------------------- Group of Dirichlet characters of modulus 432 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 [ Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field ] q + 3*q^5 + q^7 + 3*q^11 - 4*q^13 - 2*q^19 + 6*q^23 + 4*q^25 + 6*q^29 - 5*q^31 + 3*q^35 + 2*q^37 - 6*q^41 + 10*q^43 - 6*q^47 - 6*q^49 + 9*q^53 + 9*q^55 - 12*q^59 + 8*q^61 - 12*q^65 - 14*q^67 - 7*q^73 + 3*q^77 - 8*q^79 + 3*q^83 - 18*q^89 - 4*q^91 - 6*q^95 - q^97 + O(q^99) Total time: 1.429 seconds, Total memory usage: 6.00MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:12:34 2005 Input: G :=DirichletGroup(432); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:12:32 on modular [Seed = 4027612079] ------------------------------------- Group of Dirichlet characters of modulus 432 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 [ Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field ] q - 5*q^7 - 7*q^13 + q^19 - 5*q^25 + 4*q^31 - q^37 - 8*q^43 + 18*q^49 - 13*q^61 - 11*q^67 + 17*q^73 + 13*q^79 + 35*q^91 + 5*q^97 + O(q^99) Total time: 1.399 seconds, Total memory usage: 6.00MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:11:15 2005 Input: G :=DirichletGroup(432); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Sat Dec 17 2005 03:11:14 on modular [Seed = 4111826680] ------------------------------------- Group of Dirichlet characters of modulus 432 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 [ Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field ] Total time: 1.379 seconds, Total memory usage: 6.00MB '200.177' ************** MAGMA ***************** Host 200.177.28.205 (200.177.28.205) Time: Fri Dec 16 21:51:05 2005 Input: 0^0; Output: Magma V2.11-10 Fri Dec 16 2005 21:51:04 on modular [Seed = 2153823931] ------------------------------------- 1 Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.14 (212.138.47.14) Time: Fri Dec 16 21:08:24 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Fri Dec 16 2005 21:08:23 on modular [Seed = 1437951329] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x + 24021947384028469294199051557281326175896046567/88390501855296726402302\ 4091659442585600000000*z^7 - 499887372604618086734751651977842907689699\ 89/6313607275378337600164457797567447040000000*z^5 - 3578323456874350741013274769517430940050603/450971948241309828583175556\ 96910336000000*z^3 - 7781918205774041731639575981199074243/257698256137\ 8913306189574611252019200*z, y - 463586742170926250485324993944801190173/1287206326036293892078321948672\ 0000000*z^7 + 1085903501599971128651516196294585591/9194330900259242086\ 2737282048000000*z^5 + 9873044499317982446687031775259151/9381970306386\ 9817206874777600000*z^3 + 128601565742559180668707869379/11258364367664\ 3780648249733120*z, z^8 - 36195915798000255920/96199778220194782809*z^6 - 22658347845807961565600/7792182035835777407529*z^4 + 432194167813940000000/4125272842501293921633*z^2 + 16295122460522500000000/631166744902697970009849 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '155.207' ************** MAGMA ***************** Host 155.207.209.210 (155.207.209.210) Time: Fri Dec 16 19:36:19 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O,5^4); Output: Magma V2.11-10 Fri Dec 16 2005 19:36:18 on modular [Seed = 3476017489] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [5, 0, 0, 0] ] Total time: 0.320 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.210 (155.207.209.210) Time: Fri Dec 16 19:35:13 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O,2*5^4); Output: Magma V2.11-10 Fri Dec 16 2005 19:35:13 on modular [Seed = 3526544527] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [0, -5, 0, 0] ] Total time: 0.330 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.210 (155.207.209.210) Time: Fri Dec 16 19:35:00 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O,5); Output: Magma V2.11-10 Fri Dec 16 2005 19:35:00 on modular [Seed = 3577072637] ------------------------------------- [ 1, y, y^2, y^3 ] false Total time: 0.310 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.210 (155.207.209.210) Time: Fri Dec 16 19:32:52 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 5^4); Output: Magma V2.11-10 Fri Dec 16 2005 19:32:51 on modular [Seed = 3745497200] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [5, 0, 0, 0] ] Total time: 0.320 seconds, Total memory usage: 3.63MB '66.14.9' ************** MAGMA ***************** Host 66.14.95.197 (66.14.95.197) Time: Fri Dec 16 18:40:23 2005 Input: sin(3.141549); Output: Magma V2.11-10 Fri Dec 16 2005 18:40:23 on modular [Seed = 2659093989] ------------------------------------- >> sin(3.141549);; ^ User error: Identifier 'sin' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '66.14.9' ************** MAGMA ***************** Host 66.14.95.197 (66.14.95.197) Time: Fri Dec 16 18:39:33 2005 Input: sin(pi); Output: Magma V2.11-10 Fri Dec 16 2005 18:39:33 on modular [Seed = 2040097360] ------------------------------------- >> sin(pi);; ^ User error: Identifier 'pi' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '66.14.9' ************** MAGMA ***************** Host 66.14.95.197 (66.14.95.197) Time: Fri Dec 16 18:38:29 2005 Input: 5! Output: Magma V2.11-10 Fri Dec 16 2005 18:38:29 on modular [Seed = 1071592406] ------------------------------------- >> 5!; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '155.207' ************** MAGMA ***************** Host 155.207.209.110 (155.207.209.110) Time: Fri Dec 16 18:36:14 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 4*5^4); Output: Magma V2.11-10 Fri Dec 16 2005 18:36:12 on modular [Seed = 14683248] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [0, 0, -5, 0] ] Total time: 0.330 seconds, Total memory usage: 3.63MB '128.138' ************** MAGMA ***************** Host 128.138.240.175 (128.138.240.175) Time: Fri Dec 16 16:41:06 2005 Input: 1234123412341234341111/2714351253451253424 Output: Magma V2.11-10 Fri Dec 16 2005 16:41:06 on modular [Seed = 3459168674] ------------------------------------- 411374470780411447037/904783751150417808 Total time: 0.180 seconds, Total memory usage: 3.24MB '131.156' ************** MAGMA ***************** Host 131.156.3.93 (131.156.3.93) Time: Fri Dec 16 16:09:01 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Fri Dec 16 2005 16:09:01 on modular [Seed = 2040075914] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x + 24021947384028469294199051557281326175896046567/88390501855296726402302\ 4091659442585600000000*z^7 - 499887372604618086734751651977842907689699\ 89/6313607275378337600164457797567447040000000*z^5 - 3578323456874350741013274769517430940050603/450971948241309828583175556\ 96910336000000*z^3 - 7781918205774041731639575981199074243/257698256137\ 8913306189574611252019200*z, y - 463586742170926250485324993944801190173/1287206326036293892078321948672\ 0000000*z^7 + 1085903501599971128651516196294585591/9194330900259242086\ 2737282048000000*z^5 + 9873044499317982446687031775259151/9381970306386\ 9817206874777600000*z^3 + 128601565742559180668707869379/11258364367664\ 3780648249733120*z, z^8 - 36195915798000255920/96199778220194782809*z^6 - 22658347845807961565600/7792182035835777407529*z^4 + 432194167813940000000/4125272842501293921633*z^2 + 16295122460522500000000/631166744902697970009849 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '131.156' ************** MAGMA ***************** Host 131.156.3.93 (131.156.3.93) Time: Fri Dec 16 16:06:07 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; GroebnerBasis(I); Output: Magma V2.11-10 Fri Dec 16 2005 16:06:07 on modular [Seed = 2124288351] ------------------------------------- [ x + 24021947384028469294199051557281326175896046567/88390501855296726402302\ 4091659442585600000000*z^7 - 499887372604618086734751651977842907689699\ 89/6313607275378337600164457797567447040000000*z^5 - 3578323456874350741013274769517430940050603/450971948241309828583175556\ 96910336000000*z^3 - 7781918205774041731639575981199074243/257698256137\ 8913306189574611252019200*z, y - 463586742170926250485324993944801190173/1287206326036293892078321948672\ 0000000*z^7 + 1085903501599971128651516196294585591/9194330900259242086\ 2737282048000000*z^5 + 9873044499317982446687031775259151/9381970306386\ 9817206874777600000*z^3 + 128601565742559180668707869379/11258364367664\ 3780648249733120*z, z^8 - 36195915798000255920/96199778220194782809*z^6 - 22658347845807961565600/7792182035835777407529*z^4 + 432194167813940000000/4125272842501293921633*z^2 + 16295122460522500000000/631166744902697970009849 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '131.156' ************** MAGMA ***************** Host 131.156.3.93 (131.156.3.93) Time: Fri Dec 16 16:02:52 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; GroebnerBasis(I); Output: Magma V2.11-10 Fri Dec 16 2005 16:02:48 on modular [Seed = 1168452898] ------------------------------------- [ x + 24021947384028469294199051557281326175896046567/88390501855296726402302\ 4091659442585600000000*z^7 - 499887372604618086734751651977842907689699\ 89/6313607275378337600164457797567447040000000*z^5 - 3578323456874350741013274769517430940050603/450971948241309828583175556\ 96910336000000*z^3 - 7781918205774041731639575981199074243/257698256137\ 8913306189574611252019200*z, y - 463586742170926250485324993944801190173/1287206326036293892078321948672\ 0000000*z^7 + 1085903501599971128651516196294585591/9194330900259242086\ 2737282048000000*z^5 + 9873044499317982446687031775259151/9381970306386\ 9817206874777600000*z^3 + 128601565742559180668707869379/11258364367664\ 3780648249733120*z, z^8 - 36195915798000255920/96199778220194782809*z^6 - 22658347845807961565600/7792182035835777407529*z^4 + 432194167813940000000/4125272842501293921633*z^2 + 16295122460522500000000/631166744902697970009849 ] Total time: 0.230 seconds, Total memory usage: 3.34MB '222.124' ************** MAGMA ***************** Host 222.124.19.133 (222.124.19.133) Time: Fri Dec 16 14:01:39 2005 Input: if ver eq 1 then M := Matrix(8,8,[ [2, 0, 0, 0, 0, 0, 0, 0], [0, 396,-214,-386, 36, 25,-144, 426], [1,-205, -34,-196, 230, 83,-662, 19], [1,-305, 528,-358,-250, 73, 38, 277], [1, 38, -45,-282, 584, 122, -24,-476], [0, 127, 131, 119, 369,-633, 152,-275], [0, 436, -54,-138,-442, 330,-312,-350], [1, 82, 757, 102, 372, 111,-248, 258]]); else M := Matrix(8,8,[ [ 2, 0, 0, 0, 0, 0, 0, 0], [ 0, 36, 221, 5, -64, 23, -32, 352], [ 0,129,-108,-193,-285, 97, 146,-178], [ 0, 46, -89,-166, 66, -46,-374,-241], [ 0,274, -85, 254, 212, 175, 166, 36], [ 1, 99,-185, 145, 145,-400, -19, 98], [ 1, 82,-367, 197, -42, 197,-191, 80], [ 0, 23, 40, 218,-182,-214,-224,-330]]); end if; Output: Magma V2.11-10 Fri Dec 16 2005 14:01:35 on modular [Seed = 2170670481] ------------------------------------- >> if ver eq 1 then ^ User error: Identifier 'ver' has not been declared or assigned Total time: 0.230 seconds, Total memory usage: 3.24MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:55:33 2005 Input: F:=GF(2^4); DefiningPolynomial(F); P := PolynomialRing(F); f := x^3+3*x+1; Order($.1); Output: Magma V2.11-10 Fri Dec 16 2005 07:55:32 on modular [Seed = 2124383264] ------------------------------------- $.1^4 + $.1 + 1 >> Order($.1);; ^ Runtime error: Bad dollar structure Total time: 0.200 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:55:15 2005 Input: F:=GF(2^4); DefiningPolynomial(F); P := PolynomialRing(F); f := x^3+3*x+1; Order(f); Output: Magma V2.11-10 Fri Dec 16 2005 07:55:15 on modular [Seed = 2073723219] ------------------------------------- $.1^4 + $.1 + 1 >> Order(f);; ^ Runtime error in 'Order': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.200 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:52:31 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; Order(f); Output: Magma V2.11-10 Fri Dec 16 2005 07:52:31 on modular [Seed = 1252766174] ------------------------------------- >> Order(f);; ^ Runtime error in 'Order': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.200 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:52:16 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; Exponent(f); Output: Magma V2.11-10 Fri Dec 16 2005 07:52:15 on modular [Seed = 1471589427] ------------------------------------- >> Exponent(f);; ^ Runtime error in 'Exponent': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.200 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:52:08 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; Exponenet(f); Output: Magma V2.11-10 Fri Dec 16 2005 07:52:08 on modular [Seed = 1555932991] ------------------------------------- >> Exponenet(f);; ^ User error: Identifier 'Exponenet' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:51:51 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; 2*f; Output: Magma V2.11-10 Fri Dec 16 2005 07:51:50 on modular [Seed = 1505272356] ------------------------------------- 0 Total time: 0.190 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:51:29 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; 3*f; Output: Magma V2.11-10 Fri Dec 16 2005 07:51:29 on modular [Seed = 650632556] ------------------------------------- x^3 + x + 1 Total time: 0.190 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:51:18 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; 3*f; Output: Magma V2.11-10 Fri Dec 16 2005 07:51:17 on modular [Seed = 599971965] ------------------------------------- x^3 + x + 1 Total time: 0.190 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:51:09 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; f; Output: Magma V2.11-10 Fri Dec 16 2005 07:51:04 on modular [Seed = 684315403] ------------------------------------- x^3 + x + 1 Total time: 0.260 seconds, Total memory usage: 3.34MB '213.84.' ************** MAGMA ***************** Host 213.84.213.14 (213.84.213.14) Time: Fri Dec 16 02:41:40 2005 Input: FindGroupOrder2 := function (p, s) K := GF(p); v := K ! (4*s); u := K ! (s^2-5); x := u^3; b := 4*x*v; a := (v-u)^3*(3*u+v); A := a/b-2; x := x/v^3; b := x^3 + A*x^2 + x; E := EllipticCurve([0,b*A,0,b^2,0]); return FactoredOrder(E); end function; p := 140853945410621700611366248656986006762214430713643; s := 941728572; FindGroupOrder2(p, s); Output: Magma V2.11-10 Fri Dec 16 2005 02:41:38 on modular [Seed = 920066706] ------------------------------------- [ <2, 2>, <3, 1>, <5, 1>, <137, 1>, <251, 1>, <317, 1>, <331, 1>, <7307, 1>, <13339, 1>, <15667, 1>, <29209, 1>, <137209, 1>, <332933, 1>, <393571, 1>, <811351, 1> ] Total time: 2.390 seconds, Total memory usage: 5.31MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 02:29:36 2005 Input: Factorization(00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001); Output: Magma V2.11-10 Fri Dec 16 2005 02:29:36 on modular [Seed = 3994009881] ------------------------------------- [] Total time: 0.180 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 02:29:14 2005 Input: Factorization(0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Fri Dec 16 2005 02:29:14 on modular [Seed = 3808611390] ------------------------------------- >> Factorization(0000000000000000000000000000000000000000000000000000000000000 ^ Runtime error in 'Factorization': Argument 1 is not non-zero Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:56:30 2005 Input: Factorization(50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Fri Dec 16 2005 00:56:30 on modular [Seed = 616902717] ------------------------------------- [ <2, 832>, <5, 833> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:55:25 2005 Input: Factorization(500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012355); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Fri Dec 16 2005 00:55:05 on modular [Seed = 970468503] ------------------------------------- Errors: /bin/sh: line 1: 21550 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:53:55 2005 Input: Factorization(50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000005678901234567890); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Fri Dec 16 2005 00:53:34 on modular [Seed = 852699003] ------------------------------------- Errors: /bin/sh: line 1: 21541 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:06:55 2005 Input: Factorization(4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Fri Dec 16 2005 00:06:55 on modular [Seed = 2090693635] ------------------------------------- [ <2, 2351>, <5, 2349> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:06:08 2005 Input: Factorization(0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004); Output: Magma V2.11-10 Fri Dec 16 2005 00:06:07 on modular [Seed = 1437883770] ------------------------------------- [ <2, 2> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:05:06 2005 Input: Factorization(0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Fri Dec 16 2005 00:05:05 on modular [Seed = 48433683] ------------------------------------- >> Factorization(0000000000000000000000000000000000000000000000000000000000000 ^ Runtime error in 'Factorization': Argument 1 is not non-zero Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 19:21:11 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode; C3; Output: Magma V2.11-10 Thu Dec 15 2005 19:21:11 on modular [Seed = 3576995498] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.200 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 19:20:21 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode; L; Output: Magma V2.11-10 Thu Dec 15 2005 19:20:21 on modular [Seed = 3745420434] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] { (1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0), (0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0), (0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0), (0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1), (0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1), (0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1), (0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0), (0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0), (0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1), (0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0), (0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1), (0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0), (1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0), (0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0) } Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 19:17:48 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode(L); L; Output: Magma V2.11-10 Thu Dec 15 2005 19:17:47 on modular [Seed = 3644366202] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] >> C3:=LinearCode(L); ^ Runtime error in 'LinearCode': Bad argument types Argument types given: SetEnum[ModTupFldElt] { (1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0), (0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0), (0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0), (0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1), (0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1), (0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1), (0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0), (0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0), (0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1), (0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0), (0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1), (0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0), (1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0), (0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0) } Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:58:17 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); MinimumWords(C2); Output: Magma V2.11-10 Thu Dec 15 2005 18:58:17 on modular [Seed = 3193962159] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] { (1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0), (0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0), (0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0), (0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1), (0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1), (0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1), (0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0), (0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0), (0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1), (0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0), (0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1), (0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0), (1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0), (0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0) } Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:56:30 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); Output: Magma V2.11-10 Thu Dec 15 2005 18:56:30 on modular [Seed = 3092904222] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:55:19 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; Output: Magma V2.11-10 Thu Dec 15 2005 18:55:19 on modular [Seed = 2739069644] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.200 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:54:58 2005 Input: K := FiniteField(2); > C := LinearCode: S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; Output: Magma V2.11-10 Thu Dec 15 2005 18:54:58 on modular [Seed = 2924474356] ------------------------------------- >> 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>: ^ User error: bad syntax >> S, f := StandardForm(C); ^ User error: Identifier 'C' has not been declared or assigned >> D := Dual(S); ^ User error: Identifier 'S' has not been declared or assigned >> (D meet S) eq S; ^ User error: Identifier 'D' has not been declared or assigned >> M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); ^ User error: Identifier 'S' has not been declared or assigned >> M1:=EchelonForm(M); ^ User error: Identifier 'M' has not been declared or assigned >> M2:=Submatrix(M1,22,22,14,35); ^ User error: Identifier 'M1' has not been declared or assigned >> M2; ^ User error: Identifier 'M2' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:53:27 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M1; M2:=Submatrix(M1,22,22,14,35); M2; Output: Magma V2.11-10 Thu Dec 15 2005 18:53:27 on modular [Seed = 2823416407] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] true [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 0 0 0 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'65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:47:10 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M1; Output: Magma V2.11-10 Thu Dec 15 2005 18:47:10 on modular [Seed = 2675891433] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:45:08 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(S)); M1:=EchelonForm(M); M1; Output: Magma V2.11-10 Thu Dec 15 2005 18:45:08 on modular [Seed = 2591807894] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] true [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] Total time: 0.180 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:05:15 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; D := Dual(S); (D meet S) eq S; Output: Magma V2.11-10 Thu Dec 15 2005 18:05:14 on modular [Seed = 1538911993] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] true Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:03:40 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; D := Dual(S); Output: Magma V2.11-10 Thu Dec 15 2005 18:03:40 on modular [Seed = 1168633857] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] Total time: 0.190 seconds, Total memory usage: 3.34MB '152.6.1' ************** MAGMA ***************** Host 152.6.19.192 (152.6.19.192) Time: Thu Dec 15 12:35:48 2005 Input: K := FiniteField(2); > C := LinearCode; IsSelfOrthogonal(C); > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); WeightDistribution(C); Output: Magma V2.11-10 Thu Dec 15 2005 12:35:48 on modular [Seed = 3177142192] ------------------------------------- false 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] Total time: 0.310 seconds, Total memory usage: 5.51MB '199.89.' ************** MAGMA ***************** Host 199.89.64.177 (199.89.64.177) Time: Thu Dec 15 11:15:37 2005 Input: FindGroupOrder2 := function (p, s) K := GF(p); v := K ! (4*s); u := K ! (s^2-5); x := u^3; b := 4*x*v; a := (v-u)^3*(3*u+v); A := a/b-2; x := x/v^3; b := x^3 + A*x^2 + x; E := EllipticCurve([0,b*A,0,b^2,0]); return FactoredOrder(E); end function; p := 140853945410621700611366248656986006762214430713643; s := 941728572; FindGroupOrder2(p, s); Output: Magma V2.11-10 Thu Dec 15 2005 11:15:30 on modular [Seed = 1636021435] ------------------------------------- [ <2, 2>, <3, 1>, <5, 1>, <137, 1>, <251, 1>, <317, 1>, <331, 1>, <7307, 1>, <13339, 1>, <15667, 1>, <29209, 1>, <137209, 1>, <332933, 1>, <393571, 1>, <811351, 1> ] Total time: 2.439 seconds, Total memory usage: 5.31MB '129.13.' ************** MAGMA ***************** Host 129.13.186.1 (129.13.186.1) Time: Thu Dec 15 05:37:39 2005 Input: 9+100000000000000000000000000000 Output: Magma V2.11-10 Thu Dec 15 2005 05:37:39 on modular [Seed = 2338951053] ------------------------------------- 100000000000000000000000000009 Total time: 0.180 seconds, Total memory usage: 3.24MB '148.87.' ************** MAGMA ***************** Host 148.87.1.172 (148.87.1.172) Time: Thu Dec 15 00:38:27 2005 Input: for(x=1,50,print1("d(",x,")=",numdiv(x),", ")) Output: Magma V2.11-10 Thu Dec 15 2005 00:38:27 on modular [Seed = 1669407046] ------------------------------------- >> for(x=1,50,print1("d(",x,")=",numdiv(x),", ")); ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '159.18.' ************** MAGMA ***************** Host 159.18.221.196 (159.18.221.196) Time: Wed Dec 14 17:22:35 2005 Input: 3^-1%0xF5DD146F31C94C0F7BF8CE66A63E86D47FB68E7C3F00BFB2232B848ECCBB0AF5 Output: WARNING: MAGMA command contains unsafe command '%', so it will not be executed. '159.18.' ************** MAGMA ***************** Host 159.18.221.196 (159.18.221.196) Time: Wed Dec 14 17:22:14 2005 Input: 3^-1mod0xF5DD146F31C94C0F7BF8CE66A63E86D47FB68E7C3F00BFB2232B848ECCBB0AF5 Output: Magma V2.11-10 Wed Dec 14 2005 17:22:14 on modular [Seed = 3143132098] ------------------------------------- >> 3^-1mod0xF5DD146F31C94C0F7BF8CE66A63E86D47FB68E7C3F00BFB2232B848ECCBB0AF5; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '159.18.' ************** MAGMA ***************** Host 159.18.221.196 (159.18.221.196) Time: Wed Dec 14 17:22:06 2005 Input: 3^-1%0xF5DD146F31C94C0F7BF8CE66A63E86D47FB68E7C3F00BFB2232B848ECCBB0AF5 Output: WARNING: MAGMA command contains unsafe command '%', so it will not be executed. '159.18.' ************** MAGMA ***************** Host 159.18.221.196 (159.18.221.196) Time: Wed Dec 14 17:21:51 2005 Input: 3^-1 Output: Magma V2.11-10 Wed Dec 14 2005 17:21:51 on modular [Seed = 2738911795] ------------------------------------- 1/3 Total time: 0.190 seconds, Total memory usage: 3.24MB '159.18.' ************** MAGMA ***************** Host 159.18.221.196 (159.18.221.196) Time: Wed Dec 14 17:21:26 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Wed Dec 14 2005 17:21:25 on modular [Seed = 2823516450] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.200 seconds, Total memory usage: 3.24MB '147.96.' ************** MAGMA ***************** Host 147.96.20.233 (147.96.20.233) Time: Wed Dec 14 14:00:13 2005 Input: P<[x]>:=PolynomialRing(GF(2),3,"grevlex"); g1:=x[1]*x[3]+x[2]*x[3]+x[3]; g2:=x[1]*x[2]+x[1]*x[3]+x[2]*x[3]+x[1]; g3:=x[2]^2+x[2]*x[3]+x[3]^2+1; g4:=x[2]+x[3]+1; g5:=x[3]^2+x[3]; pol:=x[2]*x[3]*g4; g:=[g1,g2,g3,g4,g5]; I:=ideal; T1:=ideal; T2:=ideal; T3:=ideal; T4:=ideal; T5:=ideal; T:=[T1,T2,T3,T4,T5]; deg:=1; equis:=x cat [1]; Mons:={a:a in x} join {1}; for j in [2..deg] do Mons:={mon*vble:vble in equis, mon in Mons}; end for; Mons2:=Mons; for i in [1,2] do Mons2:={mon*vble:vble in equis, mon in Mons2}; end for; Mons2:=[mon:mon in Mons2]; polinomios:=[mon*g[i]: i in [1..5], mon in Mons | not(mon in T[i])]; dim1:=#polinomios+1; dim2:=#Mons2+1; //G:=GF(2,dim1+1); //AssertAttribute(G,"PowerPrinting",false); M:=RMatrixSpace(P,dim1,dim2); Mat:=M!0; for i in [1..dim1-1] do lista:=Monomials(polinomios[i]); for j in [1..dim2-1] do if Mons2[j] in lista then Mat[i,j]:=1; end if; end for; end for; lista:=Monomials(pol); for j in [1..dim2-1] do if Mons2[j] in lista then Mat[dim1,j]:=1; end if; end for; Mat[dim1,dim2]:=1; //Mat[dim1,dim2]:=1; B:=EchelonForm(Mat); //newpol:=&+[Mons2[j]*B[dim1,j]:j in [1..dim2-1]]; Output: Magma V2.11-10 Wed Dec 14 2005 14:00:12 on modular [Seed = 2356126809] ------------------------------------- >> B:=EchelonForm(Mat); ^ Runtime error in 'EchelonForm': Argument 1 has no echelon algorithm Total time: 0.190 seconds, Total memory usage: 3.24MB '66.50.2' ************** MAGMA ***************** Host 66.50.2.109 (66.50.2.109) Time: Wed Dec 14 11:20:13 2005 Input: 7102277187569652249869562285153253710645556161842592536300347670222640516361186363486317189963554176580264215037288238394086254424175854090684343545452440 Output: Magma V2.11-10 Wed Dec 14 2005 11:20:12 on modular [Seed = 3728595848] ------------------------------------- 7102277187569652249869562285153253710645556161842592536300347670222640516361186\ 363486317189963554176580264215037288238394086254424175854090684343545452440 Total time: 0.190 seconds, Total memory usage: 3.24MB '129.20.' ************** MAGMA ***************** Host 129.20.36.132 (129.20.36.132) Time: Wed Dec 14 10:32:34 2005 Input: K:=CyclotomicField(13); G,phi:=UnitGroup(K); G; Output: Magma V2.11-10 Wed Dec 14 2005 10:32:33 on modular [Seed = 301099370] ------------------------------------- Abelian Group isomorphic to Z/26 + Z (5 copies) Defined on 6 generators Relations: 26*G.1 = 0 Total time: 1.260 seconds, Total memory usage: 3.82MB '67.62.1' ************** MAGMA ***************** Host 67.62.112.123 (67.62.112.123) Time: Wed Dec 14 07:16:04 2005 Input: a:=1; a: Output: Magma V2.11-10 Wed Dec 14 2005 07:16:03 on modular [Seed = 3442377647] ------------------------------------- >> a:; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '83.250.' ************** MAGMA ***************** Host 83.250.129.152 (83.250.129.152) Time: Wed Dec 14 06:40:51 2005 Input: 5*5 Output: Magma V2.11-10 Wed Dec 14 2005 06:40:49 on modular [Seed = 717843850] ------------------------------------- 25 Total time: 0.260 seconds, Total memory usage: 3.24MB '217.24.' ************** MAGMA ***************** Host 217.24.144.35 (217.24.144.35) Time: Wed Dec 14 06:30:58 2005 Input: jnbkjnlml Output: Magma V2.11-10 Wed Dec 14 2005 06:30:57 on modular [Seed = 1972778086] ------------------------------------- >> jnbkjnlml; ^ User error: Identifier 'jnbkjnlml' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 22:39:18 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; L:=MinimumWords(C2); C3:= LinearCode; C3; C4:=EvenWeightCode(15); WeightDistribution(C4); Output: Magma V2.11-10 Tue Dec 13 2005 22:39:17 on modular [Seed = 670824447] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [35, 10, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 1 1 0 1 0 1 1 1] [0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1] [0 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 1 1] [0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0] [0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1] [0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <2, 105>, <4, 1365>, <6, 5005>, <8, 6435>, <10, 3003>, <12, 455>, <14, 15> ] Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 22:23:21 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; L:=MinimumWords(C2); C3:= LinearCode; C3; Output: Magma V2.11-10 Tue Dec 13 2005 22:23:21 on modular [Seed = 1038216745] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [35, 10, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 1 1 0 1 0 1 1 1] [0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1] [0 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 1 1] [0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0] [0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1] [0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 22:20:16 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; L:=MinimumWords(C2); C3:= LinearCode; C3; Output: Magma V2.11-10 Tue Dec 13 2005 22:20:16 on modular [Seed = 3323595400] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] >> C3:= LinearCode: Rhs argument 1 is invalid for this constructor >> C3;; ^ User error: Identifier 'C3' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.46.74 (200.177.46.74) Time: Tue Dec 13 22:05:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^6 mod n =", rho^6 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; conj4 := function(g) s := Eltseq(g); s[4] := -s[4]; s[5] := -s[5]; s[6] := -s[6]; return Seqelt(s, Fp12); end function; t40 := trace4(g); t41 := conj4(t40); "trace4(g) =", Eltseq(t40); "trace4(g)' =", Eltseq(t41); "sum4 =", t40 + t41; "prod4 =", t40*t41; Output: Magma V2.11-10 Tue Dec 13 2005 22:05:46 on modular [Seed = 4246811337] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^6 mod n = 1 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073, 0, 0 ] trace4(g)' = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746, 0, 0 ] sum4 = 417917298933889071442278502933205352112085215149*i + 1022991499774629604152735929191636691013539358387 prod4 = 1290910566114678872655078479583819895277125832347*i + 1222447203379043640445950747036893708432248781076 Total time: 0.870 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.46.74 (200.177.46.74) Time: Tue Dec 13 22:02:17 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^6 mod n =", rho^6 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; conj6 := function(g) s := Eltseq(g); s[4] := -s[4]; s[5] := -s[5]; s[6] := -s[6]; return Seqelt(s, Fp12); end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", Eltseq(t40); "trace4(g)' =", Eltseq(t41); "trace4(g)'' =", Eltseq(t42); "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := conj6(t60); "trace6(g) =", t60; "trace6(g)' =", t61; "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 22:02:16 on modular [Seed = 2204060549] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^6 mod n = 1 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073, 0, 0 ] trace4(g)' = [ 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603, 0, 0, 600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881, 0, 0 ] trace4(g)'' = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746, 0, 0 ] sum4 = (600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (351960067217720226471026218912124823215388842998*i + 636471885525581132299463326642541502123540418918)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (1304924158594765850898145115201098723111052031587*i + 79422959053265780304025510525997589201073687502)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 sum6 = (221528860294990275916266022509765773312510174491*i + 941515545640830953042201319471430672307688494835)*z^2 + 278611532622592714294852335288803568074723476766*i + 681994333183086402768490619461091127342359572258 prod6 = (1199217906511468853397684229168281220558780854227*i + 1030765899693116805073366946226732872257120333608)*z^4 + (378575975826376771203186488360822049552104323606*i + 472503248255084275151742862775344953200669638852)*z^2 + 230957764124266000451333381176639619321109453643*i + 684491156455614923026389854551594178711792858992 Total time: 1.100 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:59:13 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(w); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 21:58:53 on modular [Seed = 1134529040] ------------------------------------- Errors: /bin/sh: line 1: 32125 Alarm clock nice -n 19 /usr/local/bin/magma '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:58:26 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(w); Output: Magma V2.11-10 Tue Dec 13 2005 21:58:25 on modular [Seed = 1184534313] ------------------------------------- 1/t^17*w^100 + (4*t^4 + 4)/t^17*w^80 + 4/t^13*w^76 + (t^8 + 2*t^4 + 1)/t^17*w^60 + (2*t^4 + 2)/t^13*w^56 + 1/t^9*w^52 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^17*w^40 + (2*t^8 + 4*t^4 + 2)/t^13*w^36 + (2*t^4 + 2)/t^9*w^32 + 4/t^5*w^28 + (t^16 + 4*t^12 + t^8 + 4*t^4 + 1)/t^17*w^20 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^13*w^16 + (t^8 + 2*t^4 + 1)/t^9*w^12 + (4*t^4 + 4)/t^5*w^8 + 1/t*w^4 + 1/t z^125 + (t^20 + t^16 + 1)*z^25 + (2*t^20 + t^16)*z^5 + t^20*z + 4*t^21 Total time: 0.200 seconds, Total memory usage: 3.72MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:58:12 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(w); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 21:57:52 on modular [Seed = 1284539714] ------------------------------------- Errors: /bin/sh: line 1: 32114 Alarm clock nice -n 19 /usr/local/bin/magma '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:56:51 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(w); Output: Magma V2.11-10 Tue Dec 13 2005 21:56:51 on modular [Seed = 1401919902] ------------------------------------- 1/t^17*w^100 + (4*t^4 + 4)/t^17*w^80 + 4/t^13*w^76 + (t^8 + 2*t^4 + 1)/t^17*w^60 + (2*t^4 + 2)/t^13*w^56 + 1/t^9*w^52 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^17*w^40 + (2*t^8 + 4*t^4 + 2)/t^13*w^36 + (2*t^4 + 2)/t^9*w^32 + 4/t^5*w^28 + (t^16 + 4*t^12 + t^8 + 4*t^4 + 1)/t^17*w^20 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^13*w^16 + (t^8 + 2*t^4 + 1)/t^9*w^12 + (4*t^4 + 4)/t^5*w^8 + 1/t*w^4 + 1/t z^125 + (t^20 + t^16 + 1)*z^25 + (2*t^20 + t^16)*z^5 + t^20*z + 4*t^21 Total time: 0.210 seconds, Total memory usage: 3.72MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:56:42 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(w); Output: Magma V2.11-10 Tue Dec 13 2005 21:56:42 on modular [Seed = 1451924610] ------------------------------------- t^3*w^100 + 3*t^3*w^80 + 4*t^3*w^76 + 4*t^3*w^60 + 4*t^3*w^56 + t^3*w^52 + 2*t^3*w^40 + 3*t^3*w^36 + 4*t^3*w^32 + 4*t^3*w^28 + t^3*w^20 + 2*t^3*w^16 + 4*t^3*w^12 + 3*t^3*w^8 + t^3*w^4 + 1/t z^125 + (2*t^4 + 1)/t^4*z^25 + (t^4 + 2)/t^4*z^5 + 1/t^4*z + 4/t^4 Total time: 0.200 seconds, Total memory usage: 3.82MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:56:18 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x); Output: Magma V2.11-10 Tue Dec 13 2005 21:56:18 on modular [Seed = 1568774065] ------------------------------------- t^3*w^100 + 3*t^3*w^80 + 4*t^3*w^76 + 4*t^3*w^60 + 4*t^3*w^56 + t^3*w^52 + 2*t^3*w^40 + 3*t^3*w^36 + 4*t^3*w^32 + 4*t^3*w^28 + t^3*w^20 + 2*t^3*w^16 + 4*t^3*w^12 + 3*t^3*w^8 + t^3*w^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + 4/t^4 Total time: 0.200 seconds, Total memory usage: 3.53MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:56:15 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x); Output: Magma V2.11-10 Tue Dec 13 2005 21:56:15 on modular [Seed = 1622977169] ------------------------------------- t^3*w^100 + 3*t^3*w^80 + 4*t^3*w^76 + 4*t^3*w^60 + 4*t^3*w^56 + t^3*w^52 + 2*t^3*w^40 + 3*t^3*w^36 + 4*t^3*w^32 + 4*t^3*w^28 + t^3*w^20 + 2*t^3*w^16 + 4*t^3*w^12 + 3*t^3*w^8 + t^3*w^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + 4/t^4 Total time: 0.200 seconds, Total memory usage: 3.53MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:56:01 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x); Output: Magma V2.11-10 Tue Dec 13 2005 21:56:00 on modular [Seed = 1740352921] ------------------------------------- 1/t*y^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + 4/t^4 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:55:53 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x); Output: Magma V2.11-10 Tue Dec 13 2005 21:55:52 on modular [Seed = 1790353644] ------------------------------------- >> A := AffineAlgebra; ^ User error: Identifier 'w' has not been declared or assigned >> y^-1; ^ User error: Identifier 'y' has not been declared or assigned >> MinimalPolynomial(x); ^ User error: Identifier 'x' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.46.74 (200.177.46.74) Time: Tue Dec 13 21:53:18 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^6 mod n =", rho^6 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", t40; "trace4(g) =", Eltseq(t40); "trace4(g)' =", Eltseq(t41); "trace4(g)'' =", Eltseq(t42); "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", Eltseq(t60); "trace6(g)' =", Eltseq(t61); "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:53:17 on modular [Seed = 1907211090] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^6 mod n = 1 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g) = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073, 0, 0 ] trace4(g)' = [ 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603, 0, 0, 600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881, 0, 0 ] trace4(g)'' = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746, 0, 0 ] sum4 = (600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (351960067217720226471026218912124823215388842998*i + 636471885525581132299463326642541502123540418918)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = [ 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129, 0, 841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827, 0, 156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317, 0 ] trace6(g)' = [ 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129, 0, 96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515, 0, 39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208, 0 ] sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.080 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:46:47 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; Q := PolynomialRing(A); B := ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:46:47 on modular [Seed = 2074591186] ------------------------------------- >> B := ext; ^ Runtime error: No constructor provided for this type of object 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ Runtime error in 'MinimalPolynomial': Bad argument types Argument types given: RngUPolElt[RngMPolRes] Total time: 0.290 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:46:34 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; Q := PolynomialRing(A); B := ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:46:33 on modular [Seed = 2124593865] ------------------------------------- >> B := ext; ^ Runtime error: No constructor provided for this type of object 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ Runtime error in 'MinimalPolynomial': Bad argument types Argument types given: RngUPolElt[RngMPolRes] Total time: 0.300 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:45:47 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:45:47 on modular [Seed = 98156896] ------------------------------------- >> B := ext; ^ User error: Identifier 'w' has not been declared or assigned 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:45:13 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:45:13 on modular [Seed = 148163599] ------------------------------------- >> B := ext; ^ User error: bad syntax 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:44:54 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := Ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:44:54 on modular [Seed = 248695318] ------------------------------------- >> B := Ext; ^ User error: bad syntax 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:44:44 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := Ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:44:44 on modular [Seed = 365547831] ------------------------------------- >> B := Ext; ^ User error: bad syntax 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:43:39 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 21:43:19 on modular [Seed = 415555041] ------------------------------------- Errors: /bin/sh: line 1: 32038 Alarm clock nice -n 19 /usr/local/bin/magma '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:42:55 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 21:42:35 on modular [Seed = 516086715] ------------------------------------- Errors: /bin/sh: line 1: 32032 Alarm clock nice -n 19 /usr/local/bin/magma '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:41:57 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 21:41:36 on modular [Seed = 637132923] ------------------------------------- Errors: /bin/sh: line 1: 32027 Alarm clock nice -n 19 /usr/local/bin/magma '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:40:43 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:40:43 on modular [Seed = 687140157] ------------------------------------- >> B := AffineAlgebra= 0 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:40:31 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:40:31 on modular [Seed = 804514318] ------------------------------------- >> B := AffineAlgebra> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:39:31 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra, w | w^5+w-x*t/y>; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:39:31 on modular [Seed = 854521071] ------------------------------------- >> B := AffineAlgebra, w | w^5+w-x*t/y>; ^ User error: bad syntax 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:39:13 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:39:13 on modular [Seed = 971374550] ------------------------------------- >> B := AffineAlgebra; ^ Runtime error in 'AssignNames': Argument 2 should have length at most 1 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.300 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:38:49 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:38:48 on modular [Seed = 3373600590] ------------------------------------- 1/t*y^4 + 1/t $.1^5 + $.1 + 4*x*y^4 + 4*x + 4/t*y + 4*t Total time: 0.300 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:37:38 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:37:37 on modular [Seed = 3574141851] ------------------------------------- 1/t*y^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.300 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:37:22 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:37:21 on modular [Seed = 3657833604] ------------------------------------- 1/t*y^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.300 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:36:38 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:36:38 on modular [Seed = 3741522390] ------------------------------------- 1/t^17*w^100 + (4*t^4 + 4)/t^17*w^80 + 4/t^13*w^76 + (t^8 + 2*t^4 + 1)/t^17*w^60 + (2*t^4 + 2)/t^13*w^56 + 1/t^9*w^52 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^17*w^40 + (2*t^8 + 4*t^4 + 2)/t^13*w^36 + (2*t^4 + 2)/t^9*w^32 + 4/t^5*w^28 + (t^16 + 4*t^12 + t^8 + 4*t^4 + 1)/t^17*w^20 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^13*w^16 + (t^8 + 2*t^4 + 1)/t^9*w^12 + (4*t^4 + 4)/t^5*w^8 + 1/t*w^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.250 seconds, Total memory usage: 3.63MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 21:36:30 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^6 mod n =", rho^6 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", Eltseq(t40); "trace4(g)' =", Eltseq(t41); "trace4(g)'' =", Eltseq(t42); "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", Eltseq(t60); "trace6(g)' =", Eltseq(t61); "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:36:29 on modular [Seed = 3845727223] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^6 mod n = 1 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073, 0, 0 ] trace4(g)' = [ 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603, 0, 0, 600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881, 0, 0 ] trace4(g)'' = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746, 0, 0 ] sum4 = (600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (351960067217720226471026218912124823215388842998*i + 636471885525581132299463326642541502123540418918)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = [ 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129, 0, 841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827, 0, 156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317, 0 ] trace6(g)' = [ 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129, 0, 96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515, 0, 39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208, 0 ] sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.070 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:36:22 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:36:22 on modular [Seed = 3946262002] ------------------------------------- 1/t^20*w^120 + 4/t^20*w^116 + 1/t^20*w^112 + 4/t^20*w^108 + 1/t^20*w^104 + 4/t^20*w^100 + 1/t^20*w^96 + 4/t^20*w^92 + 1/t^20*w^88 + 4/t^20*w^84 + 1/t^20*w^80 + 4/t^20*w^76 + 1/t^20*w^72 + 4/t^20*w^68 + 1/t^20*w^64 + 4/t^20*w^60 + 1/t^20*w^56 + 4/t^20*w^52 + 1/t^20*w^48 + 4/t^20*w^44 + 1/t^20*w^40 + 4/t^20*w^36 + 1/t^20*w^32 + 4/t^20*w^28 + 1/t^20*w^24 + (t^4 + 1)/t^4*w^20 + (4*t^4 + 4)/t^4*w^16 + (t^4 + 1)/t^4*w^12 + (4*t^4 + 4)/t^4*w^8 + (t^4 + 1)/t^4*w^4 + 1 z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.240 seconds, Total memory usage: 3.53MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:35:19 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:35:19 on modular [Seed = 4029429544] ------------------------------------- >> B := AffineAlgebra: Rhs argument 1 is invalid for this constructor x^4*y^4 + x^4 + y^4 + 1 z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:34:52 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:34:52 on modular [Seed = 4113117702] ------------------------------------- >> B := AffineAlgebra; ^ Runtime error in AffineAlgebra< ... >: Rhs argument 1 is invalid for this constructor x^4*y^4 + x^4 + y^4 + 1 z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.200 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 21:20:51 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^6 mod n =", rho^6 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", t40; "trace4(g)' =", t41; "trace4(g)'' =", t42; "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", t60; "trace6(g)' =", t61; "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:20:50 on modular [Seed = 2170372924] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^6 mod n = 1 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g)' = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603 trace4(g)'' = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 sum4 = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (1109541557279070038674422371008660670501870047821*i + 825029738971209132845985263278243991593718471901)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208)*z^4 + (96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515)*z^2 + 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129 sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.080 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 21:15:15 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^3 mod n =", rho^3 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", t40; "trace4(g)' =", t41; "trace4(g)'' =", t42; "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", t60; "trace6(g)' =", t61; "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:15:14 on modular [Seed = 2387758467] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^3 mod n = 1461501624496790265145447380994971188499300027612 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g)' = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603 trace4(g)'' = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 sum4 = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (1109541557279070038674422371008660670501870047821*i + 825029738971209132845985263278243991593718471901)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208)*z^4 + (96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515)*z^2 + 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129 sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.070 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:14:51 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:14:50 on modular [Seed = 2471448758] ------------------------------------- x^4*y^4 + x^4 + y^4 + 1 z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 21:14:43 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^3 mod n =", rho^3 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 mod p =", zeta^3 mod p; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", t40; "trace4(g)' =", t41; "trace4(g)'' =", t42; "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", t60; "trace6(g)' =", t61; "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:14:42 on modular [Seed = 2554616757] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^3 mod n = 1461501624496790265145447380994971188499300027612 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 >> "zeta^3 mod p =", zeta^3 mod p; ^ Runtime error in 'mod': Bad argument types Argument types given: FldFinElt, FldFinElt lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g)' = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603 trace4(g)'' = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 sum4 = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (1109541557279070038674422371008660670501870047821*i + 825029738971209132845985263278243991593718471901)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208)*z^4 + (96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515)*z^2 + 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129 sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.100 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 21:07:56 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", t40; "trace4(g)' =", t41; "trace4(g)'' =", t42; "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", t60; "trace6(g)' =", t61; "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:07:54 on modular [Seed = 2909901032] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g)' = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603 trace4(g)'' = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 sum4 = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (1109541557279070038674422371008660670501870047821*i + 825029738971209132845985263278243991593718471901)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208)*z^4 + (96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515)*z^2 + 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129 sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.060 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:07:15 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+y); Output: Magma V2.11-10 Tue Dec 13 2005 21:07:15 on modular [Seed = 2993590238] ------------------------------------- x^4*y^4 + x^4 + y^4 + 1 z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (4*t^9 + 4*t + 4)/t^4 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:06:43 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+t^(-3)*y); Output: Magma V2.11-10 Tue Dec 13 2005 21:06:43 on modular [Seed = 3077278395] ------------------------------------- x^4*y^4 + x^4 + y^4 + 1 z^25 + (t^60 + t^56 + 4*t^54 + t^52 + 4*t^50 + t^48 + t^42 + 3*t^40 + 3*t^38 + t^36 + t^28 + 2*t^26 + t^24 + t^14 + t^12 + 1)/t^60*z^5 + (t^56 + 4*t^54 + t^52 + 4*t^50 + t^48 + t^42 + 3*t^40 + 3*t^38 + t^36 + t^28 + 2*t^26 + t^24 + t^14 + t^12 + 1)/t^60*z + (4*t^66 + 4*t^56 + 4*t^52 + t^50 + 4*t^48 + t^46 + 4*t^44 + 4*t^38 + 2*t^36 + 2*t^34 + 4*t^32 + 4*t^24 + 3*t^22 + 4*t^20 + 4*t^10 + 4*t^8 + 4)/t^70 Total time: 0.220 seconds, Total memory usage: 3.43MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:05:48 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+t^(-3)*y); Output: Magma V2.11-10 Tue Dec 13 2005 21:05:48 on modular [Seed = 3160441471] ------------------------------------- (4*t^2 + 4*t)/(t^3 + 2*t^2 + 3*t + 1)*x*y^2 + 4*t^2/(t^3 + 2*t^2 + 3*t + 1)*x*y + (4*t^3 + 3*t^2 + 4*t)/(t^3 + 2*t^2 + 3*t + 1)*x z^6 + (3*t + 3)/t*z^4 + (4*t^5 + 3)/t^8*z^3 + (3*t^2 + t + 3)/t^2*z^2 + (4*t^6 + 4*t^5 + t + 1)/t^9*z + (t^16 + 2*t^15 + 3*t^14 + t^13 + 3*t^10 + 2*t^5 + 1)/t^16 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:04:26 2005 Input: F := FunctionField(IntegerRing()); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+t^(-3)*y); Output: Magma V2.11-10 Tue Dec 13 2005 21:04:26 on modular [Seed = 1100840916] ------------------------------------- (-t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x*y^2 - t^2/(t^3 + 2*t^2 + 3*t + 1)*x*y + (-t^3 - 2*t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x z^6 + (3*t + 3)/t*z^4 + (-6*t^5 - 2)/t^8*z^3 + (3*t^2 + 6*t + 3)/t^2*z^2 + (-6*t^6 - 6*t^5 + 6*t + 6)/t^9*z + (t^16 + 2*t^15 + 3*t^14 + t^13 + 3*t^10 - 3*t^5 + 1)/t^16 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:04:09 2005 Input: F := FunctionField(IntegerRing()); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+t^3*y); Output: Magma V2.11-10 Tue Dec 13 2005 21:04:09 on modular [Seed = 1184528608] ------------------------------------- (-t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x*y^2 - t^2/(t^3 + 2*t^2 + 3*t + 1)*x*y + (-t^3 - 2*t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x z^6 + (3*t + 3)/t*z^4 + (-2*t^10 - 6*t^3)*z^3 + (3*t^2 + 6*t + 3)/t^2*z^2 + (6*t^10 + 6*t^9 - 6*t^3 - 6*t^2)*z + (t^23 - 3*t^16 + 3*t^9 + t^3 + 2*t^2 + 3*t + 1)/t^3 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:03:46 2005 Input: F := FunctionField(IntegerRing()); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+y); Output: Magma V2.11-10 Tue Dec 13 2005 21:03:46 on modular [Seed = 1268217334] ------------------------------------- (-t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x*y^2 - t^2/(t^3 + 2*t^2 + 3*t + 1)*x*y + (-t^3 - 2*t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x z^6 + (3*t + 3)/t*z^4 + (-2*t - 6)*z^3 + (3*t^2 + 6*t + 3)/t^2*z^2 + (6*t^2 - 6)/t*z + (t^5 - 3*t^4 + 4*t^3 + 2*t^2 + 3*t + 1)/t^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:01:59 2005 Input: F := FunctionField(IntegerRing()); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x); Output: Magma V2.11-10 Tue Dec 13 2005 21:01:59 on modular [Seed = 1368223609] ------------------------------------- (-t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x*y^2 - t^2/(t^3 + 2*t^2 + 3*t + 1)*x*y + (-t^3 - 2*t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x z^6 + (3*t + 3)/t*z^4 + (3*t^2 + 6*t + 3)/t^2*z^2 + (t^3 + 2*t^2 + 3*t + 1)/t^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 20:57:39 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t4 := trace4(g); "trace4(g) =", t4; "trace4(g)' =", t4^p; "trace4(g)'' =", t4^(p^2); t6 := trace6(g); "trace6(g) =", t6; "trace6(g)' =", t6^p; Output: Magma V2.11-10 Tue Dec 13 2005 20:57:38 on modular [Seed = 1485072245] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g)' = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603 trace4(g)'' = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208)*z^4 + (96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515)*z^2 + 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129 Total time: 1.080 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:42:28 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := ext< R | y^5+y - x >; F; Q := PolynomialRing(P); Q; G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:42:27 on modular [Seed = 1673484511] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x Univariate Polynomial Ring in z over Univariate Polynomial Ring in y over Univariate rational function field over GF(5^2) Algebraic function field defined over F by $.1^5 + $.1 + 4/x*alpha Total time: 0.310 seconds, Total memory usage: 7.89MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 20:40:50 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; "trace4(g) =", trace4(g); Output: Magma V2.11-10 Tue Dec 13 2005 20:40:49 on modular [Seed = 1756648310] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 Total time: 0.900 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:40:47 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := ext< R | y^5+y - x >; F; Q := PolynomialRing(P); G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:40:47 on modular [Seed = 1823496012] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x Algebraic function field defined over F by $.1^5 + $.1 + 4/x*alpha Total time: 0.310 seconds, Total memory usage: 7.89MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:40:29 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := ext< R | y^5+y - x >; F; Q := PolynomialRing(P); G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:40:29 on modular [Seed = 1924024640] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x >> G := ext; ^ Runtime error in '-': Bad argument types Argument types given: RngUPolElt[RngUPol[FldFunRat]], FldFunElt >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:40:12 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := ext< R | y^5+y - x >; F; Q := PolynomialRing(P); G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:40:12 on modular [Seed = 2040876474] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x >> G := ext; ^ Runtime error in '-': Bad argument types Argument types given: RngUPolElt[RngUPol[FldFunRat]], FldFunElt >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 20:40:03 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; "g[1] =", g[1]; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) g + t + s; end function; "trace4(g) =", trace4(g); Output: Magma V2.11-10 Tue Dec 13 2005 20:40:02 on modular [Seed = 2124565171] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 >> "g[1] =", g[1]; ^ Runtime error in '[]': Bad argument types (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 >> "trace4(g) =", trace4(g); ^ Runtime error: No return statement executed in user-defined function Total time: 0.860 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:39:56 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := ext< R | y^5+y - x >; F; Q := PolynomialRing(P); G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:39:56 on modular [Seed = 64960464] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x >> G := ext; ^ Runtime error in '-': Bad argument types Argument types given: RngUPolElt[RngUPol[FldFunRat]], FldFunElt >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:38:46 2005 Input: R := FunctionField(GF(5)); P := PolynomialRing(R); F := ext< R | y^2 - 1/x >; F; Q := PolynomialRing(P); G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:38:46 on modular [Seed = 348659853] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5) by y^2 + 4/x Algebraic function field defined over F by $.1^3 + 4*alpha + 4*x Total time: 0.290 seconds, Total memory usage: 7.85MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:37:32 2005 Input: R := FunctionField(GF(5)); P := PolynomialRing(R); F := ext< R | y^2 - 1/x >; F; Output: Magma V2.11-10 Tue Dec 13 2005 20:37:32 on modular [Seed = 482355327] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5) by y^2 + 4/x Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:37:26 2005 Input: R := FunctionField(GF(5)); F := ext< R | y^2 - 1/x >; F; Output: Magma V2.11-10 Tue Dec 13 2005 20:37:26 on modular [Seed = 570234184] ------------------------------------- >> F := ext< R | y^2 - 1/x >; ^ User error: Identifier 'y' has not been declared or assigned >> F; ^ User error: Identifier 'F' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:37:09 2005 Input: R := FunctionField(GF(5)); P := PolynomialRing(R); F := ext< R | y^2 - 1/x >; F; Output: Magma V2.11-10 Tue Dec 13 2005 20:37:09 on modular [Seed = 653922908] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5) by y^2 + 4/x Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:35:29 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := FunctionField(y^5+y - x); F; Output: Magma V2.11-10 Tue Dec 13 2005 20:35:28 on modular [Seed = 737615604] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x Total time: 0.180 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:34:56 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := FunctionField(y^2 - 1/x); F; Output: Magma V2.11-10 Tue Dec 13 2005 20:34:56 on modular [Seed = 820774351] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^2 + 4/x Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:34:10 2005 Input: R := FunctionField(GF(5)); P := PolynomialRing(R); F := FunctionField(y^2 - 1/x); F; Output: Magma V2.11-10 Tue Dec 13 2005 20:34:09 on modular [Seed = 904462491] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5) by y^2 + 4/x Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:27:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; "g[1] =", g[1]; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ Output: Magma V2.11-10 Tue Dec 13 2005 20:27:46 on modular [Seed = 1021315601] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 >> "g[1] =", g[1]; ^ Runtime error in '[]': Bad argument types Total time: 0.610 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:26:47 2005 Input: FunctionField(FiniteField(11)) Output: Magma V2.11-10 Tue Dec 13 2005 20:26:46 on modular [Seed = 3323654440] ------------------------------------- Univariate rational function field over GF(11) Variables: $.1 Total time: 0.180 seconds, Total memory usage: 3.24MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:25:00 2005 Input: FunctionField(Integers()) Output: Magma V2.11-10 Tue Dec 13 2005 20:25:00 on modular [Seed = 3239966357] ------------------------------------- Univariate rational function field over Integer Ring Variables: $.1 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:20:26 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 20:20:06 on modular [Seed = 3373660599] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 -- 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- 7 -- 8 -- 9 -- 10 -- 11 -- 12 -- 13 -- 14 -- 15 -- 16 -- 17 -- 18 -- 19 -- 20 -- 21 -- 22 -- 23 -- 24 -- 25 -- 26 -- 27 -- 28 -- 29 -- 30 -- 31 -- 32 -- 33 -- 34 -- 35 -- 36 -- 37 -- 38 -- 39 -- 40 -- 41 -- 42 -- 43 -- 44 -- 45 -- 46 -- 47 -- 48 Errors: /bin/sh: line 1: 31729 Alarm clock nice -n 19 /usr/local/bin/magma '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:15:58 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 20:15:38 on modular [Seed = 3490505170] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 -- 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- 7 -- 8 -- 9 -- 10 -- 11 -- 12 -- 13 -- 14 -- 15 -- 16 -- 17 -- 18 -- 19 -- 20 -- 21 -- 22 -- 23 -- 24 -- 25 -- 26 -- 27 -- 28 -- 29 -- 30 -- 31 -- 32 -- 33 -- 34 -- 35 -- 36 -- 37 -- 38 -- 39 -- 40 -- 41 -- 42 -- 43 -- 44 -- 45 -- 46 -- 47 -- 48 Errors: /bin/sh: line 1: 31720 Alarm clock nice -n 19 /usr/local/bin/magma '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:14:45 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 20:14:25 on modular [Seed = 3674728722] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 Errors: /bin/sh: line 1: 31715 Alarm clock nice -n 19 /usr/local/bin/magma '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:13:26 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:13:25 on modular [Seed = 3862615147] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) >> if bit(r, i) eq 1 then ^ User error: Identifier 'r' has not been declared or assigned >> g := tate(G, Gt); ^ User error: Identifier 'tate' has not been declared or assigned g = function(U, V, Q) ... end function >> w := tate(u*G, v*Gt); ^ User error: Identifier 'tate' has not been declared or assigned Success! Total time: 0.260 seconds, Total memory usage: 3.34MB '70.172.' ************** MAGMA ***************** Host 70.172.216.115 (70.172.216.115) Time: Tue Dec 13 20:13:13 2005 Input: factor 74037563479561712828046796097429573142593188889231 28908493623263897276503402826627689199641962511784 39958943305021275853701189680982867331732731089309 00552505116877063299072396380786710086096962537934 650563796359 Output: Magma V2.11-10 Tue Dec 13 2005 20:13:13 on modular [Seed = 3778925424] ------------------------------------- >> factor 74037563479561712828046796097429573142593188889231 ^ User error: bad syntax >> 39958943305021275853701189680982867331732731089309 ^ User error: bad syntax >> 650563796359; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '70.172.' ************** MAGMA ***************** Host 70.172.216.115 (70.172.216.115) Time: Tue Dec 13 20:12:56 2005 Input: fac 74037563479561712828046796097429573142593188889231 28908493623263897276503402826627689199641962511784 39958943305021275853701189680982867331732731089309 00552505116877063299072396380786710086096962537934 650563796359 Output: Magma V2.11-10 Tue Dec 13 2005 20:12:56 on modular [Seed = 3946305867] ------------------------------------- >> fac 74037563479561712828046796097429573142593188889231 ^ User error: bad syntax >> 39958943305021275853701189680982867331732731089309 ^ User error: bad syntax >> 650563796359; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:12:28 2005 Input: K := GF(16) Output: Magma V2.11-10 Tue Dec 13 2005 20:12:28 on modular [Seed = 4096313008] ------------------------------------- Total time: 0.180 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:12:24 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(r) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:12:24 on modular [Seed = 4029465264] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) >> for i := length(r) - 2 to 0 by -1 do ^ User error: Identifier 'r' has not been declared or assigned >> g := tate(G, Gt); ^ User error: Identifier 'tate' has not been declared or assigned g = function(U, V, Q) ... end function >> w := tate(u*G, v*Gt); ^ User error: Identifier 'tate' has not been declared or assigned Success! Total time: 0.230 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:10:50 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; */ for j in [1..100] do u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:10:50 on modular [Seed = 4213688793] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (76*i + 99)*z^5 + (3*i + 57)*z^4 + (24*i + 88)*z^3 + (78*i + 63)*z^2 + (58*i + 84)*z + 23*i + 29 Success! Total time: 0.220 seconds, Total memory usage: 3.53MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:09:33 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; */ for j in [1..100] do u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:09:32 on modular [Seed = 2254091634] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (76*i + 99)*z^5 + (3*i + 57)*z^4 + (24*i + 88)*z^3 + (78*i + 63)*z^2 + (58*i + 84)*z + 23*i + 29 >> if w ne s or w ne h then ^ User error: Identifier 's' has not been declared or assigned Success! Total time: 0.190 seconds, Total memory usage: 3.53MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:01:22 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:01:22 on modular [Seed = 3077282413] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (76*i + 99)*z^5 + (3*i + 57)*z^4 + (24*i + 88)*z^3 + (78*i + 63)*z^2 + (58*i + 84)*z + 23*i + 29 Success! Total time: 0.240 seconds, Total memory usage: 3.53MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:00:15 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then "***", i; f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:00:15 on modular [Seed = 2993594133] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) *** 5 *** 0 g = (76*i + 99)*z^5 + (3*i + 57)*z^4 + (24*i + 88)*z^3 + (78*i + 63)*z^2 + (58*i + 84)*z + 23*i + 29 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 Success! Total time: 0.250 seconds, Total memory usage: 3.53MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 19:53:53 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then "***", i; f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 19:53:53 on modular [Seed = 2090881637] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) *** 6 *** 5 *** 0 g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 *** 6 *** 5 *** 0 *** 6 *** 5 *** 0 *** 6 *** 5 *** 0 *** 6 *** 5 *** 0 Failure! u = 2 e(u*P, Q) = (38*i + 16)*z^5 + (101*i + 58)*z^4 + (43*i + 45)*z^3 + (102*i + 83)*z^2 + (61*i + 73)*z + 20*i + 35 e(P, u*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 e(P, Q)^u = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.190 seconds, Total memory usage: 3.53MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 19:16:04 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; L:=MinimumWords(C2); C3:=LinearCode(L); C3; Output: Magma V2.11-10 Tue Dec 13 2005 19:16:04 on modular [Seed = 587072862] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] >> C3:=LinearCode(L); ^ Runtime error in 'LinearCode': Bad argument types Argument types given: SetEnum[ModTupFldElt] >> C3;; ^ User error: Identifier 'C3' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 19:11:47 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; MinimumWords(C2); Output: Magma V2.11-10 Tue Dec 13 2005 19:11:47 on modular [Seed = 753929303] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] { (0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0), (0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0), (1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0), (0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0), (0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1), (0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0), (1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0) } Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:52:22 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; Output: Magma V2.11-10 Tue Dec 13 2005 18:52:22 on modular [Seed = 3256803149] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:46:56 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); Output: Magma V2.11-10 Tue Dec 13 2005 18:46:55 on modular [Seed = 3423654554] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:42:13 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; Output: Magma V2.11-10 Tue Dec 13 2005 18:42:12 on modular [Seed = 3591031227] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:38:53 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M:=EchelonForm(M); M2:=Submatrix(M,22,22,14,35); M2; Output: Magma V2.11-10 Tue Dec 13 2005 18:38:53 on modular [Seed = 3879468883] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:37:30 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M:=EchelonForm(M); M2:=Submatrix(M,22,22,35,14); M2; Output: Magma V2.11-10 Tue Dec 13 2005 18:37:30 on modular [Seed = 4013160428] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1] [0 1 0 0 0 0 0 0 0 0 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 1 0 0 0 1 1 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:37:09 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M:=EchelonForm(M); M2:=Submathrix(M,22,22,35,14); M2; Output: Magma V2.11-10 Tue Dec 13 2005 18:37:08 on modular [Seed = 3895783669] ------------------------------------- >> M2:=Submathrix(M,22,22,35,14); ^ User error: Identifier 'Submathrix' has not been declared or assigned >> M2;; ^ User error: Identifier 'M2' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:32:27 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M:=EchelonForm(M); M; Output: Magma V2.11-10 Tue Dec 13 2005 18:32:26 on modular [Seed = 4063161281] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 0 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 0 0] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 1 0 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:30:17 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M:=EchelonForm(M); Output: Magma V2.11-10 Tue Dec 13 2005 18:30:17 on modular [Seed = 4280542543] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:27:47 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1]>; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); EchelonForm(M); Output: Magma V2.11-10 Tue Dec 13 2005 18:27:47 on modular [Seed = 2170395211] ------------------------------------- [1 0 0 0 0 1 1 1] [0 1 0 0 0 1 0 1] [0 0 1 0 0 0 1 1] [0 0 0 1 0 0 0 0] [0 0 0 0 1 1 1 0] [0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:15:06 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1]>; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); C := LinearCode(M); GeneratorMatrix(C); Output: Magma V2.11-10 Tue Dec 13 2005 18:15:06 on modular [Seed = 2504622934] ------------------------------------- [1 0 0 0 0 1 1 1] [0 1 0 0 0 1 0 1] [0 0 1 0 0 0 1 1] [0 0 0 1 0 0 0 0] [0 0 0 0 1 1 1 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:11:56 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M; Output: Magma V2.11-10 Tue Dec 13 2005 18:11:56 on modular [Seed = 2672003482] ------------------------------------- [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:11:38 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); Output: Magma V2.11-10 Tue Dec 13 2005 18:11:37 on modular [Seed = 2554629768] ------------------------------------- Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:06:45 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; GeneratorMatrix(C); ParityCheckMatrix(C); Output: Magma V2.11-10 Tue Dec 13 2005 18:06:44 on modular [Seed = 3026756489] ------------------------------------- [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:06:09 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; GeneratorMatrix(C); Output: Magma V2.11-10 Tue Dec 13 2005 18:06:09 on modular [Seed = 2976751977] ------------------------------------- [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:05:35 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; C; Output: Magma V2.11-10 Tue Dec 13 2005 18:05:35 on modular [Seed = 3127290428] ------------------------------------- [8, 4, 4] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 17:53:01 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; Output: Magma V2.11-10 Tue Dec 13 2005 17:53:01 on modular [Seed = 64950519] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] Total time: 0.180 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:55:57 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:55:57 on modular [Seed = 1807159056] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 2 e(u*P, Q) = (38*i + 16)*z^5 + (101*i + 58)*z^4 + (43*i + 45)*z^3 + (102*i + 83)*z^2 + (61*i + 73)*z + 20*i + 35 e(P, u*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 e(P, Q)^u = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:55:01 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); if w ne s then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:55:01 on modular [Seed = 1857161805] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 2 e(u*P, Q) = (38*i + 16)*z^5 + (101*i + 58)*z^4 + (43*i + 45)*z^3 + (102*i + 83)*z^2 + (61*i + 73)*z + 20*i + 35 e(P, u*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 Total time: 0.190 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:52:25 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ v := 1; for u in [1..n-1] do w := tate(u*G, v*Gt); s := g^((u*v) mod n); if w ne s then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P,v*Q) = " * Sprint(w); print "e(P,Q)^(uv) = " * Sprint(s); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:52:24 on modular [Seed = 2024021302] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 2 v = 1 e(u*P,v*Q) = (38*i + 16)*z^5 + (101*i + 58)*z^4 + (43*i + 45)*z^3 + (102*i + 83)*z^2 + (61*i + 73)*z + 20*i + 35 e(P,Q)^(uv) = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:51:09 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for v, u in [1..n-1] do w := tate(u*G, v*Gt); s := g^((u*v) mod n); if w ne s then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P,v*Q) = " * Sprint(w); print "e(P,Q)^(uv) = " * Sprint(s); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:51:08 on modular [Seed = 1907168363] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 2 v = 1 e(u*P,v*Q) = (38*i + 16)*z^5 + (101*i + 58)*z^4 + (43*i + 45)*z^3 + (102*i + 83)*z^2 + (61*i + 73)*z + 20*i + 35 e(P,Q)^(uv) = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:49:25 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u, v in [1..n-1] do w := tate(u*G, v*Gt); s := g^((u*v) mod n); if w ne s then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P,v*Q) = " * Sprint(w); print "e(P,Q)^(uv) = " * Sprint(s); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:49:25 on modular [Seed = 1957171847] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 1 v = 2 e(u*P,v*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 e(P,Q)^(uv) = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:48:24 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do for v in [1..n-1] do w := tate(u*G, v*Gt); s := g^((u*v) mod n); if w ne s then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P,v*Q) = " * Sprint(w); print "e(P,Q)^(uv) = " * Sprint(s); quit; end if; end for; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:48:23 on modular [Seed = 198667706] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 1 v = 2 e(u*P,v*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 e(P,Q)^(uv) = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:46:16 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do for v in [1..n-1] do w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:46:16 on modular [Seed = 31287103] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure: u = 1 , v = 2 , e(u*P,v*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 , e(P,Q)^(uv) = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.190 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:45:31 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [0..n-1] do for v in [0..n-1] do w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:45:30 on modular [Seed = 81290216] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 tate( P: (0 : 1 : 0), Qt: (0 : 1 : 0) ) >> Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; ^ Runtime error in '!': Illegal coercion Success! Total time: 0.190 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:44:16 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:44:16 on modular [Seed = 466058429] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure: u = 88 , v = 67 , e(u*P,v*Q) = (100*i + 78)*z^5 + (36*i + 83)*z^4 + (30*i + 95)*z^3 + (93*i + 66)*z^2 + (21*i + 39)*z + 74*i + 33 , e(P,Q)^(uv) = (28*i + 78)*z^5 + (62*i + 21)*z^4 + (82*i + 32)*z^3 + (36*i + 57)*z^2 + (69*i + 26)*z + 95*i + 78 Total time: 0.190 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:39:51 2005 Input: p := 103;//1461501624496790265145448589920785493717258890819; n := 97;//1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := Fp!2; mu := 2 + i;//1 + i; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; b := Fp!12;//Fp!3; y0 := Fp!61;//Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; "G =", G; Et := EllipticCurve([0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:39:51 on modular [Seed = 298676640] ------------------------------------- G = (1 : 61 : 1) Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure: u = 2 , v = 18 , e(u*P,v*Q) = (50*i + 27)*z^5 + (94*i + 26)*z^4 + (96*i + 54)*z^3 + (19*i + 50)*z^2 + (61*i + 86)*z + 74*i + 83 , e(P,Q)^(uv) = (85*i + 97)*z^5 + (46*i + 29)*z^4 + (25*i + 30)*z^3 + (5*i + 40)*z^2 + (59*i + 45)*z + 71*i + 56 Total time: 0.190 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:39:14 2005 Input: p := 103;//1461501624496790265145448589920785493717258890819; n := 97;//1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := Fp!2; mu := 2 + i;//1 + i; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; b := Fp!12;//Fp!3; y0 := Fp!61;//Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:39:14 on modular [Seed = 348681280] ------------------------------------- xt = 4 g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure: u = 77 , v = 45 , e(u*P,v*Q) = (99*i + 33)*z^5 + (i + 79)*z^4 + (25*i + 30)*z^3 + (24*i + 89)*z^2 + (8*i + 48)*z + 71*i + 56 , e(P,Q)^(uv) = (45*i + 90)*z^5 + (29*i + 53)*z^4 + (91*i + 98)*z^3 + (68*i + 94)*z^2 + (10*i + 44)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:37:09 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := Fp!2; mu := 1 + i; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:37:08 on modular [Seed = 1005017833] ------------------------------------- g = (788575332791159297943382330888091507234373160984*i + 1058591418264462827430151667682426387205715653646)*z^5 + (1322305885575807535232020102800928943661744848278*i + 1409518745655936467893014453594650508665852052962)*z^4 + (92691491674328017873297655882559873492848083281*i + 833387513557317502960994534404369304220347785370)*z^3 + (494268977073817690135764886501032821621331118299*i + 1229571523737605236254885852230274558531837523170)*z^2 + (808544033252198825980179735588413753558824714098*i + 1078734709554906413376551726137377859976359534003)*z + 901493422496026382770929617713581877034333521597*i + 179130707045747991601837568997307929180024496359 Failure: u = 946857947927091542053549758428905646280859693128 , v = 466000635125302880497731311434509156520464968357 , e(u*P,v*Q) = (1200099635135265526944287342088491944694417283646*i + 1014718590403343121885833198299782572285854256079)*z^5 + (471917906168926683696779282454416809308080942623*i + 812644610572628849110985110390868588547305952867)*z^4 + (1324320698252798975422880297923770527123745467545*i + 1369801539108789273837312698785305468703191114557)*z^3 + (912504363855611966661376277297362740269326191111*i + 1380551513148965958602464026524016649810176191967)*z^2 + (1105893840184059270391354805924659396077920064041*i + 1288971271799831078521824062316836734081548434254)*z + 597889428879138809242066013037790879345607008490*i + 1169462561227871511533990611160523461698537007163 , e(P,Q)^(uv) = (1279762112918260549426510071012241774233201710837*i + 1038569981133317417069888654877080847608690012705)*z^5 + (1209071413845908584505709653556653498685569554252*i + 367978315180861868580674923069718724352829282487)*z^4 + (1092679439520606553629045996698544887309903354323*i + 1432439429241915293719430050078044125231284711989)*z^3 + (720945178705548260866392701230718571277060392582*i + 965355359457859959621310893985759282425870986789)*z^2 + (237204670011952913218038313811081503208054984454*i + 1073500538414130670572005779575000303603195535920)*z + 37244205619087872590547375873385141388703250234*i + 1082282633627808574389817278679504033474868288016 Total time: 1.050 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:36:11 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; K := GF(p); b := K!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([K|0,b]); G := E![1,y]; until IsZero(n*G); "b =", b; "G =", G; lambda := K!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; K2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; Output: Magma V2.11-10 Tue Dec 13 2005 14:36:10 on modular [Seed = 1055023010] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Total time: 0.190 seconds, Total memory usage: 3.43MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:32:16 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; K := GF(p); b := K!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([K|0,b]); G := E![1,y]; until IsZero(n*G); "b =", b; "G =", G; Output: Magma V2.11-10 Tue Dec 13 2005 14:32:15 on modular [Seed = 887641789] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) Total time: 0.180 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:32:00 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; K := GF(p); b := K!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([k|0,b]); G := E![1,y]; until IsZero(n*G); "b =", b; "G =", G; Output: Magma V2.11-10 Tue Dec 13 2005 14:32:00 on modular [Seed = 3390482509] ------------------------------------- u = 1 p = 103 n = 97 >> E := EllipticCurve([k|0,b]); ^ User error: Identifier 'k' has not been declared or assigned b = 0 >> "G =", G; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:30:09 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Output: Magma V2.11-10 Tue Dec 13 2005 14:30:08 on modular [Seed = 3474173368] ------------------------------------- u = 1 p = 103 n = 97 Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:28:50 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Output: Magma V2.11-10 Tue Dec 13 2005 14:28:49 on modular [Seed = 3289949954] ------------------------------------- u = -1 p = 19 n = 13 Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:25:25 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; u := 0; while true do u +:= 1; t := 6*u^2 + 1; p := Evaluate(P, -u); if not IsProbablePrime(p) then continue; end if; n := p + 1 - t; if IsProbablePrime(n) then break; end if; p := Evaluate(P, u); if not IsProbablePrime(p) then continue; end if; n := p + 1 - t; if IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Output: Magma V2.11-10 Tue Dec 13 2005 14:25:24 on modular [Seed = 3657873865] ------------------------------------- u = 1 p = 19 n = 13 Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:17:13 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:17:12 on modular [Seed = 4113148154] ------------------------------------- g = (672926291705630967202066259032693986482885729835*i + 402910206232327437715296922238359106511543237173)*z^5 + (1322305885575807535232020102800928943661744848278*i + 1409518745655936467893014453594650508665852052962)*z^4 + (1368810132822462247272150934038225620224410807538*i + 628114110939472762184454055516416189496911105449)*z^3 + (494268977073817690135764886501032821621331118299*i + 1229571523737605236254885852230274558531837523170)*z^2 + (652957591244591439165268854332371740158434176721*i + 382766914941883851768896863783407633740899356816)*z + 901493422496026382770929617713581877034333521597*i + 179130707045747991601837568997307929180024496359 Failure: u = 787544600137698273840913483523578412686940791799 , v = 633390288056570917236749546569673252543054620654 , e(u*P,v*Q) = (1230799145370778246341173515929335566649536643006*i + 480750625874954609025504843133359714744867086605)*z^5 + (1042350571388618446487630159957378489367482972903*i + 603997525618261102690174964818300174247773796868)*z^4 + (372555237390881172830113633133348948005361857273*i + 795790796595629198651221834435852032328843762695)*z^3 + (294507650957009594258002783265863173832662354454*i + 608683974417992602158011087154420253986790025605)*z^2 + (7609910714021721107059555931969091666598777225*i + 945198760430716673304112293252204618143489684435)*z + 147738933904891503161234155216849327398150311466*i + 11975084496371106465181782342706929746861740417 , e(P,Q)^(uv) = (436069547787440145692125043594555141196201225212*i + 104917863794017341921655052505855786976213992405)*z^5 + (1273611771612726823696618599863708218222654331128*i + 58016250495606580058214502460360819634176535723)*z^4 + (650807760957229285905208902992656562655973656443*i + 522387035783265890583694065207579752375558876825)*z^3 + (935118843989569600386157522041707999880163319815*i + 769360049790716169376779087226867932064575695394)*z^2 + (1137422255938782465848831089043436888213998499734*i + 255708389714534001533432868391992802173796862328)*z + 61611046909036180490590110705335451925853639750*i + 146857281263717838984053265699458256927236573517 Total time: 1.100 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:15:53 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; x_U := U[0]; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:15:53 on modular [Seed = 2370949971] ------------------------------------- tate( P: (1 : 2 : 1), Qt: (8 : 645238442624673913635245604741558906439788126004*i + 11... ) miller( r: 1461501624496790265145447380994971188499300027613, P: (1 : 2 : 1), Q: (8*z^2 : (645238442624673913635245604741558906439788126004*i... ) g( U: (1 : 2 : 1), V: (1 : 2 : 1), Q: (8*z^2 : (645238442624673913635245604741558906439788126004*i... ) >> x_U := U[0]; ^ Runtime error in '[]': Argument 2 (0) should be in the range [1 .. 3] g = function(U, V, Q) ... end function tate( P: (1168716806992048472600575426428280493482442262392 : 9145640..., Qt: (1398335284650143722212829649205089955966390414241*i + 51796... ) miller( r: 1461501624496790265145447380994971188499300027613, P: (1168716806992048472600575426428280493482442262392 : 9145640..., Q: ((1398335284650143722212829649205089955966390414241*i + 5179... ) g( U: (1168716806992048472600575426428280493482442262392 : 9145640..., V: (1168716806992048472600575426428280493482442262392 : 9145640..., Q: ((1398335284650143722212829649205089955966390414241*i + 5179... ) >> x_U := U[0]; ^ Runtime error in '[]': Argument 2 (0) should be in the range [1 .. 3] Success! Total time: 0.230 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:14:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:14:46 on modular [Seed = 2187254803] ------------------------------------- g = (1294412285019402306363363178405640025835588490573*i + 110083851862711115164294604712404064123786212316)*z^5 + (1461000921493363626807176655087143303615070519998*i + 179329319464877725831768652574235563708218294754)*z^4 + (1380159870452351478229999574688122314163279894835*i + 608853380181358438321396844362562256927503461575)*z^3 + (1096834514295262897057490471840451625632598605569*i + 556024354574832251790797618637935038587303588339)*z^2 + (1312769194087813280589806690561007885216498370207*i + 585387117445278704678873601298024025030517986150)*z + 737166616423533515609953322860296363171325419696*i + 327478258348140990171576024019950781712369617012 Failure: u = 1125474438691666154889619531753276137514054616092 , v = 74355561060828237152603481423932850628604339121 , e(u*P,v*Q) = (429751061675843899340369203815082176545408527718*i + 909427573279119657754805669761900114071342873652)*z^5 + (1247734288733857748021692265553075464695937579208*i + 743816016218828928608456547179207548076982979307)*z^4 + (487557256047992095318968005263397437474936254189*i + 37857860423732375651447446890944269006478056096)*z^3 + (1163989855974890736075862642655486087920170035473*i + 203749602481885254344251451474004759409452749236)*z^2 + (1030453876983867035177297133699795873715007249920*i + 660735836157490264065099937197471719042581607777)*z + 81803167227496355034247834271195469737156426272*i + 956207628639205541360074132469925387897212870999 , e(P,Q)^(uv) = (456387365657668154214383519085672897734254356213*i + 901464030134534466212711635931829911205601723280)*z^5 + (1457502911931788790996440618804285616966470935812*i + 167907066249828659721266469744918999176634507737)*z^4 + (1117081125135662063752403398732529743718000625159*i + 1166482808557132720766303763649223676515724544694)*z^3 + (45409092783112768705197353118716020133117497726*i + 1140195580893668606687433084773043246385384621649)*z^2 + (797400106022357958571873126411052707015877860255*i + 1144849651621705114749107513915197993111731626826)*z + 398139448757274963177977906845541386205679308934*i + 1011464188230029570370071800294314999300195638774 Total time: 1.080 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:13:45 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:13:44 on modular [Seed = 2270943704] ------------------------------------- g = (1414620835193373657646613880233559577407734609545*i + 352489175314961464607671418528001817189092256541)*z^5 + (904802261065103105936668720505067133656667597074*i + 1343248922269291340494538393490943353845819222256)*z^4 + (1025700017505207783921241027028089077317301724291*i + 1267567142236668396047190065512596226428512971416)*z^3 + (283587609846347412879109527640610607003328134432*i + 250487955904041646310509260856358041239663937723)*z^2 + (110054808378578108265445578093874268052578876998*i + 202041545854984388890408940644851972139794936772)*z + 257093848091331658607960702914357017287219708568*i + 652950073647454119068347296682657355787566777234 Failure: u = 658677412291089462095414946131774452312281593714 , v = 1216437558465192851244742711765770083951252389636 , e(u*P,v*Q) = (1399072020171236105190679320673870340346132836857*i + 557339706670292369068837859617582039559121726559)*z^5 + (87632175260928283896022294886624356298791691273*i + 353631789820014829475382783849498669504077254473)*z^4 + (1167181401062038608632643315708112439190409866068*i + 1201319255475829211069929657476285565774671872992)*z^3 + (1314520118761252810746572930401672772734452109507*i + 692190026558869491946574975355715885291753444864)*z^2 + (30302970564993022126258376903270909561081118427*i + 981711504087328555212568427163427828202637031781)*z + 650156321266501086248612693725874390933511156097*i + 86504195952951481401572117229146789328835422166 , e(P,Q)^(uv) = (213093635849167276598922493416793532239780363365*i + 176954146803611932474107371546291585496100314567)*z^5 + (423996100507151792116824541957513700044018239694*i + 83422655284377911844489375762414059048084741713)*z^4 + (1422886585162015559920092658102846713850101330779*i + 458574078442061740914292994408151093044898187067)*z^3 + (273126339780048501849982943960562129337069997055*i + 1303685246506966196047520334746858504911291400698)*z^2 + (1304770219556858465763105537968022378874250386356*i + 416455068376200642044392053732047037130013150572)*z + 398741370625816481193912713678168496900587107922*i + 581927151108935829460509086487928349289055788193 Total time: 1.060 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 13:05:57 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); Submatrix(S, 1, 22, 21, 35); Output: Magma V2.11-10 Tue Dec 13 2005 13:05:57 on modular [Seed = 804479389] ------------------------------------- >> Submatrix(S, 1, 22, 21, 35); ^ Runtime error in 'Submatrix': Bad argument types Argument types given: CodeLinFld, RngIntElt, RngIntElt, RngIntElt, RngIntElt Total time: 0.200 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 13:03:43 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); ExtractBlock(S, 1, 22, 21, 35); Output: Magma V2.11-10 Tue Dec 13 2005 13:03:42 on modular [Seed = 653945327] ------------------------------------- >> ExtractBlock(S, 1, 22, 21, 35); ^ Runtime error in 'ExtractBlock': Bad argument types Argument types given: CodeLinFld, RngIntElt, RngIntElt, RngIntElt, RngIntElt Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 13:01:22 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; Output: Magma V2.11-10 Tue Dec 13 2005 13:01:22 on modular [Seed = 1021342585] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 13:00:59 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); Output: Magma V2.11-10 Tue Dec 13 2005 13:00:59 on modular [Seed = 837647965] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 12:41:53 2005 Input: M:=[1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1]; Transpose(M); Output: Magma V2.11-10 Tue Dec 13 2005 12:41:53 on modular [Seed = 3624180745] ------------------------------------- >> M:=[1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] ^ User error: bad syntax >> [1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] ^ User error: bad syntax >> [1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] ^ User error: bad syntax >> [1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] ^ User error: bad syntax >> [1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] ^ User error: bad syntax >> [1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] ^ User error: bad syntax >> [1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] ^ User error: bad syntax >> [1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] ^ User error: bad syntax >> [1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] ^ User error: bad syntax >> [1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] ^ User error: bad syntax >> [1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] ^ User error: bad syntax >> [1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] ^ User error: bad syntax >> [1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] ^ User error: bad syntax >> [0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] ^ User error: bad syntax >> [0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] ^ User error: bad syntax >> [0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] ^ User error: bad syntax >> [0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] ^ User error: bad syntax >> [0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] ^ User error: bad syntax >> [0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] ^ User error: bad syntax >> [0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] ^ User error: bad syntax >> [0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1]; ^ User error: bad syntax >> Transpose(M);; ^ User error: Identifier 'M' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 13 12:10:52 2005 Input: count:=0; m:=5; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to q-1 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DP:=Points(Jac)[ii]; if HasOrder(DP,8) then for jj:=1 to Floor(J) do DQ:=Points(Jac)[jj]; if HasOrder(DQ,8) then WeilPairing(DP,DQ,8); end if; end for; //print DD, "has order 8, a=", a," b= ", b; end if; end for; count; end if; count:=0; end if; end for; end for; print "done"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 12:10:32 on modular [Seed = 2893082388] ------------------------------------- 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Errors: /bin/sh: line 1: 29596 Alarm clock nice -n 19 /usr/local/bin/magma '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 13 12:08:28 2005 Input: count:=0; m:=5; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to q-1 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DP:=Points(Jac)[ii]; if HasOrder(DP,8) then for jj:=1 to Floor(J) do DQ:=Points(Jac)[jj]; if HasOrder(DQ,8) then WeilPairing(DP,DQ,3); end if; end for; //print DD, "has order 8, a=", a," b= ", b; end if; end for; count; end if; count:=0; end if; end for; end for; print "done"; Output: Magma V2.11-10 Tue Dec 13 2005 12:08:27 on modular [Seed = 3127310721] ------------------------------------- >> WeilPairing(DP,DQ,3); ^ Runtime error in 'WeilPairing': divisors must be m-torsion elements done Total time: 0.780 seconds, Total memory usage: 3.72MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 13 12:08:14 2005 Input: m:=5; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to q-1 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DP:=Points(Jac)[ii]; if HasOrder(DP,8) then for jj:=1 to Floor(J) do DQ:=Points(Jac)[jj]; if HasOrder(DQ,8) then WeilPairing(DP,DQ,3); end if; end for; //print DD, "has order 8, a=", a," b= ", b; end if; end for; count; end if; count:=0; end if; end for; end for; print "done"; Output: Magma V2.11-10 Tue Dec 13 2005 12:08:14 on modular [Seed = 3043494267] ------------------------------------- >> count; ^ User error: Identifier 'count' has not been declared or assigned done Total time: 0.190 seconds, Total memory usage: 3.34MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 13 12:07:59 2005 Input: m:=5; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to q-1 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DP:=Points(Jac)[ii]; if HasOrder(DP,8) then for jj:=1 to Floor(J) do DQ:=Points(Jac)[jj]; if HasOrder(DQ,8) then WeilPairing(DP,DQ,3); end if end for //print DD, "has order 8, a=", a," b= ", b; end if; end for; count; end if; count:=0; end if; end for; end for; print "done"; Output: Magma V2.11-10 Tue Dec 13 2005 12:07:58 on modular [Seed = 1318318736] ------------------------------------- >> end for ^ User error: bad syntax >> end if; ^ User error: bad syntax >> end for; ^ User error: bad syntax >> count; ^ User error: Identifier 'count' has not been declared or assigned >> end if; ^ User error: bad syntax >> end if; ^ User error: bad syntax >> end for; ^ User error: bad syntax >> end for; ^ User error: bad syntax done Total time: 0.180 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 11:58:23 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; Output: Magma V2.11-10 Tue Dec 13 2005 11:58:22 on modular [Seed = 449291396] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] Total time: 0.190 seconds, Total memory usage: 3.34MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:39:59 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; I:=ideal; Output: Magma V2.11-10 Tue Dec 13 2005 11:39:59 on modular [Seed = 804427072] ------------------------------------- >> I:=ideal;; ^ Runtime error: No constructor provided for this type of object Total time: 0.190 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:38:59 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; I:=Ideal; Output: Magma V2.11-10 Tue Dec 13 2005 11:38:59 on modular [Seed = 1055097901] ------------------------------------- >> I:=Ideal;; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:37:27 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; Decomposition(K); Output: Magma V2.11-10 Tue Dec 13 2005 11:37:27 on modular [Seed = 921406913] ------------------------------------- >> Decomposition(K);; ^ Runtime error in 'Decomposition': Bad argument types Argument types given: FldFin Total time: 0.180 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:36:34 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; Decomposition(K,f); Output: Magma V2.11-10 Tue Dec 13 2005 11:36:33 on modular [Seed = 837585552] ------------------------------------- >> Decomposition(K,f);; ^ Runtime error in 'Decomposition': Bad argument types Argument types given: FldFin, RngUPolElt[FldFin] Total time: 0.180 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:34:06 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; Decomposition(K,f); Output: Magma V2.11-10 Tue Dec 13 2005 11:34:05 on modular [Seed = 3440408833] ------------------------------------- >> Decomposition(K,f);; ^ Runtime error in 'Decomposition': Bad argument types Argument types given: FldFin, RngUPolElt[FldFin] Total time: 0.190 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:32:13 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; Decomposition(f); Output: Magma V2.11-10 Tue Dec 13 2005 11:32:13 on modular [Seed = 3340399483] ------------------------------------- >> Decomposition(f);; ^ Runtime error in 'Decomposition': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.190 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:31:11 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; Decomposition(f); Output: Magma V2.11-10 Tue Dec 13 2005 11:31:11 on modular [Seed = 3223155902] ------------------------------------- >> K:=ext; ^ User error: Identifier 'definingPolynomial' has not been declared or assigned >> Decomposition(f);; ^ Runtime error in 'Decomposition': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.180 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:29:51 2005 Input: R:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K:=ext; Decomposition(f); Output: Magma V2.11-10 Tue Dec 13 2005 11:29:51 on modular [Seed = 3540936221] ------------------------------------- >> Decomposition(f);; ^ User error: Identifier 'f' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 11:26:17 2005 Input: > K := FiniteField(2); > E := LinearCode; > WeightDistribution(E); Output: Magma V2.11-10 Tue Dec 13 2005 11:26:17 on modular [Seed = 3979368693] ------------------------------------- [ <0, 1>, <2, 105>, <4, 1365>, <6, 5005>, <8, 6435>, <10, 3003>, <12, 455>, <14, 15> ] Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:54:53 2005 Input: > K := FiniteField(2); > C := LinearCode; > D:=Dual(C); > (D meet C) eq C; Output: Magma V2.11-10 Tue Dec 13 2005 10:54:53 on modular [Seed = 2354127216] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:52:23 2005 Input: > K := FiniteField(2); > C := LinearCode; > D:=Dual(C); > (d meet C) eq C; Output: Magma V2.11-10 Tue Dec 13 2005 10:52:22 on modular [Seed = 2220435843] ------------------------------------- >> (d meet C) eq C; ^ User error: Identifier 'd' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:40:50 2005 Input: > K := FiniteField(2); > K56 := VectorSpace(K, 56); > C := LinearCode(sub); > D:=Hull(C); > WeightDistribution(D); Output: Magma V2.11-10 Tue Dec 13 2005 10:40:50 on modular [Seed = 2671916844] ------------------------------------- >> D:=Hull(C); ^ User error: Identifier 'Hull' has not been declared or assigned >> WeightDistribution(D); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:35:53 2005 Input: > K := FiniteField(2); > K56 := VectorSpace(K, 56); > C := LinearCode(sub; > D:=Hull(C); > WeightDistribution(D); Output: Magma V2.11-10 Tue Dec 13 2005 10:35:53 on modular [Seed = 2876637891] ------------------------------------- >> 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; ^ User error: bad syntax >> D:=Hull(C); ^ User error: Identifier 'C' has not been declared or assigned >> WeightDistribution(D); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:25:19 2005 Input: K := FiniteField(2); > C := LinearCode; > D:=Hull(C); > WeightDistribution(D); Output: Magma V2.11-10 Tue Dec 13 2005 10:25:19 on modular [Seed = 2726233676] ------------------------------------- >> D:=Hull(C); ^ User error: Identifier 'Hull' has not been declared or assigned >> WeightDistribution(D); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:23:39 2005 Input: K := FiniteField(2); > C := LinearCode; D:=Hull(C); WeightDistribution(D); Output: Magma V2.11-10 Tue Dec 13 2005 10:23:39 on modular [Seed = 3210877941] ------------------------------------- >> D:=Hull(C); ^ User error: Identifier 'Hull' has not been declared or assigned >> WeightDistribution(D); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.210 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:20:00 2005 Input: K := FiniteField(2); > C := LinearCode; C; WeightDistribution(C); Output: Magma V2.11-10 Tue Dec 13 2005 10:19:57 on modular [Seed = 820876674] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] Total time: 0.240 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:16:57 2005 Input: K := FiniteField(2); > C := LinearCode; E := Hull(C); E; WeightDistribution(E); Output: Magma V2.11-10 Tue Dec 13 2005 10:16:56 on modular [Seed = 2671917382] ------------------------------------- >> E := Hull(C); ^ User error: Identifier 'Hull' has not been declared or assigned >> E; ^ User error: Identifier 'E' has not been declared or assigned >> WeightDistribution(E); ^ User error: Identifier 'E' has not been declared or assigned Total time: 0.220 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:01:43 2005 Input: K := FiniteField(2); > C := LinearCode; D := Dual(C); WeightDistribution(C); time WeightDistribution(D); Output: Magma V2.11-10 Tue Dec 13 2005 10:01:42 on modular [Seed = 4096349653] ------------------------------------- [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] [ <0, 1>, <8, 1155>, <10, 34496>, <12, 539280>, <14, 5480640>, <16, 40029297>, <18, 200963840>, <20, 755023808>, <22, 2027262720>, <24, 4185896715>, <26, 6289084032>, <28, 7351106400>, <30, 6289084032>, <32, 4185896715>, <34, 2027262720>, <36, 755023808>, <38, 200963840>, <40, 40029297>, <42, 5480640>, <44, 539280>, <46, 34496>, <48, 1155>, <56, 1> ] Time: 0.020 Total time: 0.250 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 10:00:16 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, "e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 10:00:15 on modular [Seed = 164932481] ------------------------------------- g = (1414620835193373657646613880233559577407734609545*i + 352489175314961464607671418528001817189092256541)*z^5 + (904802261065103105936668720505067133656667597074*i + 1343248922269291340494538393490943353845819222256)*z^4 + (1025700017505207783921241027028089077317301724291*i + 1267567142236668396047190065512596226428512971416)*z^3 + (283587609846347412879109527640610607003328134432*i + 250487955904041646310509260856358041239663937723)*z^2 + (110054808378578108265445578093874268052578876998*i + 202041545854984388890408940644851972139794936772)*z + 257093848091331658607960702914357017287219708568*i + 652950073647454119068347296682657355787566777234 Failure: u = 505653466152672793617695632599288515920022369214 , v = 813309314074995504442616266927015985129671509525 e(u*P,v*Q) = (229450318520897930832122824950759136162186600899*i + 171458666541069865322322784204702213676643928600)*z^5 + (1123336210201860974010594496403076935746616156201*i + 382509869157557354897123314935302020523067652238)*z^4 + (195956015357740925670293015392589736976005190419*i + 1201060680146586880867752998079193262658782520318)*z^3 + (1400183652328377308210652428671451994834327508147*i + 885783581485769151127555190406908658009200060139)*z^2 + (241990578305321134124934647372233092586143696750*i + 353283743758609339844557914979526193600469875967)*z + 69301698278701722581645298580459933625029213350*i + 522130179428839937361552509967162010658032702262 , e(P,Q)^(uv) = (1233480993554786834008498621906536758921898377565*i + 1190015354691216118850119495146059439731677746380)*z^5 + (921756594327110983814290309172804473768320708717*i + 738420790495346685113640877223616173148157182434)*z^4 + (1448167631698704692933635090700677473693607566307*i + 467911266345108570245537574526004995346892189176)*z^3 + (819222557435172257255808600604347664799809662153*i + 691163176033377731593142083640280484981447850039)*z^2 + (1178974401966800441343583985193986104164302821005*i + 276511259554730579432106796424120788700936371162)*z + 646813776638959146887930847368281873288578357077*i + 1043047602020859885512585427844085790318024216544 Total time: 1.080 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 09:47:58 2005 Input: K := FiniteField(2); > D := LinearCode; > aut := AutomorphismGroup(D); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(D); Output: Magma V2.11-10 Tue Dec 13 2005 09:47:57 on modular [Seed = 1552007222] ------------------------------------- 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } false Total time: 0.280 seconds, Total memory usage: 5.51MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 09:47:01 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Tue Dec 13 2005 09:47:01 on modular [Seed = 1385156544] ------------------------------------- 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } false Total time: 0.250 seconds, Total memory usage: 5.51MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 09:43:19 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Tue Dec 13 2005 09:43:18 on modular [Seed = 1468713510] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:41:20 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); if tate(u*G, v*Gt) ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v; quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 09:41:19 on modular [Seed = 1840313237] ------------------------------------- g = (46880789303416607498834709687225916309524281274*i + 1109012449181828800537777171392783676528166634278)*z^5 + (904802261065103105936668720505067133656667597074*i + 1343248922269291340494538393490943353845819222256)*z^4 + (435801606991582481224207562892696416399957166528*i + 193934482260121869098258524408189267288745919403)*z^3 + (283587609846347412879109527640610607003328134432*i + 250487955904041646310509260856358041239663937723)*z^2 + (1351446816118212156880003011826911225664680013821*i + 1259460078641805876255039649275933521577463954047)*z + 257093848091331658607960702914357017287219708568*i + 652950073647454119068347296682657355787566777234 Failure: u = 1026132414533535312271167998032452174431467642015 , v = 607317891842996534038173466812558720729177303044 Total time: 1.060 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:30:18 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; Output: Magma V2.11-10 Tue Dec 13 2005 09:30:15 on modular [Seed = 1924131826] ------------------------------------- g = (1414620835193373657646613880233559577407734609545*i + 352489175314961464607671418528001817189092256541)*z^5 + (904802261065103105936668720505067133656667597074*i + 1343248922269291340494538393490943353845819222256)*z^4 + (1025700017505207783921241027028089077317301724291*i + 1267567142236668396047190065512596226428512971416)*z^3 + (283587609846347412879109527640610607003328134432*i + 250487955904041646310509260856358041239663937723)*z^2 + (110054808378578108265445578093874268052578876998*i + 202041545854984388890408940644851972139794936772)*z + 257093848091331658607960702914357017287219708568*i + 652950073647454119068347296682657355787566777234 g^(p^1-1): true g^(p^2-1): true g^(p^3-1): true g^(p^4-1): true g^(p^5-1): true g^(p^6-1): true g^(p^7-1): true g^(p^8-1): true g^(p^9-1): true g^(p^10-1): true g^(p^11-1): true g^(p^12-1): true g^n = 1 Total time: 2.600 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:28:11 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := i + 2; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; "P =", P; "Q' =", Qt; "Q =", Q; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; for j in [1..12] do print "g^(p^" * Sprint(j) * "-1) = " * Sprint(g^(p^j-1)); end for; "g^n =", g^n; Output: Magma V2.11-10 Tue Dec 13 2005 09:28:08 on modular [Seed = 2655192982] ------------------------------------- P = (1 : 2 : 1) Q' = (i + 2 : 1097950022558348869462494891919937327680507168109*i + 1374283222983568961963002763601919595127481124512 : 1) Q = ((i + 2)*z^2 : (1097950022558348869462494891919937327680507168109*i + 1374283222983568961963002763601919595127481124512)*z^3 : 1) g = (123909007522715632304234432926990441159731420430*i + 637508010256500182107280699368993348477341084485)*z^5 + (688703507104756931070001853914995131005373541312*i + 1002621352581475249329708955843984496240073153982)*z^4 + (608846424657090324998119234825851616974125548764*i + 771482020725020581050629871627202529613821639823)*z^3 + (1392434018234511176428044389335844610478592711217*i + 473331107538064635840791020785366158595476919127)*z^2 + (73970378918706542820629598671516884756426271661*i + 365943832704743787495421878796345795274953421973)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^1-1) = (748100618927567348520741622873551546451532782022*i + 1428699130224630511049471936323405187567302837671)*z^5 + (1202680812184493569478300721391313577308119778263*i + 313447253330080409810020430322639687095227202514)*z^4 + (896962745306077156991764130250048502701811544250*i + 75030114724933747382754664337864814318468370719)*z^3 + (333682831282116263631143017468030399561367002231*i + 518405572876237815324490177632651602018332790765)*z^2 + (1112317314850298844232882970368687655422664826708*i + 560782544915172117676888817418082341837354667733)*z + 1202838306203487510142528951450976199443662125144*i + 795069427093050539880893267448617311116978407546 g^(p^2-1) = (1122685762559073106420486640172303029900241318667*i + 1206761843969459490842516022375418025481873864298)*z^5 + (3817024364748341053470651752544249413211110060*i + 92074379894423550543412574098909660792084933016)*z^4 + (608846424657090324998119234825851616974125548764*i + 771482020725020581050629871627202529613821639823)*z^3 + (477186809706685400548284839616627303398814604581*i + 1353293254655497653064288440912729947389627707084)*z^2 + (708592972296574786248902668360725858769675156568*i + 566475776092548476468859447971519767492233615991)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^3-1) = (1357106999709464496564779036298599057498303412530*i + 1345682320959266667240129620417040749623418914188)*z^5 + (936139532752018606210117538341071166608350863727*i + 1112461501477852426997296904293113682769958766774)*z^4 + (1445405172445096850777244515807849246142141374340*i + 396466674385106777111606626855996450744375790171)*z^3 + (1461250712521757914096163977919390965791567384854*i + 244970246429598896709919226014995935882868586820)*z^2 + (542551411651261070353031322827794497138348015296*i + 429807684156175367374279713695611598069895859239)*z + 70531838254713320311910681453696149021611168678*i + 1398834529239549175114023695374662216300177982250 g^(p^4-1) = (1221509504491967374311557208672867933832795980673*i + 200467472176772056987212698380766546792219162971)*z^5 + (208862746409985936264596906005246300650626057668*i + 565357618624448102009812399714319141941210262495)*z^4 + (865756177282313545985213216550036708550476280949*i + 475475525215527373084617709122508974760880884335)*z^3 + (102466015727909809069678183573961098275241875184*i + 940151235303014027110245043307168921216332322768)*z^2 + (928922738858698562216100850689154274539784497877*i + 382002999584106301024558712410303674095683347609)*z + 1220974658126123565670019751888593761044993991185*i + 211811947018249923223576976718524137383767937965 g^(p^5-1) = (594480147410955283734684832159447099240412938228*i + 1007447468906095875580426526654019572773339214368)*z^5 + (1152066755303591353030630678326466707035079216340*i + 651594961867943713451573799740600135483141099859)*z^4 + (49501392213291562027375381267081647404991522810*i + 820710015151236425244004586118888126082585441078)*z^3 + (42188653754860119450135480418886642353674596443*i + 562408702302581793941475002165818938687417258094)*z^2 + (1091572228871275931758790486093603710299273846570*i + 577436592715763274817514347647367029829638669007)*z + 1168565180281449525586662304102209555431570678308*i + 503582600681258840386198926408111711693060441073 g^(p^6-1) = (205768684348313624077451572794465353568914781421*i + 613950226393615708182994901900063381875511405659)*z^5 + (1428633672987200370786940114820930632812523417902*i + 395314511348759603485431448487808080038157695517)*z^4 + (881979103651631189249974992863662825116960349537*i + 1044381106242866386024323569275011144406191731131)*z^3 + (377475555156278737030694625815440794829353520650*i + 60549325767510300874857944491779135450365424950)*z^2 + (904476028311783023806480875092919429317592811440*i + 654508681633842315960851796564179216185709460867)*z + 826543425518806376820411130172467469135354246707*i + 546674685375929320723432354108401487302836191956 g^(p^7-1) = (334999190429734450302032977239633601705995714949*i + 285499081478684180346094108130235958500726170807)*z^5 + (1357641212310828158490838523101409841505272979404*i + 678284707222532143244512307089503927268688656266)*z^4 + (447713965858424633050352197649395842462816234715*i + 523942895907349573254753909344390113077006213603)*z^3 + (395448289533461064020784297720245760316432679996*i + 256449527939203012159253238775693631557658125017)*z^2 + (825189319832069956822612513847208165005287290855*i + 333752408118167725005777483986738208826843204644)*z + 292936444215340739558786285818575938285688212511*i + 503582600681258840386198926408111711693060441073 g^(p^8-1) = (128794472520144390825537958245889556576532668457*i + 682817146795319322173041394863892996788054910205)*z^5 + (712683874015386917422759238642315687393680373361*i + 1227924079489556276578945161248737876531470377844)*z^4 + (595745447214476719160235373370748785166782609870*i + 986026099281262892060830880798276518956378006484)*z^3 + (375673708125247397107051660343678979770619117775*i + 114357758639702456761704170918133809768845643944)*z^2 + (1411889287093050769181126017526789790323252735684*i + 1105315081259072298561198519568055599119405178147)*z + 1220974658126123565670019751888593761044993991185*i + 211811947018249923223576976718524137383767937965 g^(p^9-1) = (1223692038621048430153872436913389348975101235031*i + 474621643755870221635939468034787852720049039479)*z^5 + (936139532752018606210117538341071166608350863727*i + 349040123018937838148151685627671810947300124045)*z^4 + (1376144652516737512570670115379753945711985528623*i + 1373311158685001552683394130170484212465308688527)*z^3 + (244970246429598896709919226014995935882868586820*i + 1461250712521757914096163977919390965791567384854)*z^2 + (51496949206574234593005076323843447627546727734*i + 22372145650816023655116449432656061648752517842)*z + 1390969786242076944833537908467089344695647722141*i + 1398834529239549175114023695374662216300177982250 g^(p^10-1) = (214906854415001526420727516821492022657286151722*i + 1078733394767620857341100458097159613475302832855)*z^5 + (768981093027284993021976084253246113298674239447*i + 366805892020891465272327059977891336685100803821)*z^4 + (608846424657090324998119234825851616974125548764*i + 771482020725020581050629871627202529613821639823)*z^3 + (1053382421052383953314567950889099073557110465840*i + 1096378886800018241385817718143474881449413155427)*z^2 + (678938273281508936075916322888542750191157462590*i + 529082015699498001181167263152919930950071852855)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^11-1) = (548580783601665930616001142054617430470218498646*i + 1451628029083941079148995842838738809076287723525)*z^5 + (2545939521711331370283095426719602351824833799*i + 1344046085896870808486244472445559740184244945852)*z^4 + (837618043175865668846772504097006035270479350279*i + 786494693152924360802801900817858926731509706423)*z^3 + (1402153388011088316650991466733555001741554598260*i + 247983074468381684089738348912296989265022486477)*z^2 + (1185906070039435994832468343291972442612922376685*i + 305123309751637633980829646230021828644012617207)*z + 258663318293302755002919638469809294273596765675*i + 795069427093050539880893267448617311116978407546 g^(p^12-1) = 1 g^n = 1 Total time: 2.589 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:25:43 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; //"P =", P; "Q' =", Qt; "Q =", Q; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^j-1) eq 1); "g^n =", g^n; Output: Magma V2.11-10 Tue Dec 13 2005 09:25:40 on modular [Seed = 2538212611] ------------------------------------- g = (46880789303416607498834709687225916309524281274*i + 1109012449181828800537777171392783676528166634278)*z^5 + (904802261065103105936668720505067133656667597074*i + 1343248922269291340494538393490943353845819222256)*z^4 + (435801606991582481224207562892696416399957166528*i + 193934482260121869098258524408189267288745919403)*z^3 + (283587609846347412879109527640610607003328134432*i + 250487955904041646310509260856358041239663937723)*z^2 + (1351446816118212156880003011826911225664680013821*i + 1259460078641805876255039649275933521577463954047)*z + 257093848091331658607960702914357017287219708568*i + 652950073647454119068347296682657355787566777234 g^(p^1-1): true g^(p^2-1): true g^(p^3-1): true g^(p^4-1): true g^(p^5-1): true g^(p^6-1): true g^(p^7-1): true g^(p^8-1): true g^(p^9-1): true g^(p^10-1): true g^(p^11-1): true >> print "g^(p^12-1): " * Sprint(g^(p^j-1) eq 1); ^ User error: Identifier 'j' has not been declared or assigned g^n = 1 Total time: 2.640 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:21:25 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := i + 2; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; "P =", P; "Q' =", Qt; "Q =", Q; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; for j in [1..12] do print "g^(p^" * Sprint(j) * "-1) = " * Sprint(g^(p^j-1)); end for; "g^n =", g^n; Output: Magma V2.11-10 Tue Dec 13 2005 09:21:17 on modular [Seed = 2943497218] ------------------------------------- P = (1 : 2 : 1) Q' = (i + 2 : 363551601938441395682953698000848166036751722710*i + 87218401513221303182445826318865898589777766307 : 1) Q = ((i + 2)*z^2 : (363551601938441395682953698000848166036751722710*i + 87218401513221303182445826318865898589777766307)*z^3 : 1) g = (1337592616974074632841214156993795052557527470389*i + 823993614240290083038167890551792145239917806334)*z^5 + (688703507104756931070001853914995131005373541312*i + 1002621352581475249329708955843984496240073153982)*z^4 + (852655199839699940147329355094933876743133342055*i + 690019603771769684094818718293582964103437250996)*z^3 + (1392434018234511176428044389335844610478592711217*i + 473331107538064635840791020785366158595476919127)*z^2 + (1387531245578083722324818991249268608960832619158*i + 1095557791792046477650026711124439698442305468846)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^1-1) = (713401005569222916624706967047233947265726108797*i + 32802494272159754095976653597380306149956053148)*z^5 + (1202680812184493569478300721391313577308119778263*i + 313447253330080409810020430322639687095227202514)*z^4 + (564538879190713108153684459670736991015447346569*i + 1386471509771856517762693925582920679398790520100)*z^3 + (333682831282116263631143017468030399561367002231*i + 518405572876237815324490177632651602018332790765)*z^2 + (349184309646491420912565619552097838294594064111*i + 900719079581618147468559772502703151879904223086)*z + 1202838306203487510142528951450976199443662125144*i + 795069427093050539880893267448617311116978407546 g^(p^2-1) = (338815861937717158724961949748482463817017572152*i + 254739780527330774302932567545367468235385026521)*z^5 + (3817024364748341053470651752544249413211110060*i + 92074379894423550543412574098909660792084933016)*z^4 + (852655199839699940147329355094933876743133342055*i + 690019603771769684094818718293582964103437250996)*z^3 + (477186809706685400548284839616627303398814604581*i + 1353293254655497653064288440912729947389627707084)*z^2 + (752908652200215478896545921560059634947583734251*i + 895025848404241788676589141949265726225025274828)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^3-1) = (104394624787325768580669553622186436218955478289*i + 115819303537523597905318969503744744093839976631)*z^5 + (936139532752018606210117538341071166608350863727*i + 1112461501477852426997296904293113682769958766774)*z^4 + (16096452051693414368204074112936247575117516479*i + 1065034950111683488033841963064789042972883100648)*z^3 + (1461250712521757914096163977919390965791567384854*i + 244970246429598896709919226014995935882868586820)*z^2 + (918950212845529194792417267092990996578910875523*i + 1031693940340614897771168876225173895647363031580)*z + 70531838254713320311910681453696149021611168678*i + 1398834529239549175114023695374662216300177982250 g^(p^4-1) = (239992120004822890833891381247917559884462910146*i + 1261034152320018208158235891540018946925039727848)*z^5 + (208862746409985936264596906005246300650626057668*i + 565357618624448102009812399714319141941210262495)*z^4 + (595745447214476719160235373370748785166782609870*i + 986026099281262892060830880798276518956378006484)*z^3 + (102466015727909809069678183573961098275241875184*i + 940151235303014027110245043307168921216332322768)*z^2 + (532578885638091702929347739231631219177474392942*i + 1079498624912683964120889877510481819621575543210)*z + 1220974658126123565670019751888593761044993991185*i + 211811947018249923223576976718524137383767937965 g^(p^5-1) = (867021477085834981410763757761338394476845952591*i + 454054155590694389565022063266765920943919676451)*z^5 + (1152066755303591353030630678326466707035079216340*i + 651594961867943713451573799740600135483141099859)*z^4 + (1412000232283498703118073208653703846312267368009*i + 640791609345553839901444003801897367634673449741)*z^3 + (42188653754860119450135480418886642353674596443*i + 562408702302581793941475002165818938687417258094)*z^2 + (369929395625514333386658103827181783417985044249*i + 884065031781026990327934242273418463887620221812)*z + 1168565180281449525586662304102209555431570678308*i + 503582600681258840386198926408111711693060441073 g^(p^6-1) = (1255732940148476641067997017126320140148344109398*i + 847551398103174556962453688020722111841747485160)*z^5 + (1428633672987200370786940114820930632812523417902*i + 395314511348759603485431448487808080038157695517)*z^4 + (579522520845159075895473597057122668600298541282*i + 417120518253923879121125020645774349311067159688)*z^3 + (377475555156278737030694625815440794829353520650*i + 60549325767510300874857944491779135450365424950)*z^2 + (557025596185007241338967714827866064399666079379*i + 806992942862947949184596793356606277531549429952)*z + 826543425518806376820411130172467469135354246707*i + 546674685375929320723432354108401487302836191956 g^(p^7-1) = (1126502434067055814843415612681151892011263175870*i + 1176002543018106084799354481790549535216532720012)*z^5 + (1357641212310828158490838523101409841505272979404*i + 678284707222532143244512307089503927268688656266)*z^4 + (1013787658638365632095096392271389651254442656104*i + 937558728589440691890694680576395380640252677216)*z^3 + (395448289533461064020784297720245760316432679996*i + 256449527939203012159253238775693631557658125017)*z^2 + (636312304664720308322836076073577328711971599964*i + 1127749216378622540139671105934047284890415686175)*z + 292936444215340739558786285818575938285688212511*i + 503582600681258840386198926408111711693060441073 g^(p^8-1) = (1332707151976645874319910631674895937140726222362*i + 778684477701470942972407195056892496929203980614)*z^5 + (712683874015386917422759238642315687393680373361*i + 1227924079489556276578945161248737876531470377844)*z^4 + (865756177282313545985213216550036708550476280949*i + 475475525215527373084617709122508974760880884335)*z^3 + (375673708125247397107051660343678979770619117775*i + 114357758639702456761704170918133809768845643944)*z^2 + (49612337403739495964322572393995703394006155135*i + 356186543237717966584250070352729894597853712672)*z + 1220974658126123565670019751888593761044993991185*i + 211811947018249923223576976718524137383767937965 g^(p^9-1) = (237809585875741834991576153007396144742157655788*i + 986879980740920043509509121885997640997209851340)*z^5 + (936139532752018606210117538341071166608350863727*i + 349040123018937838148151685627671810947300124045)*z^4 + (85356971980052752574778474541031548005273362196*i + 88190465811788712462054459750301281251950202292)*z^3 + (244970246429598896709919226014995935882868586820*i + 1461250712521757914096163977919390965791567384854)*z^2 + (1410004675290216030552443513596942046089712163085*i + 1439129478845974241490332140488129432068506372977)*z + 1390969786242076944833537908467089344695647722141*i + 1398834529239549175114023695374662216300177982250 g^(p^10-1) = (1246594770081788738724721073099293471059972739097*i + 382768229729169407804348131823625880241956057964)*z^5 + (768981093027284993021976084253246113298674239447*i + 366805892020891465272327059977891336685100803821)*z^4 + (852655199839699940147329355094933876743133342055*i + 690019603771769684094818718293582964103437250996)*z^3 + (1053382421052383953314567950889099073557110465840*i + 1096378886800018241385817718143474881449413155427)*z^2 + (782563351215281329069532267032242743526101428229*i + 932419608797292263964281326767865562767187037964)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^11-1) = (912920840895124334529447447866168063247040392173*i + 9873595412849185996452747082046684640971167294)*z^5 + (2545939521711331370283095426719602351824833799*i + 1344046085896870808486244472445559740184244945852)*z^4 + (623883581320924596298676085823779458446779540540*i + 675006931343865904342646689102926566985749184396)*z^3 + (1402153388011088316650991466733555001741554598260*i + 247983074468381684089738348912296989265022486477)*z^2 + (275595554457354270312980246628813051104336514134*i + 1156378314745152631164618943690763665073246273612)*z + 258663318293302755002919638469809294273596765675*i + 795069427093050539880893267448617311116978407546 g^(p^12-1) = 1 g^n = 1 Total time: 2.680 seconds, Total memory usage: 3.34MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 09:09:16 2005 Input: R:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K:=ext; CompositionFactors(155); Output: Magma V2.11-10 Tue Dec 13 2005 09:09:15 on modular [Seed = 3010340552] ------------------------------------- >> CompositionFactors(155);; ^ Runtime error in 'CompositionFactors': Bad argument types Argument types given: RngIntElt Total time: 0.190 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 09:07:43 2005 Input: R:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K:=ext; CompositionFactors(definingPolynomial); Output: Magma V2.11-10 Tue Dec 13 2005 09:07:43 on modular [Seed = 3741468928] ------------------------------------- >> CompositionFactors(definingPolynomial);; ^ Runtime error in 'CompositionFactors': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.180 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 09:04:55 2005 Input: R:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K:=ext; Output: Magma V2.11-10 Tue Dec 13 2005 09:04:55 on modular [Seed = 3657911469] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:03:36 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := i + 2; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; "P =", P; "Q' =", Qt; "Q =", Q; return miller(n, P, Q)^chi; end function; "miller =", miller(n, G, Gt); "tate =", tate(G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 09:03:36 on modular [Seed = 3574088221] ------------------------------------- miller = 162471042542776450109546911262159626540194874282*i + 1254993595675210729223411613343751283964425868707 P = (1 : 2 : 1) Q' = (i + 2 : 363551601938441395682953698000848166036751722710*i + 87218401513221303182445826318865898589777766307 : 1) Q = ((i + 2)*z^2 : (363551601938441395682953698000848166036751722710*i + 87218401513221303182445826318865898589777766307)*z^3 : 1) tate = (1337592616974074632841214156993795052557527470389*i + 823993614240290083038167890551792145239917806334)*z^5 + (688703507104756931070001853914995131005373541312*i + 1002621352581475249329708955843984496240073153982)*z^4 + (852655199839699940147329355094933876743133342055*i + 690019603771769684094818718293582964103437250996)*z^3 + (1392434018234511176428044389335844610478592711217*i + 473331107538064635840791020785366158595476919127)*z^2 + (1387531245578083722324818991249268608960832619158*i + 1095557791792046477650026711124439698442305468846)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 Total time: 0.630 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:59:32 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := i + 2; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; z := (p^k - 1) div n; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Q) return miller(n, P, Q)^z; end function; "P =", G; "Q =", Gt; "miller =", miller(n, G, Gt); "tate =", tate(G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:59:31 on modular [Seed = 4013064028] ------------------------------------- P = (1 : 2 : 1) Q = (i + 2 : 363551601938441395682953698000848166036751722710*i + 87218401513221303182445826318865898589777766307 : 1) miller = 162471042542776450109546911262159626540194874282*i + 1254993595675210729223411613343751283964425868707 tate = 1 Total time: 0.260 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:58:31 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := i + 2; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; z := (p^k - 1) div n; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Q) return miller(n, P, Q)^z; end function; "miller =", miller(n, G, Gt); "tate =", tate(G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:58:31 on modular [Seed = 3929504285] ------------------------------------- miller = 694976144959185627384757159853755106109815530714*i + 1311958513305109301215342597778813598163540723157 tate = 1 Total time: 0.250 seconds, Total memory usage: 3.34MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 08:58:30 2005 Input: R:=PolynomialRing(GF(2)); Output: Magma V2.11-10 Tue Dec 13 2005 08:58:23 on modular [Seed = 3845682666] ------------------------------------- Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:50:40 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; z := (p^k - 1) div n; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Q) return miller(n, P, Q)^z; end function; "miller =", miller(n, G, Gt); "tate =", tate(G, Gt); Roots(24+25*i, 3); xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; Output: Magma V2.11-10 Tue Dec 13 2005 08:50:18 on modular [Seed = 4179913490] ------------------------------------- miller = 34978940859506703873739429950455443570933508707*i + 411686203016070577518067565293188378259628682113 tate = 1 >> Roots(24+25*i, 3); ^ Runtime error in 'Roots': Bad argument types Argument types given: FldFinElt, RngIntElt xt = i + 2 Total time: 0.300 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:42:17 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; z := (p^k - 1) div n; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Q) return miller(n, P, Q)^z; end function; "miller =", miller(n, P, Q); "tate =", tate(G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:42:01 on modular [Seed = 4046224683] ------------------------------------- >> "miller =", miller(n, P, Q); ^ User error: Identifier 'P' has not been declared or assigned tate = 1 Total time: 0.270 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:37:54 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) // assert (q^k - 1) mod r eq 0; z := (q^k - 1) div r; return miller(r, P, Q)^z; end function; tate(n, p, 12, G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:37:53 on modular [Seed = 516133855] ------------------------------------- 1 Total time: 0.240 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:36:36 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) m := miller(r, P, Q); assert (q^k - 1) mod r eq 0; z := (q^k - 1) div r; return m^z; end function; tate(n, p, 12, G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:36:33 on modular [Seed = 415601929] ------------------------------------- 1 Total time: 0.260 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:34:42 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; tate(n, p, 12, G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:34:23 on modular [Seed = 332307842] ------------------------------------- tate( r: 1461501624496790265145447380994971188499300027613, q: 1461501624496790265145448589920785493717258890819, k: 12, P: (1 : 2 : 1), Q: (8 : 645238442624673913635245604741558906439788126004*i + 11... ) >> return miller(r, P, Q)^((q^k - 1)/r); ^ Runtime error in '^': Bad argument types Argument types given: FldFinElt, FldFinElt Total time: 0.290 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:32:15 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; tate(n, p, 12, G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:31:57 on modular [Seed = 770745107] ------------------------------------- tate( r: 1461501624496790265145447380994971188499300027613, q: 1461501624496790265145448589920785493717258890819, k: 12, P: (1 : 2 : 1), Q: (8 : 816263181872116351510202985179226587277470764815*i + 29... ) miller( r: 1461501624496790265145447380994971188499300027613, P: (1 : 2 : 1), Q: (8 : 816263181872116351510202985179226587277470764815*i + 29... ) >> if bit(r, i) = 1 then ^ Runtime error in if: Logical expected Total time: 0.319 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:19:36 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := 2; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; Gt := Et![xt, Sqrt(xt^3 + b/xi)]; Gt; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:19:24 on modular [Seed = 620342536] ------------------------------------- xt = 8 (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) Total time: 0.260 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:17:14 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); Gt := Et![1, Sqrt(1 + b/xi)]; Gt; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:17:11 on modular [Seed = 1038137242] ------------------------------------- >> Gt := Et![1, Sqrt(1 + b/xi)]; ^ Runtime error in 'Sqrt': Argument has no square root >> Gt; ^ User error: Identifier 'Gt' has not been declared or assigned Total time: 0.240 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:15:08 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); Et; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:15:05 on modular [Seed = 871282970] ------------------------------------- Elliptic Curve defined by y^2 = x^3 + (24*i + 1461501624496790265145448589920785493717258890795) over GF(1461501624496790265145448589920785493717258890819^2) Total time: 0.230 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:14:46 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; G; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:14:43 on modular [Seed = 1318306941] ------------------------------------- (1 : 2 : 1) Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:13:39 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; G; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:13:35 on modular [Seed = 1234485958] ------------------------------------- (1 : 2 : 1) Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:12:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; E := EllipticCurve([0, b]); y0 := -Sqrt(1 + b); G := E![1, y0]; G; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:12:44 on modular [Seed = 1552013280] ------------------------------------- (1 : 2 : 1) Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:12:00 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; E := EllipticCurve([0, b]); y0 := -Sqrt(1 + b); E; y0; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:11:57 on modular [Seed = 1468718266] ------------------------------------- Elliptic Curve defined by y^2 = x^3 + 3 over GF(1461501624496790265145448589920785493717258890819) 2 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:11:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; E := EllipticCurve([0, b]); y0 := p - Sqrt(1 + b); E; y0; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:11:44 on modular [Seed = 1385159090] ------------------------------------- Elliptic Curve defined by y^2 = x^3 + 3 over GF(1461501624496790265145448589920785493717258890819) 2 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:11:19 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; E := EllipticCurve([0, b]); y0 := Sqrt(1 + b); E; y0; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:11:16 on modular [Seed = 1840309865] ------------------------------------- Elliptic Curve defined by y^2 = x^3 + 3 over GF(1461501624496790265145448589920785493717258890819) 1461501624496790265145448589920785493717258890817 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:10:59 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := 3; E := EllipticCurve([0, b]); y0 := Sqrt(1 + b); E; y0; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:10:55 on modular [Seed = 1756750663] ------------------------------------- Elliptic Curve defined by y^2 = x^3 + 3 over Rational Field 2.00000000000000000000000000000000000000 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:08:18 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; for r in [0..64] do r, ":", length(r); end for; Output: Magma V2.11-10 Tue Dec 13 2005 08:08:11 on modular [Seed = 2007692220] ------------------------------------- 0 : 0 1 : 1 2 : 2 3 : 2 4 : 3 5 : 3 6 : 3 7 : 3 8 : 4 9 : 4 10 : 4 11 : 4 12 : 4 13 : 4 14 : 4 15 : 4 16 : 5 17 : 5 18 : 5 19 : 5 20 : 5 21 : 5 22 : 5 23 : 5 24 : 5 25 : 5 26 : 5 27 : 5 28 : 5 29 : 5 30 : 5 31 : 5 32 : 6 33 : 6 34 : 6 35 : 6 36 : 6 37 : 6 38 : 6 39 : 6 40 : 6 41 : 6 42 : 6 43 : 6 44 : 6 45 : 6 46 : 6 47 : 6 48 : 6 49 : 6 50 : 6 51 : 6 52 : 6 53 : 6 54 : 6 55 : 6 56 : 6 57 : 6 58 : 6 59 : 6 60 : 6 61 : 6 62 : 6 63 : 6 64 : 7 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:06:55 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := t - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; for r in [0..64] do r, ":", length(r); end for; Output: Magma V2.11-10 Tue Dec 13 2005 08:06:48 on modular [Seed = 2387806870] ------------------------------------- >> for i := t - 1 to 0 by -1 do ^ User error: Identifier 't' has not been declared or assigned >> return miller(r, P, Q)^((q^k - 1)/r); ^ User error: Identifier 'miller' has not been declared or assigned 0 : 0 1 : 1 2 : 2 3 : 2 4 : 3 5 : 3 6 : 3 7 : 3 8 : 4 9 : 4 10 : 4 11 : 4 12 : 4 13 : 4 14 : 4 15 : 4 16 : 5 17 : 5 18 : 5 19 : 5 20 : 5 21 : 5 22 : 5 23 : 5 24 : 5 25 : 5 26 : 5 27 : 5 28 : 5 29 : 5 30 : 5 31 : 5 32 : 6 33 : 6 34 : 6 35 : 6 36 : 6 37 : 6 38 : 6 39 : 6 40 : 6 41 : 6 42 : 6 43 : 6 44 : 6 45 : 6 46 : 6 47 : 6 48 : 6 49 : 6 50 : 6 51 : 6 52 : 6 53 : 6 54 : 6 55 : 6 56 : 6 57 : 6 58 : 6 59 : 6 60 : 6 61 : 6 62 : 6 63 : 6 64 : 7 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:03:03 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := t – 1 to 0 by –1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:02:46 on modular [Seed = 2287275406] ------------------------------------- >> for i := t ^Ö 1 to 0 by ^Ö1 do ^ User error: Unknown character (ASCII value 150) >> f := f^2*g(A, A, Q); A := 2*A; ^ User error: Identifier 'f' has not been declared or assigned >> f := f^2*g(A, A, Q); A := 2*A; ^ User error: Identifier 'A' has not been declared or assigned >> if bit(r, i) = 1 then ^ User error: Identifier 'r' has not been declared or assigned >> end for; ^ User error: bad syntax >> return f; ^ User error: A 'return' can only be used inside a procedure or function >> end function; ^ User error: bad syntax Total time: 0.270 seconds, Total memory usage: 3.24MB '213.78.' ************** MAGMA ***************** Host 213.78.42.15 (213.78.42.15) Time: Tue Dec 13 06:18:14 2005 Input: 10! Output: Magma V2.11-10 Tue Dec 13 2005 06:18:13 on modular [Seed = 298647435] ------------------------------------- >> 10!; ^ User error: bad syntax Total time: 0.220 seconds, Total memory usage: 3.24MB '130.83.' ************** MAGMA ***************** Host 130.83.244.131 (130.83.244.131) Time: Tue Dec 13 06:14:16 2005 Input: F2 := FiniteField(2); P := PolynomialRing(F2); p := x^230 + x^50 + x^33 + x^32 + 1; F := ext< F2 | p >; a := z; E := EllipticCurve([1, 0, 0, 0, a]); time #E; FactoredOrder(E); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 06:13:32 on modular [Seed = 1234478843] ------------------------------------- 1725436586697640946858688965569256371777165413914956878562009594085376 Time: 0.170 Errors: /bin/sh: line 1: 11302 Alarm clock nice -n 19 /usr/local/bin/magma '159.149' ************** MAGMA ***************** Host 159.149.2.252 (159.149.2.252) Time: Tue Dec 13 04:03:09 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Tue Dec 13 2005 04:02:39 on modular [Seed = 2420989386] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.370 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Mon Dec 12 18:17:50 2005 Input: Factorization(100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 12 2005 18:17:30 on modular [Seed = 1890399814] ------------------------------------- Errors: /bin/sh: line 1: 32588 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Mon Dec 12 18:16:29 2005 Input: Factorization(10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Mon Dec 12 2005 18:16:28 on modular [Seed = 2141337743] ------------------------------------- [ <2, 709>, <5, 709> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '80.201.' ************** MAGMA ***************** Host 80.201.74.36 (80.201.74.36) Time: Mon Dec 12 15:19:29 2005 Input: factor(111111111111111111111111111111111) Output: Magma V2.11-10 Mon Dec 12 2005 15:19:29 on modular [Seed = 2170493422] ------------------------------------- >> factor(111111111111111111111111111111111) ^ User error: Identifier 'factor' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Mon Dec 12 14:29:29 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; Output: Magma V2.11-10 Mon Dec 12 2005 14:29:29 on modular [Seed = 1418385651] ------------------------------------- Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Mon Dec 12 14:29:20 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (u eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; Output: Magma V2.11-10 Mon Dec 12 2005 14:29:20 on modular [Seed = 1535498229] ------------------------------------- >> if IsZero(U) or IsZero(V) or (u eq -V) or IsZero(Q) then ^ User error: Identifier 'u' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Mon Dec 12 14:06:22 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; Output: Magma V2.11-10 Mon Dec 12 2005 14:06:21 on modular [Seed = 2057630569] ------------------------------------- Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Mon Dec 12 14:04:05 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ext; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ext; Output: Magma V2.11-10 Mon Dec 12 2005 14:04:05 on modular [Seed = 14584475] ------------------------------------- >> Fp2 := ext; ^ User error: bad syntax >> mu := 1 + i; ^ User error: Identifier 'i' has not been declared or assigned >> xi := 1/(-8 + 8*i); ^ User error: Identifier 'i' has not been declared or assigned >> Fp12 := ext; ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '131.188' ************** MAGMA ***************** Host 131.188.166.152 (131.188.166.152) Time: Mon Dec 12 11:58:49 2005 Input: 1+2; Output: Magma V2.11-10 Mon Dec 12 2005 11:58:49 on modular [Seed = 248570239] ------------------------------------- 3 Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 07:14:00 2005 Input: G :=DirichletGroup(15); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],80);Parent($1); Output: Magma V2.11-10 Mon Dec 12 2005 07:14:00 on modular [Seed = 586942663] ------------------------------------- Group of Dirichlet characters of modulus 15 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field, Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field ] q + q^2 - 3*q^3 - 3*q^4 + 5*q^5 - 3*q^6 - 7*q^8 + 9*q^9 + 5*q^10 + 9*q^12 - 15*q^15 + 5*q^16 - 14*q^17 + 9*q^18 - 22*q^19 - 15*q^20 + 34*q^23 + 21*q^24 + 25*q^25 - 27*q^27 - 15*q^30 + 2*q^31 + 33*q^32 - 14*q^34 - 27*q^36 - 22*q^38 - 35*q^40 + 45*q^45 + 34*q^46 - 14*q^47 - 15*q^48 + 49*q^49 + 25*q^50 + 42*q^51 - 86*q^53 - 27*q^54 + 66*q^57 + 45*q^60 - 118*q^61 + 2*q^62 + 13*q^64 + 42*q^68 - 102*q^69 - 63*q^72 - 75*q^75 + 66*q^76 + 98*q^79 + O(q^80) Power series ring in q over Rational Field Total time: 0.280 seconds, Total memory usage: 4.81MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 07:13:46 2005 Input: G :=DirichletGroup(15); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[12],80);Parent($1); Output: Magma V2.11-10 Mon Dec 12 2005 07:13:42 on modular [Seed = 670633276] ------------------------------------- Group of Dirichlet characters of modulus 15 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field, Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field ] >> qEigenform(D[12],80);Parent($1);; ^ Runtime error in '[]': Sequence element 12 not defined Set of sequences over Power Structure of ModSym Total time: 0.300 seconds, Total memory usage: 4.42MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:27:07 2005 Input: G :=DirichletGroup(75,CyclotomicField(4)); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[2],80);Parent($1); Output: Magma V2.11-10 Mon Dec 12 2005 06:27:06 on modular [Seed = 1054876634] ------------------------------------- Group of Dirichlet characters of modulus 75 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3 ] 15 4 [ Modular symbols space of level 75, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2, Modular symbols space of level 75, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2 ] q + a*q^2 + zeta_4*a*q^3 + zeta_4*q^4 - 3*q^6 - zeta_4*a*q^8 - 3*zeta_4*q^9 - a*q^12 + 5*q^16 - 4*a*q^17 - 3*zeta_4*a*q^18 + 4*zeta_4*q^19 + 2*zeta_4*a*q^23 + 3*zeta_4*q^24 + 3*a*q^27 - 8*q^31 + 3*a*q^32 - 12*zeta_4*q^34 + 3*q^36 + 4*zeta_4*a*q^38 - 6*q^46 + 6*a*q^47 + 5*zeta_4*a*q^48 + 7*zeta_4*q^49 + 12*q^51 - 8*zeta_4*a*q^53 + 9*zeta_4*q^54 - 4*a*q^57 + 2*q^61 - 8*a*q^62 - zeta_4*q^64 - 4*zeta_4*a*q^68 - 6*zeta_4*q^69 - 3*a*q^72 - 4*q^76 - 16*zeta_4*q^79 + O(q^80) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 4 and degree 2 with modulus a^2 - 3*zeta_4 Total time: 0.380 seconds, Total memory usage: 5.04MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:26:16 2005 Input: G :=DirichletGroup(75,CyclotomicField(4)); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],80);Parent($1); Output: Magma V2.11-10 Mon Dec 12 2005 06:26:16 on modular [Seed = 620378386] ------------------------------------- Group of Dirichlet characters of modulus 75 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3 ] 15 4 [ Modular symbols space of level 75, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2, Modular symbols space of level 75, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2 ] q - zeta_4*a*q^3 - 2*zeta_4*q^4 + a*q^7 + 3*zeta_4*q^9 - 2*a*q^12 + 3*zeta_4*a*q^13 - 4*q^16 + zeta_4*q^19 - 3*q^21 + 3*a*q^27 - 2*zeta_4*a*q^28 + 7*q^31 + 6*q^36 - 4*a*q^37 - 9*zeta_4*q^39 - 7*zeta_4*a*q^43 + 4*zeta_4*a*q^48 + 4*zeta_4*q^49 + 6*a*q^52 + a*q^57 - 13*q^61 + 3*zeta_4*a*q^63 + 8*zeta_4*q^64 - 9*a*q^67 + 8*zeta_4*a*q^73 + 2*q^76 - 4*zeta_4*q^79 + O(q^80) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 4 and degree 2 with modulus a^2 + 3*zeta_4 Total time: 0.400 seconds, Total memory usage: 5.04MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:23:10 2005 Input: G :=DirichletGroup(45,CyclotomicField(4)); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],80);Parent($1); Output: Magma V2.11-10 Mon Dec 12 2005 06:23:10 on modular [Seed = 1940745603] ------------------------------------- Group of Dirichlet characters of modulus 45 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3 ] 15 4 [ Modular symbols space of level 45, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2 ] q + a*q^2 - zeta_4*q^4 + (2*zeta_4 - 1)*a*q^5 + (2*zeta_4 - 2)*q^7 - 3*zeta_4*a*q^8 + (-zeta_4 - 2)*q^10 + (-2*zeta_4 - 2)*a*q^11 + (zeta_4 + 1)*q^13 + (2*zeta_4 - 2)*a*q^14 + q^16 + 4*a*q^17 + (zeta_4 + 2)*a*q^20 + (-2*zeta_4 + 2)*q^22 + 4*zeta_4*a*q^23 + (-3*zeta_4 + 4)*q^25 + (zeta_4 + 1)*a*q^26 + (2*zeta_4 + 2)*q^28 + (-3*zeta_4 + 3)*a*q^29 - 4*q^31 - 5*a*q^32 + 4*zeta_4*q^34 + (-6*zeta_4 - 2)*a*q^35 + (-zeta_4 + 1)*q^37 + (6*zeta_4 - 3)*q^40 + (zeta_4 + 1)*a*q^41 + (-8*zeta_4 - 8)*q^43 + (2*zeta_4 - 2)*a*q^44 - 4*q^46 - 8*a*q^47 - zeta_4*q^49 + (-3*zeta_4 + 4)*a*q^50 + (-zeta_4 + 1)*q^52 + 4*zeta_4*a*q^53 + (6*zeta_4 + 2)*q^55 + (6*zeta_4 + 6)*a*q^56 + (3*zeta_4 + 3)*q^58 + (-6*zeta_4 + 6)*a*q^59 + 8*q^61 - 4*a*q^62 - 7*zeta_4*q^64 + (zeta_4 - 3)*a*q^65 + (-4*zeta_4 + 4)*q^67 - 4*zeta_4*a*q^68 + (-2*zeta_4 + 6)*q^70 + (4*zeta_4 + 4)*a*q^71 + (zeta_4 + 1)*q^73 + (-zeta_4 + 1)*a*q^74 + 8*a*q^77 + 12*zeta_4*q^79 + O(q^80) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 4 and degree 2 with modulus a^2 - zeta_4 Total time: 0.340 seconds, Total memory usage: 4.98MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:21:51 2005 Input: G :=DirichletGroup(45,CyclotomicField(4)); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],300); Output: Magma V2.11-10 Mon Dec 12 2005 06:21:50 on modular [Seed = 2024171096] ------------------------------------- Group of Dirichlet characters of modulus 45 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3 ] 15 4 [ Modular symbols space of level 45, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2 ] q + a*q^2 - zeta_4*q^4 + (2*zeta_4 - 1)*a*q^5 + (2*zeta_4 - 2)*q^7 - 3*zeta_4*a*q^8 + (-zeta_4 - 2)*q^10 + (-2*zeta_4 - 2)*a*q^11 + (zeta_4 + 1)*q^13 + (2*zeta_4 - 2)*a*q^14 + q^16 + 4*a*q^17 + (zeta_4 + 2)*a*q^20 + (-2*zeta_4 + 2)*q^22 + 4*zeta_4*a*q^23 + (-3*zeta_4 + 4)*q^25 + (zeta_4 + 1)*a*q^26 + (2*zeta_4 + 2)*q^28 + (-3*zeta_4 + 3)*a*q^29 - 4*q^31 - 5*a*q^32 + 4*zeta_4*q^34 + (-6*zeta_4 - 2)*a*q^35 + (-zeta_4 + 1)*q^37 + (6*zeta_4 - 3)*q^40 + (zeta_4 + 1)*a*q^41 + (-8*zeta_4 - 8)*q^43 + (2*zeta_4 - 2)*a*q^44 - 4*q^46 - 8*a*q^47 - zeta_4*q^49 + (-3*zeta_4 + 4)*a*q^50 + (-zeta_4 + 1)*q^52 + 4*zeta_4*a*q^53 + (6*zeta_4 + 2)*q^55 + (6*zeta_4 + 6)*a*q^56 + (3*zeta_4 + 3)*q^58 + (-6*zeta_4 + 6)*a*q^59 + 8*q^61 - 4*a*q^62 - 7*zeta_4*q^64 + (zeta_4 - 3)*a*q^65 + (-4*zeta_4 + 4)*q^67 - 4*zeta_4*a*q^68 + (-2*zeta_4 + 6)*q^70 + (4*zeta_4 + 4)*a*q^71 + (zeta_4 + 1)*q^73 + (-zeta_4 + 1)*a*q^74 + 8*a*q^77 + 12*zeta_4*q^79 + (2*zeta_4 - 1)*a*q^80 + (zeta_4 - 1)*q^82 + 4*zeta_4*a*q^83 + (-4*zeta_4 - 8)*q^85 + (-8*zeta_4 - 8)*a*q^86 + (-6*zeta_4 - 6)*q^88 + (9*zeta_4 - 9)*a*q^89 - 4*q^91 + 4*a*q^92 - 8*zeta_4*q^94 + (11*zeta_4 - 11)*q^97 - zeta_4*a*q^98 + (-4*zeta_4 - 3)*q^100 + (-11*zeta_4 - 11)*a*q^101 + (10*zeta_4 + 10)*q^103 + (-3*zeta_4 + 3)*a*q^104 - 4*q^106 + 4*a*q^107 + (6*zeta_4 + 2)*a*q^110 + (2*zeta_4 - 2)*q^112 - 14*zeta_4*a*q^113 + (-8*zeta_4 + 4)*q^115 + (-3*zeta_4 - 3)*a*q^116 + (6*zeta_4 + 6)*q^118 + (8*zeta_4 - 8)*a*q^119 + 3*q^121 + 8*a*q^122 + 4*zeta_4*q^124 + (11*zeta_4 + 2)*a*q^125 + (-10*zeta_4 + 10)*q^127 + 3*zeta_4*a*q^128 + (-3*zeta_4 - 1)*q^130 + (10*zeta_4 + 10)*a*q^131 + (-4*zeta_4 + 4)*a*q^134 + 12*q^136 + 10*a*q^137 - 12*zeta_4*q^139 + (2*zeta_4 - 6)*a*q^140 + (4*zeta_4 - 4)*q^142 - 4*zeta_4*a*q^143 + (3*zeta_4 - 9)*q^145 + (zeta_4 + 1)*a*q^146 + (-zeta_4 - 1)*q^148 + (3*zeta_4 - 3)*a*q^149 + 8*q^151 + 8*zeta_4*q^154 + (-8*zeta_4 + 4)*a*q^155 + (5*zeta_4 - 5)*q^157 + 12*zeta_4*a*q^158 + (5*zeta_4 + 10)*q^160 + (-8*zeta_4 - 8)*a*q^161 + (-8*zeta_4 - 8)*q^163 + (-zeta_4 + 1)*a*q^164 - 4*q^166 - 20*a*q^167 - 11*zeta_4*q^169 + (-4*zeta_4 - 8)*a*q^170 + (8*zeta_4 - 8)*q^172 - 14*zeta_4*a*q^173 + (14*zeta_4 - 2)*q^175 + (-2*zeta_4 - 2)*a*q^176 + (-9*zeta_4 - 9)*q^178 + (-18*zeta_4 + 18)*a*q^179 - 16*q^181 - 4*a*q^182 + 12*zeta_4*q^184 + (3*zeta_4 + 1)*a*q^185 + (-8*zeta_4 + 8)*q^187 + 8*zeta_4*a*q^188 + (16*zeta_4 + 16)*a*q^191 + (zeta_4 + 1)*q^193 + (11*zeta_4 - 11)*a*q^194 - q^196 - 14*a*q^197 + (-12*zeta_4 - 9)*a*q^200 + (-11*zeta_4 + 11)*q^202 + 12*zeta_4*a*q^203 + (-3*zeta_4 - 1)*q^205 + (10*zeta_4 + 10)*a*q^206 + (zeta_4 + 1)*q^208 - 4*q^211 + 4*a*q^212 + 4*zeta_4*q^214 + (-8*zeta_4 + 24)*a*q^215 + (-8*zeta_4 + 8)*q^217 + (-2*zeta_4 + 6)*q^220 + (4*zeta_4 + 4)*a*q^221 + (10*zeta_4 + 10)*q^223 + (-10*zeta_4 + 10)*a*q^224 + 14*q^226 - 8*a*q^227 + 6*zeta_4*q^229 + (-8*zeta_4 + 4)*a*q^230 + (-9*zeta_4 + 9)*q^232 + 4*zeta_4*a*q^233 + (8*zeta_4 + 16)*q^235 + (-6*zeta_4 - 6)*a*q^236 + (-8*zeta_4 - 8)*q^238 + (12*zeta_4 - 12)*a*q^239 - 10*q^241 + 3*a*q^242 - 8*zeta_4*q^244 + (zeta_4 + 2)*a*q^245 + 12*zeta_4*a*q^248 + (2*zeta_4 - 11)*q^250 + (-14*zeta_4 - 14)*a*q^251 + (8*zeta_4 + 8)*q^253 + (-10*zeta_4 + 10)*a*q^254 - 17*q^256 - 2*a*q^257 + 4*zeta_4*q^259 + (3*zeta_4 + 1)*a*q^260 + (10*zeta_4 - 10)*q^262 + 4*zeta_4*a*q^263 + (-8*zeta_4 + 4)*q^265 + (-4*zeta_4 - 4)*q^268 + (9*zeta_4 - 9)*a*q^269 - 16*q^271 + 4*a*q^272 + 10*zeta_4*q^274 + (-2*zeta_4 - 14)*a*q^275 + (11*zeta_4 - 11)*q^277 - 12*zeta_4*a*q^278 + (-18*zeta_4 - 6)*q^280 + (7*zeta_4 + 7)*a*q^281 + (-8*zeta_4 - 8)*q^283 + (-4*zeta_4 + 4)*a*q^284 + 4*q^286 - 4*a*q^287 - zeta_4*q^289 + (3*zeta_4 - 9)*a*q^290 + (-zeta_4 + 1)*q^292 - 14*zeta_4*a*q^293 + (6*zeta_4 - 18)*q^295 + (-3*zeta_4 - 3)*a*q^296 + (-3*zeta_4 - 3)*q^298 + (4*zeta_4 - 4)*a*q^299 + O(q^300) Total time: 0.550 seconds, Total memory usage: 7.41MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:21:37 2005 Input: G :=DirichletGroup(45,CyclotomicField(4)); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],300); Output: Magma V2.11-10 Mon Dec 12 2005 06:21:36 on modular [Seed = 2141282641] ------------------------------------- Group of Dirichlet characters of modulus 45 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3 ] >> Y :=X[16]; Conductor(Y); Order(Y); ^ Runtime error in '[]': Sequence element 16 not defined >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 2, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],300);; ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:02:56 2005 Input: G :=DirichletGroup(15); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],300); Output: Magma V2.11-10 Mon Dec 12 2005 06:02:55 on modular [Seed = 3026890500] ------------------------------------- Group of Dirichlet characters of modulus 15 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field, Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field ] q + q^2 - 3*q^3 - 3*q^4 + 5*q^5 - 3*q^6 - 7*q^8 + 9*q^9 + 5*q^10 + 9*q^12 - 15*q^15 + 5*q^16 - 14*q^17 + 9*q^18 - 22*q^19 - 15*q^20 + 34*q^23 + 21*q^24 + 25*q^25 - 27*q^27 - 15*q^30 + 2*q^31 + 33*q^32 - 14*q^34 - 27*q^36 - 22*q^38 - 35*q^40 + 45*q^45 + 34*q^46 - 14*q^47 - 15*q^48 + 49*q^49 + 25*q^50 + 42*q^51 - 86*q^53 - 27*q^54 + 66*q^57 + 45*q^60 - 118*q^61 + 2*q^62 + 13*q^64 + 42*q^68 - 102*q^69 - 63*q^72 - 75*q^75 + 66*q^76 + 98*q^79 + 25*q^80 + 81*q^81 + 154*q^83 - 70*q^85 + 45*q^90 - 102*q^92 - 6*q^93 - 14*q^94 - 110*q^95 - 99*q^96 + 49*q^98 - 75*q^100 + 42*q^102 - 86*q^106 + 106*q^107 + 81*q^108 - 22*q^109 - 206*q^113 + 66*q^114 + 170*q^115 + 105*q^120 + 121*q^121 - 118*q^122 - 6*q^124 + 125*q^125 - 119*q^128 - 135*q^135 + 98*q^136 + 226*q^137 - 102*q^138 - 262*q^139 + 42*q^141 + 45*q^144 - 147*q^147 - 75*q^150 - 238*q^151 + 154*q^152 - 126*q^153 + 10*q^155 + 98*q^158 + 258*q^159 + 165*q^160 + 81*q^162 + 154*q^166 - 254*q^167 + 169*q^169 - 70*q^170 - 198*q^171 + 154*q^173 - 135*q^180 + 122*q^181 + 354*q^183 - 238*q^184 - 6*q^186 + 42*q^188 - 110*q^190 - 39*q^192 - 147*q^196 - 374*q^197 - 142*q^199 - 175*q^200 - 126*q^204 + 306*q^207 + 362*q^211 + 258*q^212 + 106*q^214 + 189*q^216 - 22*q^218 + 225*q^225 - 206*q^226 - 134*q^227 - 198*q^228 + 218*q^229 + 170*q^230 + 34*q^233 - 70*q^235 - 294*q^237 - 75*q^240 - 478*q^241 + 121*q^242 - 243*q^243 + 354*q^244 + 245*q^245 - 14*q^248 - 462*q^249 + 125*q^250 + 210*q^255 - 171*q^256 + 466*q^257 - 446*q^263 - 430*q^265 - 135*q^270 + 482*q^271 - 70*q^272 + 226*q^274 + 306*q^276 - 262*q^278 + 18*q^279 + 42*q^282 + 330*q^285 + 297*q^288 - 93*q^289 + 394*q^293 - 147*q^294 + O(q^300) Total time: 0.500 seconds, Total memory usage: 7.23MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:02:39 2005 Input: G :=DirichletGroup(15); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],300); Output: Magma V2.11-10 Mon Dec 12 2005 06:02:39 on modular [Seed = 3110576665] ------------------------------------- Group of Dirichlet characters of modulus 15 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 1 1 [] >> qEigenform(D[1],300);; ^ Runtime error in '[]': Sequence element 1 not defined Total time: 0.230 seconds, Total memory usage: 4.59MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:02:30 2005 Input: G :=DirichletGroup(15); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],300); Output: Magma V2.11-10 Mon Dec 12 2005 06:02:30 on modular [Seed = 3194004759] ------------------------------------- Group of Dirichlet characters of modulus 15 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] >> Y :=X[16]; Conductor(Y); Order(Y); ^ Runtime error in '[]': Sequence element 16 not defined >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 3, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],300);; ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.34MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:20:10 2005 Input: Factorization(40^2+40+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:20:10 on modular [Seed = 1401770230] ------------------------------------- [ <41, 2> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:59 2005 Input: Factorization(41^2+41+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:59 on modular [Seed = 1485200264] ------------------------------------- [ <41, 1>, <43, 1> ] Total time: 0.180 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:51 2005 Input: Factorization(40^2+40+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:50 on modular [Seed = 1568890500] ------------------------------------- [ <41, 2> ] Total time: 0.180 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:42 2005 Input: Factorization(39^2+39+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:42 on modular [Seed = 1117536643] ------------------------------------- [ <1601, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:34 2005 Input: Factorization(31^2+31+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:34 on modular [Seed = 1201226894] ------------------------------------- [ <1033, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:23 2005 Input: Factorization(26^2+26+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:23 on modular [Seed = 1284657080] ------------------------------------- [ <743, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:15 2005 Input: Factorization(19^2+19+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:15 on modular [Seed = 1907323533] ------------------------------------- [ <421, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:10 2005 Input: Factorization(9^2+9+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:09 on modular [Seed = 1990751636] ------------------------------------- [ <131, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:03 2005 Input: Factorization(4^2+4+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:03 on modular [Seed = 2074441875] ------------------------------------- [ <61, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:18:55 2005 Input: Factorization(3^2+3+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:18:55 on modular [Seed = 1623090032] ------------------------------------- [ <53, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:18:47 2005 Input: Factorization(2^2+2+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:18:47 on modular [Seed = 1706780285] ------------------------------------- [ <47, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:18:40 2005 Input: Factorization(1^2+1+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:18:40 on modular [Seed = 1790208373] ------------------------------------- [ <43, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:18:31 2005 Input: 1^2+1+41 Output: Magma V2.11-10 Mon Dec 12 2005 04:18:31 on modular [Seed = 1873898599] ------------------------------------- 43 Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:12:46 2005 Input: e^(Pi 163^(1/2)) Output: Magma V2.11-10 Mon Dec 12 2005 04:12:46 on modular [Seed = 2671895545] ------------------------------------- >> e^(Pi 163^(1/2)); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:12:36 2005 Input: e^(pi 163^(1/2)) Output: Magma V2.11-10 Mon Dec 12 2005 04:12:35 on modular [Seed = 2220554472] ------------------------------------- >> e^(pi 163^(1/2)); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '70.228.' ************** MAGMA ***************** Host 70.228.67.16 (70.228.67.16) Time: Mon Dec 12 02:46:25 2005 Input: A := Matrix(2,2,[1,2,3,4]) SmithForm(A) Output: Magma V2.11-10 Mon Dec 12 2005 02:46:25 on modular [Seed = 1150955005] ------------------------------------- >> SmithForm(A); ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '70.228.' ************** MAGMA ***************** Host 70.228.67.16 (70.228.67.16) Time: Mon Dec 12 02:45:59 2005 Input: A := Matrix(2,2,[1,2,3,4]) Output: Magma V2.11-10 Mon Dec 12 2005 02:45:59 on modular [Seed = 1334912656] ------------------------------------- Total time: 0.180 seconds, Total memory usage: 3.24MB '70.228.' ************** MAGMA ***************** Host 70.228.67.16 (70.228.67.16) Time: Mon Dec 12 02:45:28 2005 Input: A := Matrix(2,2,[1,2,3,4]) Output: Magma V2.11-10 Mon Dec 12 2005 02:45:28 on modular [Seed = 1251222458] ------------------------------------- Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sun Dec 11 15:31:08 2005 Input: G :=DirichletGroup(1155); G; X :=Elements(G); X; Output: Magma V2.11-10 Sun Dec 11 2005 15:31:07 on modular [Seed = 3026942808] ------------------------------------- Group of Dirichlet characters of modulus 1155 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '155.207' ************** MAGMA ***************** Host 155.207.209.204 (155.207.209.204) Time: Sun Dec 11 13:10:04 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 4*5^4); Output: Magma V2.11-10 Sun Dec 11 2005 13:10:03 on modular [Seed = 1251304016] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [8, 0, -9, 4], [4, -8, -1, 2], [-8, 0, 7, 0], [4, 8, -1, -2], [8, 0, -9, -4], [4, -8, -3, 6], [0, 0, 5, 0], [4, 8, -3, -6], [8, 0, -1, 0] ] Total time: 0.400 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.204 (155.207.209.204) Time: Sun Dec 11 13:09:51 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 4); Output: Magma V2.11-10 Sun Dec 11 2005 13:09:51 on modular [Seed = 2007523193] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [0, 0, 1, 0] ] Total time: 0.340 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.204 (155.207.209.204) Time: Sun Dec 11 13:08:48 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 4*5^4); Output: Magma V2.11-10 Sun Dec 11 2005 13:08:48 on modular [Seed = 2554835146] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [8, 0, -9, 4], [4, -8, -1, 2], [-8, 0, 7, 0], [4, 8, -1, -2], [8, 0, -9, -4], [4, -8, -3, 6], [0, 0, 5, 0], [4, 8, -3, -6], [8, 0, -1, 0] ] Total time: 0.400 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.204 (155.207.209.204) Time: Sun Dec 11 13:08:33 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 3); Output: Magma V2.11-10 Sun Dec 11 2005 13:08:33 on modular [Seed = 2521415381] ------------------------------------- [ 1, y, y^2, y^3 ] false Total time: 0.310 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.204 (155.207.209.204) Time: Sun Dec 11 13:08:13 2005 Input: R := PolynomialRing(Integers()); > > f :=x^8 - 20*x^6 + 98*x^4 - 76*x^2 + 1; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 3); Output: Magma V2.11-10 Sun Dec 11 2005 13:08:12 on modular [Seed = 2371003875] ------------------------------------- [ 1, y, y^2, y^3, 1/8*(y^4 + 4*y^3 + 6*y^2 + 4*y + 7), 1/8*(y^5 + 6*y^3 + 4*y^2 + 7*y + 4), 1/56*(y^6 + 28*y^3 + 35*y^2 + 28*y + 22), 1/112*(y^7 + y^6 + 7*y^5 + 7*y^4 + 21*y^3 + 77*y^2 + 71*y + 15) ] false Total time: 0.520 seconds, Total memory usage: 3.90MB '84.167.' ************** MAGMA ***************** Host 84.167.225.156 (84.167.225.156) Time: Sun Dec 11 10:16:20 2005 Input: F2 := FiniteField(2); P := PolynomialRing(F2); p := x^120 + x^41 + x^35 + x^32 + 1; F := ext< F2 | p >; a := z; E := EllipticCurve([1, 0, 0, 0, a]); time #E; FactoredOrder(E); Output: Magma V2.11-10 Sun Dec 11 2005 10:16:19 on modular [Seed = 3707952194] ------------------------------------- 1329227995784915872847313096830369792 Time: 0.290 [ <2, 14>, <26113, 1>, <368539681, 1>, <8430212384348387921, 1> ] Total time: 0.620 seconds, Total memory usage: 3.82MB '85.226.' ************** MAGMA ***************** Host 85.226.120.223 (85.226.120.223) Time: Sun Dec 11 09:41:05 2005 Input: R := PolynomialRing(RationalField(),1); I := ideal< R | (x - 1)^2*x>; S := R/I; S; IsNilpotent((x-1)^2); Output: Magma V2.11-10 Sun Dec 11 2005 09:41:04 on modular [Seed = 1468542603] ------------------------------------- Affine Algebra of rank 1 over Rational Field Lexicographical Order Variables: x Quotient relations: [ x^3 - 2*x^2 + x ] Magma: Internal error Please mail this entire run [**WITH THE FOLLOWING LINES**] to magma-bugs@maths.usyd.edu.au Version: 2.11-10 Link date: Thu Nov 4 20:39:55 EST 2004 Machine type: x86_64-linux Initial seed: 1468542603 Time to this point: 0.19 Memory usage: 3.24MB Internal error in glue_ring_elt_is_nilpotent() at ring/elt_glue.c, line 367 Total time: 0.190 seconds, Total memory usage: 3.24MB '85.226.' ************** MAGMA ***************** Host 85.226.120.223 (85.226.120.223) Time: Sun Dec 11 09:40:50 2005 Input: R := PolynomialRing(RationalField(),1); I := ideal< R | (x - 1)^2*x>; S := R/I; S; Output: Magma V2.11-10 Sun Dec 11 2005 09:40:49 on modular [Seed = 1518674807] ------------------------------------- Affine Algebra of rank 1 over Rational Field Lexicographical Order Variables: x Quotient relations: [ x^3 - 2*x^2 + x ] Total time: 0.190 seconds, Total memory usage: 3.24MB '85.226.' ************** MAGMA ***************** Host 85.226.120.223 (85.226.120.223) Time: Sun Dec 11 09:40:43 2005 Input: R := PolynomialRing(RationalField(),1); I := ideal< R | (x - 1)^2*x>; S := R/I; Output: Magma V2.11-10 Sun Dec 11 2005 09:40:42 on modular [Seed = 1568810091] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '85.226.' ************** MAGMA ***************** Host 85.226.120.223 (85.226.120.223) Time: Sun Dec 11 09:40:14 2005 Input: R := PolyonialRing(RationalField(),1); I := ideal< R | (x - 1)^2*x>; S := R/I; Output: Magma V2.11-10 Sun Dec 11 2005 09:40:13 on modular [Seed = 2793004982] ------------------------------------- >> R := PolyonialRing(RationalField(),1); ^ User error: Identifier 'PolyonialRing' has not been declared or assigned >> I := ideal< R | (x - 1)^2*x>; ^ User error: Identifier 'R' has not been declared or assigned >> S := R/I;; ^ User error: Identifier 'R' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '67.62.1' ************** MAGMA ***************** Host 67.62.112.123 (67.62.112.123) Time: Sun Dec 11 07:14:56 2005 Input: c:=10; e:=180; r:=c*e; r; Output: Magma V2.11-10 Sun Dec 11 2005 07:14:55 on modular [Seed = 3390411179] ------------------------------------- 1800 Total time: 0.190 seconds, Total memory usage: 3.24MB '67.62.1' ************** MAGMA ***************** Host 67.62.112.123 (67.62.112.123) Time: Sun Dec 11 06:40:38 2005 Input: t:=11111111111; y:=11111111111; g:=t*y; g; Output: Magma V2.11-10 Sun Dec 11 2005 06:40:37 on modular [Seed = 2220611561] ------------------------------------- 123456790120987654321 Total time: 0.240 seconds, Total memory usage: 3.24MB '67.62.1' ************** MAGMA ***************** Host 67.62.112.123 (67.62.112.123) Time: Sun Dec 11 06:34:58 2005 Input: c:=11; e:=18; r:=c*e; r; Output: Magma V2.11-10 Sun Dec 11 2005 06:34:58 on modular [Seed = 2421153066] ------------------------------------- 198 Total time: 0.200 seconds, Total memory usage: 3.24MB '67.62.1' ************** MAGMA ***************** Host 67.62.112.123 (67.62.112.123) Time: Sun Dec 11 06:34:47 2005 Input: c:=11; e:=11; r:=c*e; r; Output: Magma V2.11-10 Sun Dec 11 2005 06:34:47 on modular [Seed = 2604979732] ------------------------------------- 121 Total time: 0.200 seconds, Total memory usage: 3.24MB '217.24.' ************** MAGMA ***************** Host 217.24.144.35 (217.24.144.35) Time: Sun Dec 11 06:33:09 2005 Input: c:=11; e:=11; r:=c*e; r; Output: Magma V2.11-10 Sun Dec 11 2005 06:33:09 on modular [Seed = 3778981068] ------------------------------------- 121 Total time: 0.200 seconds, Total memory usage: 3.24MB '217.24.' ************** MAGMA ***************** Host 217.24.144.35 (217.24.144.35) Time: Sun Dec 11 06:27:13 2005 Input: c:=123; e:=11; r:=c*e; r; Output: Magma V2.11-10 Sun Dec 11 2005 06:27:11 on modular [Seed = 3657798797] ------------------------------------- 1353 Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sun Dec 11 04:38:47 2005 Input: G :=DirichletGroup(175,CyclotomicField(4));G; X :=Elements(G);X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],104);Parent($1); Output: Magma V2.11-10 Sun Dec 11 2005 04:38:47 on modular [Seed = 921601736] ------------------------------------- Group of Dirichlet characters of modulus 175 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.1^2, $.1^3, $.2, $.1*$.2, $.1^2*$.2, $.1^3*$.2 ] 35 4 [ Modular symbols space of level 175, weight 2, character $.1*$.2, and dimension 2 over Cyclotomic Field of order 4 and degree 2, Modular symbols space of level 175, weight 2, character $.1*$.2, and dimension 2 over Cyclotomic Field of order 4 and degree 2, Modular symbols space of level 175, weight 2, character $.1*$.2, and dimension 2 over Cyclotomic Field of order 4 and degree 2, Modular symbols space of level 175, weight 2, character $.1*$.2, and dimension 4 over Cyclotomic Field of order 4 and degree 2 ] q - 1/3*a*q^3 - 2*zeta_4*q^4 + 1/3*zeta_4*a*q^7 + 4*zeta_4*q^9 - 3*q^11 + 2/3*zeta_4*a*q^12 - 1/3*a*q^13 - 4*q^16 - zeta_4*a*q^17 + 7*q^21 - 1/3*zeta_4*a*q^27 + 2/3*a*q^28 - 9*zeta_4*q^29 + a*q^33 + 8*q^36 + 7*zeta_4*q^39 + 6*zeta_4*q^44 - zeta_4*a*q^47 + 4/3*a*q^48 - 7*zeta_4*q^49 - 21*q^51 + 2/3*zeta_4*a*q^52 - 4/3*a*q^63 + 8*zeta_4*q^64 - 2*a*q^68 + 12*q^71 + 4/3*a*q^73 - zeta_4*a*q^77 + zeta_4*q^79 + 5*q^81 - 2*a*q^83 - 14*zeta_4*q^84 + 3*zeta_4*a*q^87 + 7*q^91 + 7/3*zeta_4*a*q^97 - 12*zeta_4*q^99 - 1/3*a*q^103 + O(q^104) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 4 and degree 2 with modulus a^2 - 63*zeta_4 Total time: 0.530 seconds, Total memory usage: 5.46MB '83.194.' ************** MAGMA ***************** Host 83.194.162.38 (83.194.162.38) Time: Sun Dec 11 03:35:56 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Sun Dec 11 2005 03:35:55 on modular [Seed = 4147073094] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.190 seconds, Total memory usage: 3.24MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Sat Dec 10 16:55:59 2005 Input: Q:=GaloisField(35098201); P:=PolynomialRing(Q,2); I:=ideal; Groebner(I); I; Output: Magma V2.11-10 Sat Dec 10 2005 16:55:56 on modular [Seed = 3290220920] ------------------------------------- Ideal of Polynomial ring of rank 2 over GF(35098201) Lexicographical Order Variables: x, y Dimension 0 Groebner basis: [ x + 33784728*y^33 + 15744019*y^32 + 14466235*y^31 + 14937582*y^30 + 9988153*y^29 + 24849537*y^28 + 13827463*y^27 + 10851940*y^26 + 25333828*y^25 + 29238403*y^24 + 35087366*y^23 + 3185785*y^22 + 12125255*y^21 + 11305600*y^20 + 713800*y^19 + 11882241*y^18 + 23388419*y^17 + 12677392*y^16 + 20159861*y^15 + 31143912*y^14 + 33062327*y^13 + 11580434*y^12 + 10629964*y^11 + 14094725*y^10 + 30606411*y^9 + 20913610*y^8 + 23355486*y^7 + 32139384*y^6 + 35026862*y^5 + 11038274*y^4 + 26690476*y^3 + 752845*y^2 + 9514380*y + 16409093, y^34 + 34*y^33 + 561*y^32 + 5984*y^31 + 46376*y^30 + 278256*y^29 + 1344904*y^28 + 5379616*y^27 + 18156204*y^26 + 17353055*y^25 + 25833537*y^24 + 5312152*y^23 + 21881025*y^22 + 15430534*y^21 + 23145801*y^20 + 30861068*y^19 + 27872968*y^18 + 17124952*y^17 + 27873223*y^16 + 30858484*y^15 + 23140973*y^14 + 15452362*y^13 + 21951303*y^12 + 5362540*y^11 + 25762851*y^10 + 17184126*y^9 + 18014271*y^8 + 5332492*y^7 + 1358334*y^6 + 297330*y^5 + 54824*y^4 + 7259*y^3 + 714*y^2 + 34*y + 1 ] Total time: 0.240 seconds, Total memory usage: 3.43MB '84.59.7' ************** MAGMA ***************** Host 84.59.79.231 (84.59.79.231) Time: Sat Dec 10 11:35:50 2005 Input: factor(1111111111111) Output: Magma V2.11-10 Sat Dec 10 2005 11:35:50 on modular [Seed = 3895502914] ------------------------------------- >> factor(1111111111111); ^ User error: Identifier 'factor' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '86.138.' ************** MAGMA ***************** Host 86.138.172.89 (86.138.172.89) Time: Sat Dec 10 11:04:37 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Sat Dec 10 2005 11:04:37 on modular [Seed = 654285311] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.200 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:31:44 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; assert (Vh ne 0) or (Uh eq 0 and Vh eq 0 and Vh eq -2*Qh); else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; assert (Vl ne 0) or (Uh eq -Ql and Vh eq 0 and Vl eq -2*Ql); end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; assert Uh*Vl eq 0; return Uh, Vl; end function; for i, j in [1..10] do i, ":", Lucas(j, i); end for; Output: Magma V2.11-10 Sat Dec 10 2005 10:31:43 on modular [Seed = 1940571360] ------------------------------------- 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 2 : 0 -2 2 : 0 -4 2 : 0 -6 2 : 0 -8 2 : 0 -10 2 : 0 -12 2 : 0 -14 2 : 0 -16 2 : 0 -18 2 : 0 -20 3 : -1 0 3 : -2 0 3 : -3 0 3 : -4 0 3 : -5 0 3 : -6 0 3 : -7 0 3 : -8 0 3 : -9 0 3 : -10 0 4 : 0 2 4 : 0 8 4 : 0 18 4 : 0 32 4 : 0 50 4 : 0 72 4 : 0 98 4 : 0 128 4 : 0 162 4 : 0 200 5 : 1 0 5 : 4 0 5 : 9 0 5 : 16 0 5 : 25 0 5 : 36 0 5 : 49 0 5 : 64 0 5 : 81 0 5 : 100 0 6 : 0 -2 6 : 0 -16 6 : 0 -54 6 : 0 -128 6 : 0 -250 6 : 0 -432 6 : 0 -686 6 : 0 -1024 6 : 0 -1458 6 : 0 -2000 7 : -1 0 7 : -8 0 7 : -27 0 7 : -64 0 7 : -125 0 7 : -216 0 7 : -343 0 7 : -512 0 7 : -729 0 7 : -1000 0 8 : 0 2 8 : 0 32 8 : 0 162 8 : 0 512 8 : 0 1250 8 : 0 2592 8 : 0 4802 8 : 0 8192 8 : 0 13122 8 : 0 20000 9 : 1 0 9 : 16 0 9 : 81 0 9 : 256 0 9 : 625 0 9 : 1296 0 9 : 2401 0 9 : 4096 0 9 : 6561 0 9 : 10000 0 10 : 0 -2 10 : 0 -64 10 : 0 -486 10 : 0 -2048 10 : 0 -6250 10 : 0 -15552 10 : 0 -33614 10 : 0 -65536 10 : 0 -118098 10 : 0 -200000 Total time: 0.200 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:28:35 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; assert Uh*Vl eq 0; return Uh, Vl; end function; for i, j in [1..10] do i, ":", Lucas(j, i); end for; Output: Magma V2.11-10 Sat Dec 10 2005 10:28:34 on modular [Seed = 2141636625] ------------------------------------- 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 2 : 0 -2 2 : 0 -4 2 : 0 -6 2 : 0 -8 2 : 0 -10 2 : 0 -12 2 : 0 -14 2 : 0 -16 2 : 0 -18 2 : 0 -20 3 : -1 0 3 : -2 0 3 : -3 0 3 : -4 0 3 : -5 0 3 : -6 0 3 : -7 0 3 : -8 0 3 : -9 0 3 : -10 0 4 : 0 2 4 : 0 8 4 : 0 18 4 : 0 32 4 : 0 50 4 : 0 72 4 : 0 98 4 : 0 128 4 : 0 162 4 : 0 200 5 : 1 0 5 : 4 0 5 : 9 0 5 : 16 0 5 : 25 0 5 : 36 0 5 : 49 0 5 : 64 0 5 : 81 0 5 : 100 0 6 : 0 -2 6 : 0 -16 6 : 0 -54 6 : 0 -128 6 : 0 -250 6 : 0 -432 6 : 0 -686 6 : 0 -1024 6 : 0 -1458 6 : 0 -2000 7 : -1 0 7 : -8 0 7 : -27 0 7 : -64 0 7 : -125 0 7 : -216 0 7 : -343 0 7 : -512 0 7 : -729 0 7 : -1000 0 8 : 0 2 8 : 0 32 8 : 0 162 8 : 0 512 8 : 0 1250 8 : 0 2592 8 : 0 4802 8 : 0 8192 8 : 0 13122 8 : 0 20000 9 : 1 0 9 : 16 0 9 : 81 0 9 : 256 0 9 : 625 0 9 : 1296 0 9 : 2401 0 9 : 4096 0 9 : 6561 0 9 : 10000 0 10 : 0 -2 10 : 0 -64 10 : 0 -486 10 : 0 -2048 10 : 0 -6250 10 : 0 -15552 10 : 0 -33614 10 : 0 -65536 10 : 0 -118098 10 : 0 -200000 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:27:19 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; assert Uh*Vl eq 0; return Uh, Vl; end function; for i in [1..100] do i, ":", Lucas(5, i); end for; Output: Magma V2.11-10 Sat Dec 10 2005 10:27:18 on modular [Seed = 1639502331] ------------------------------------- 1 : 1 0 2 : 0 -10 3 : -5 0 4 : 0 50 5 : 25 0 6 : 0 -250 7 : -125 0 8 : 0 1250 9 : 625 0 10 : 0 -6250 11 : -3125 0 12 : 0 31250 13 : 15625 0 14 : 0 -156250 15 : -78125 0 16 : 0 781250 17 : 390625 0 18 : 0 -3906250 19 : -1953125 0 20 : 0 19531250 21 : 9765625 0 22 : 0 -97656250 23 : -48828125 0 24 : 0 488281250 25 : 244140625 0 26 : 0 -2441406250 27 : -1220703125 0 28 : 0 12207031250 29 : 6103515625 0 30 : 0 -61035156250 31 : -30517578125 0 32 : 0 305175781250 33 : 152587890625 0 34 : 0 -1525878906250 35 : -762939453125 0 36 : 0 7629394531250 37 : 3814697265625 0 38 : 0 -38146972656250 39 : -19073486328125 0 40 : 0 190734863281250 41 : 95367431640625 0 42 : 0 -953674316406250 43 : -476837158203125 0 44 : 0 4768371582031250 45 : 2384185791015625 0 46 : 0 -23841857910156250 47 : -11920928955078125 0 48 : 0 119209289550781250 49 : 59604644775390625 0 50 : 0 -596046447753906250 51 : -298023223876953125 0 52 : 0 2980232238769531250 53 : 1490116119384765625 0 54 : 0 -14901161193847656250 55 : -7450580596923828125 0 56 : 0 74505805969238281250 57 : 37252902984619140625 0 58 : 0 -372529029846191406250 59 : -186264514923095703125 0 60 : 0 1862645149230957031250 61 : 931322574615478515625 0 62 : 0 -9313225746154785156250 63 : -4656612873077392578125 0 64 : 0 46566128730773925781250 65 : 23283064365386962890625 0 66 : 0 -232830643653869628906250 67 : -116415321826934814453125 0 68 : 0 1164153218269348144531250 69 : 582076609134674072265625 0 70 : 0 -5820766091346740722656250 71 : -2910383045673370361328125 0 72 : 0 29103830456733703613281250 73 : 14551915228366851806640625 0 74 : 0 -145519152283668518066406250 75 : -72759576141834259033203125 0 76 : 0 727595761418342590332031250 77 : 363797880709171295166015625 0 78 : 0 -3637978807091712951660156250 79 : -1818989403545856475830078125 0 80 : 0 18189894035458564758300781250 81 : 9094947017729282379150390625 0 82 : 0 -90949470177292823791503906250 83 : -45474735088646411895751953125 0 84 : 0 454747350886464118957519531250 85 : 227373675443232059478759765625 0 86 : 0 -2273736754432320594787597656250 87 : -1136868377216160297393798828125 0 88 : 0 11368683772161602973937988281250 89 : 5684341886080801486968994140625 0 90 : 0 -56843418860808014869689941406250 91 : -28421709430404007434844970703125 0 92 : 0 284217094304040074348449707031250 93 : 142108547152020037174224853515625 0 94 : 0 -1421085471520200371742248535156250 95 : -710542735760100185871124267578125 0 96 : 0 7105427357601001858711242675781250 97 : 3552713678800500929355621337890625 0 98 : 0 -35527136788005009293556213378906250 99 : -17763568394002504646778106689453125 0 100 : 0 177635683940025046467781066894531250 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:27:02 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; assert Uh*Vl eq 0; return Uh, Vl; end function; for i in [1, 100] do i, ":", Lucas(5, i); end for; Output: Magma V2.11-10 Sat Dec 10 2005 10:27:01 on modular [Seed = 1690162413] ------------------------------------- 1 : 1 0 100 : 0 177635683940025046467781066894531250 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:26:01 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0 and Uh eq -Ql); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; assert Uh*Vl eq 0; return Uh, Vl; end function; for i in [1, 100] do i, ":", Lucas(5, i); end for; Output: Magma V2.11-10 Sat Dec 10 2005 10:26:01 on modular [Seed = 1874255786] ------------------------------------- 1 : 1 0 Lucas( Q: 5, k: 100 ) >> assert (Vl eq 0 and Uh eq 0) or (Vh eq 0 and Uh eq -Ql); ^ Runtime error in assert: Assertion failed Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:24:16 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0 and Uh eq -Ql); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 1); Lucas(5, 2); Lucas(5, 3); Lucas(5, 4); Lucas(5, 5); Lucas(5, 6); Lucas(5, 7); Lucas(5, 8); Output: Magma V2.11-10 Sat Dec 10 2005 10:24:16 on modular [Seed = 1384899134] ------------------------------------- 1 0 0 -10 -5 0 0 50 25 0 0 -250 -125 0 0 1250 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:23:11 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 1); Lucas(5, 2); Lucas(5, 3); Lucas(5, 4); Lucas(5, 5); Lucas(5, 6); Lucas(5, 7); Lucas(5, 8); Output: Magma V2.11-10 Sat Dec 10 2005 10:23:10 on modular [Seed = 1535306441] ------------------------------------- 1 0 0 -10 -5 0 0 50 25 0 0 -250 Lucas( Q: 5, k: 7 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq ^ Runtime error in assert: Assertion failed 0 1250 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:22:25 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); end for; assert Vl*Vh eq 0; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; //Lucas(5, 1); //Lucas(5, 2); Lucas(5, 3); Lucas(5, 4); Lucas(5, 5); Lucas(5, 6); Lucas(5, 7); Lucas(5, 8); Output: Magma V2.11-10 Sat Dec 10 2005 10:22:25 on modular [Seed = 1552280993] ------------------------------------- -5 0 Lucas( Q: 5, k: 4 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql ^ Runtime error in assert: Assertion failed 25 0 0 -250 Lucas( Q: 5, k: 7 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq ^ Runtime error in assert: Assertion failed Lucas( Q: 5, k: 8 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql ^ Runtime error in assert: Assertion failed Total time: 0.200 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:21:29 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); end for; assert Vl*Vh eq 0; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 1); Lucas(5, 2); Lucas(5, 3); Lucas(5, 4); Lucas(5, 5); Output: Magma V2.11-10 Sat Dec 10 2005 10:21:29 on modular [Seed = 3010069127] ------------------------------------- Lucas( Q: 5, k: 1 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql ^ Runtime error in assert: Assertion failed Lucas( Q: 5, k: 2 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql ^ Runtime error in assert: Assertion failed -5 0 Lucas( Q: 5, k: 4 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql ^ Runtime error in assert: Assertion failed 25 0 Total time: 0.190 seconds, Total memory usage: 3.34MB '80.184.' ************** MAGMA ***************** Host 80.184.165.238 (80.184.165.238) Time: Sat Dec 10 10:21:23 2005 Input: gcd(105;384) Output: Magma V2.11-10 Sat Dec 10 2005 10:21:23 on modular [Seed = 3027043722] ------------------------------------- >> gcd(105;384); ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:14:32 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl eq 0 or Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); end for; Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 3); Output: Magma V2.11-10 Sat Dec 10 2005 10:14:31 on modular [Seed = 2521769228] ------------------------------------- -5 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:10:40 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl eq 0 or Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; end for; Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 3); Output: Magma V2.11-10 Sat Dec 10 2005 10:10:40 on modular [Seed = 2554410021] ------------------------------------- -5 0 Total time: 0.210 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:09:26 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; end for; Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 3); Output: Magma V2.11-10 Sat Dec 10 2005 10:09:25 on modular [Seed = 2638492021] ------------------------------------- -5 0 Total time: 0.200 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:07:32 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; TestBit := function(k, j) return (2^j and k) ne 0; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do Ql *:= Qh; if TestBit(k, j) then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; end for; Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 3); Output: Magma V2.11-10 Sat Dec 10 2005 10:07:31 on modular [Seed = 4045954447] ------------------------------------- Lucas( Q: 5, k: 3 ) TestBit( k: 3, j: 1 ) >> return (2^j and k) ne 0; ^ Runtime error: Expected a logical for the 'and' operator Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sat Dec 10 08:40:00 2005 Input: K := FiniteField(2); > C := LinearCode; IsSelfOrthogonal(C); WeightDistribution(C); Output: Magma V2.11-10 Sat Dec 10 2005 08:38:02 on modular [Seed = 2023874932] ------------------------------------- false [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] Total time: 0.100 seconds, Total memory usage: 3.34MB '219.108' ************** MAGMA ***************** Host 219.108.73.204 (219.108.73.204) Time: Sat Dec 10 06:55:26 2005 Input: V:=EvenWeightCode(27); e1:=CharacteristicVector(VectorSpace(GF(2),27),{1}); // for v1 in V do v1:=Random(V); x:=v1+e1; v:=Vector(GF(2),56,[0: i in [1..28]] cat [1] cat Eltseq(x)); v; Weight(v); Output: Magma V2.11-10 Sat Dec 10 2005 06:55:10 on modular [Seed = 1267911207] ------------------------------------- (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 1 0) 16 Total time: 0.180 seconds, Total memory usage: 3.24MB '219.108' ************** MAGMA ***************** Host 219.108.73.204 (219.108.73.204) Time: Sat Dec 10 06:54:52 2005 Input: V:=EvenWeightCode(27); e1:=CharacteristicVector(VectorSpace(GF(2),27),{1}); // for v1 in V do v1:=Random(V); x:=v1+e1; v:=Vector(GF(2),56,[0: i in [1..28]] cat [1] cat Eltseq(x)); v; Output: Magma V2.11-10 Sat Dec 10 2005 06:54:42 on modular [Seed = 2959684093] ------------------------------------- (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0) Total time: 0.200 seconds, Total memory usage: 3.24MB '219.108' ************** MAGMA ***************** Host 219.108.73.204 (219.108.73.204) Time: Sat Dec 10 06:54:47 2005 Input: V:=EvenWeightCode(27); e1:=CharacteristicVector(VectorSpace(GF(2),27),{1}); // for v1 in V do v1:=Random(V); x:=v1+e1; v:=Vector(GF(2),56,[0: i in [1..28]] cat [1] cat Eltseq(x)); v; Output: Magma V2.11-10 Sat Dec 10 2005 06:54:08 on modular [Seed = 3010081052] ------------------------------------- (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 0) Total time: 0.180 seconds, Total memory usage: 3.24MB '219.108' ************** MAGMA ***************** Host 219.108.73.204 (219.108.73.204) Time: Sat Dec 10 06:53:02 2005 Input: V:=EvenWeightCode(27); e1:=CharacteristicVector(VectorSpace(GF(2),27),{1}); // for v1 in V do v1:=Randam(V); x:=v1+e1; v:=Vector(GF(2),56,[0: i in [1..28]] cat [1] cat Eltseq(x)); v; Output: Magma V2.11-10 Sat Dec 10 2005 06:52:59 on modular [Seed = 3143776623] ------------------------------------- >> v1:=Randam(V); ^ User error: Identifier 'Randam' has not been declared or assigned >> x:=v1+e1; ^ User error: Identifier 'v1' has not been declared or assigned >> v:=Vector(GF(2),56,[0: i in [1..28]] cat [1] cat Eltseq(x)); ^ User error: Identifier 'x' has not been declared or assigned >> v; ^ User error: Identifier 'v' has not been declared or assigned Total time: 0.220 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 10 03:43:11 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 6, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 10 2005 03:42:50 on modular [Seed = 3177475070] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 Errors: /bin/sh: line 1: 2804 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 10 03:21:17 2005 Input: G :=DirichletGroup(610); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 6, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Sat Dec 10 2005 03:21:10 on modular [Seed = 3256553084] ------------------------------------- Group of Dirichlet characters of modulus 610 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 1 1 Current total memory usage: 68.2MB, failed memory request: 29.7MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned Total time: 3.600 seconds, Total memory usage: 68.18MB '69.175.' ************** MAGMA ***************** Host 69.175.68.230 (69.175.68.230) Time: Fri Dec 9 12:20:25 2005 Input: x = 1 + 2 Output: Magma V2.11-10 Fri Dec 9 2005 12:20:25 on modular [Seed = 1201518809] ------------------------------------- >> x = 1 + 2; ^ User error: Identifier 'x' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '87.69.5' ************** MAGMA ***************** Host 87.69.58.225 (87.69.58.225) Time: Fri Dec 9 12:12:21 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Fri Dec 9 2005 12:12:20 on modular [Seed = 2621859112] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.200 seconds, Total memory usage: 3.24MB '152.6.1' ************** MAGMA ***************** Host 152.6.19.192 (152.6.19.192) Time: Fri Dec 9 11:40:12 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 11:40:12 on modular [Seed = 2170253740] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.200 seconds, Total memory usage: 3.24MB '152.6.1' ************** MAGMA ***************** Host 152.6.19.192 (152.6.19.192) Time: Fri Dec 9 11:39:02 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 11:39:01 on modular [Seed = 4196403870] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.200 seconds, Total memory usage: 3.24MB '152.6.1' ************** MAGMA ***************** Host 152.6.19.192 (152.6.19.192) Time: Fri Dec 9 10:46:16 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); WeightDistribution(C); Output: Magma V2.11-10 Fri Dec 9 2005 10:46:15 on modular [Seed = 804637353] ------------------------------------- 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } false [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] Total time: 0.310 seconds, Total memory usage: 5.51MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Fri Dec 9 09:21:10 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); WeightDistribution(C); Output: Magma V2.11-10 Fri Dec 9 2005 09:21:09 on modular [Seed = 2554480911] ------------------------------------- 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } false [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] Total time: 0.300 seconds, Total memory usage: 5.51MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Fri Dec 9 09:16:23 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 09:16:23 on modular [Seed = 2404736378] ------------------------------------- 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } false Total time: 0.270 seconds, Total memory usage: 5.51MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Fri Dec 9 08:49:20 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 08:49:19 on modular [Seed = 3996395251] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.200 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Fri Dec 9 08:48:35 2005 Input: K := FiniteField(2); > C := LinearCode [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] > [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 08:48:35 on modular [Seed = 3829279893] ------------------------------------- >> [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] ^ User error: bad syntax >> [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0]>; ^ User error: bad syntax >> aut := AutomorphismGroup(C); ^ User error: Identifier 'C' has not been declared or assigned >> Order(aut); ^ User error: Identifier 'aut' has not been declared or assigned >> FactoredOrder(aut); ^ User error: Identifier 'aut' has not been declared or assigned >> CompositionFactors(aut); ^ User error: Identifier 'aut' has not been declared or assigned >> Generators(aut); ^ User error: Identifier 'aut' has not been declared or assigned >> IsSelfOrthogonal(C);; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Fri Dec 9 08:46:50 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 08:46:50 on modular [Seed = 3812174443] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.190 seconds, Total memory usage: 3.24MB '141.20.' ************** MAGMA ***************** Host 141.20.57.85 (141.20.57.85) Time: Fri Dec 9 08:35:26 2005 Input: H := PermutationGroup< 5 | (1,2,3,5),(1,2,3,4) >; H; FPGroup(H); Output: Magma V2.11-10 Fri Dec 9 2005 08:35:26 on modular [Seed = 3474261139] ------------------------------------- Permutation group H acting on a set of cardinality 5 (1, 2, 3, 5) (1, 2, 3, 4) Finitely presented group on 2 generators Relations $.1^4 = Id($) $.2^4 = Id($) ($.2^-1 * $.1)^3 = Id($) ($.1^-1, $.2^-1)^2 = Id($) $.2^-1 * $.1^-2 * $.2^-1 * $.1^2 * $.2^2 * $.1^-1 * $.2^-1 = Id($) Total time: 0.190 seconds, Total memory usage: 3.34MB '141.20.' ************** MAGMA ***************** Host 141.20.57.85 (141.20.57.85) Time: Fri Dec 9 08:31:58 2005 Input: H := PermutationGroup< 9 | (1,2,4)(5,6,8)(3,9,7), (4,5,6)(7,9,8) >; H; FPGroup(H); Output: Magma V2.11-10 Fri Dec 9 2005 08:31:57 on modular [Seed = 3390049989] ------------------------------------- Permutation group H acting on a set of cardinality 9 (1, 2, 4)(3, 9, 7)(5, 6, 8) (4, 5, 6)(7, 9, 8) Finitely presented group on 2 generators Relations $.1^-3 = Id($) $.2^-3 = Id($) ($.1^-1 * $.2^-1)^4 = Id($) $.1^-1 * $.2^-1 * $.1 * $.2^-1 * $.1^-1 * $.2^-1 * $.1 * $.2 * $.1^-1 * $.2 * $.1 * $.2 = Id($) Total time: 0.190 seconds, Total memory usage: 3.34MB '141.20.' ************** MAGMA ***************** Host 141.20.57.85 (141.20.57.85) Time: Fri Dec 9 08:31:32 2005 Input: H := PermutationGroup< 9 | (1,2,4)(5,6,8)(3,9,7), (4,5,6)(7,9,8) >; H; Output: Magma V2.11-10 Fri Dec 9 2005 08:31:32 on modular [Seed = 3323724464] ------------------------------------- Permutation group H acting on a set of cardinality 9 (1, 2, 4)(3, 9, 7)(5, 6, 8) (4, 5, 6)(7, 9, 8) Total time: 0.200 seconds, Total memory usage: 3.24MB '141.20.' ************** MAGMA ***************** Host 141.20.57.85 (141.20.57.85) Time: Fri Dec 9 08:30:49 2005 Input: H := PermutationGroup< 9 | (1,2,4)(5,6,8)(3,9,7), (4,5,6)(7,9,8) >; Output: Magma V2.11-10 Fri Dec 9 2005 08:30:49 on modular [Seed = 1038612854] ------------------------------------- Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 08:00:24 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*b)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*b*(b+c-1)*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 08:00:24 on modular [Seed = 2254223954] ------------------------------------- [ , , , ] [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:59:55 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*b)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a*(a+c-1)*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:59:54 on modular [Seed = 1267824573] ------------------------------------- [ , , , ] [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:57:36 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a*(a+c-1)*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:57:35 on modular [Seed = 1622826083] ------------------------------------- [ , , , ] [ , ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:57:26 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a*(a+c-1)*(a+b-1)*(a+b+c-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:57:26 on modular [Seed = 1789941344] ------------------------------------- [ , , , ] [ ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:56:29 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a*(a+c-1)*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:56:28 on modular [Seed = 2057324474] ------------------------------------- [ , , , ] [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:56:13 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*(a+c-1)*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:56:13 on modular [Seed = 64569042] ------------------------------------- [ , , , ] [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:55:35 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*(a+c-1)*(a*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:55:34 on modular [Seed = 181417968] ------------------------------------- [ , , , ] >> print Factorization(a*(a+c-1)*(a*(a+b-1)-(4*a*b*c-a-b-c+1));; ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:55:18 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*(a+c-1)*(a+b+c-1)*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:55:17 on modular [Seed = 331951770] ------------------------------------- [ ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:55:08 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*(a+c-1)*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:55:08 on modular [Seed = 365897861] ------------------------------------- [ ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:54:40 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a^2*b^2*c^3*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:54:40 on modular [Seed = 482746800] ------------------------------------- [ ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:54:23 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a^3*b^3*c^3*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:54:23 on modular [Seed = 637474918] ------------------------------------- [ ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:54:03 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*b*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:54:02 on modular [Seed = 771166331] ------------------------------------- [ ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:52:46 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*b*c*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:52:45 on modular [Seed = 1021706525] ------------------------------------- [ ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:49:19 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a^2*b^2*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:49:19 on modular [Seed = 3608013221] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:48:58 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a^2*b^2*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:57 on modular [Seed = 3724861658] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:40 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(1/2*a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:40 on modular [Seed = 3778797556] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:33 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-1/2*a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:33 on modular [Seed = 3945911799] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:26 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-1/4*a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:26 on modular [Seed = 3979858890] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:20 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(1/4*a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:20 on modular [Seed = 4146973150] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:13 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(1/4*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:13 on modular [Seed = 4163028680] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:07 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-1/4*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:06 on modular [Seed = 2186853560] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:01 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-1/2*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:01 on modular [Seed = 2271066555] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:47:55 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(1/2*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:47:54 on modular [Seed = 2303964750] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:47:36 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(16*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:47:35 on modular [Seed = 2471078985] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:47:27 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(8*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:47:27 on modular [Seed = 2505026167] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:47:13 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(6*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:47:13 on modular [Seed = 2672140403] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:47:05 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(6*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:47:05 on modular [Seed = 2709233237] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:46:59 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(8*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:58 on modular [Seed = 2876347476] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:42 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-2*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:41 on modular [Seed = 2910294561] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:34 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-2*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:34 on modular [Seed = 3077408800] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:29 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:28 on modular [Seed = 3093464865] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:20 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:19 on modular [Seed = 1117289956] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:12 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-a*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:12 on modular [Seed = 1201502446] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:05 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:04 on modular [Seed = 1234401016] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:45:41 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-4*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:45:41 on modular [Seed = 1351249688] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:45:33 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-4*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:45:32 on modular [Seed = 1435462148] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:45:24 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-2*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:45:23 on modular [Seed = 1602576385] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:45:18 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:45:18 on modular [Seed = 1639669250] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:45:05 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:45:05 on modular [Seed = 1806783495] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:58 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a^2*b^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:58 on modular [Seed = 1823888155] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:51 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a^2*b^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:51 on modular [Seed = 1991002376] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:45 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a^2*b^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:44 on modular [Seed = 2074166395] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:37 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a^2*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:36 on modular [Seed = 2108113524] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:12 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:12 on modular [Seed = 131938342] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:06 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:06 on modular [Seed = 164836898] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:59 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:59 on modular [Seed = 315108640] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:53 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:53 on modular [Seed = 399321123] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:47 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:47 on modular [Seed = 415377190] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:41 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:40 on modular [Seed = 499590620] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:18 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:17 on modular [Seed = 670900223] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:11 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:11 on modular [Seed = 686952650] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:42:47 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:42:46 on modular [Seed = 854067913] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:39:45 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:39:44 on modular [Seed = 3239836143] ------------------------------------- [ , , , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:39:28 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(2*a-1)*(a+b+c-1); print Factorization(s^2-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:39:28 on modular [Seed = 3340632057] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:39:22 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(2*a-1)*(a+b+c-1); print Factorization(s^2-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:39:21 on modular [Seed = 3440899082] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:39:13 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(a-b)*(a+b+c-1); print Factorization(s^2-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:39:13 on modular [Seed = 3490375967] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:38:58 2005 Input: K:=PolynomialRing(RationalField(),3); s:=a-b; print Factorization(s^2-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:38:54 on modular [Seed = 3812219395] ------------------------------------- [ ] Total time: 0.240 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:30:06 2005 Input: F := GF(8); a :=PrimitiveElement(F); (1+a)^(-1); Output: Magma V2.11-10 Fri Dec 9 2005 00:30:05 on modular [Seed = 4213288604] ------------------------------------- F.1^4 Total time: 0.190 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:29:43 2005 Input: F := GF(8); a :=PrimitiveElement(F); a^(-1); Output: Magma V2.11-10 Fri Dec 9 2005 00:29:43 on modular [Seed = 2254230523] ------------------------------------- F.1^6 Total time: 0.190 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:29:07 2005 Input: F := GF(8); a :=PrimitiveElement(F); Inverse(a); Output: Magma V2.11-10 Fri Dec 9 2005 00:29:07 on modular [Seed = 2371344532] ------------------------------------- >> Inverse(a);; ^ Runtime error in 'Inverse': Argument 1 must be a variable reference (use ~) Argument types given: FldFinElt Total time: 0.190 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:28:02 2005 Input: F := GF(8); a :=PrimitiveElement(F); Trace(a+1); Output: Magma V2.11-10 Fri Dec 9 2005 00:28:02 on modular [Seed = 2337398099] ------------------------------------- 1 Total time: 0.180 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:26:31 2005 Input: F := GF(8); a :=PrimitiveElement(F); Trace(a); Output: Magma V2.11-10 Fri Dec 9 2005 00:26:31 on modular [Seed = 2521612331] ------------------------------------- 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:25:50 2005 Input: F := GF(8); a :=PrimitiveElement(F); a; Output: Magma V2.11-10 Fri Dec 9 2005 00:25:50 on modular [Seed = 2638728185] ------------------------------------- F.1 Total time: 0.180 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:25:13 2005 Input: F := GF(8); F; Output: Magma V2.11-10 Fri Dec 9 2005 00:25:12 on modular [Seed = 2554515098] ------------------------------------- Finite field of size 2^3 Total time: 0.180 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:24:46 2005 Input: F := GF(8) Output: Magma V2.11-10 Fri Dec 9 2005 00:24:46 on modular [Seed = 2810048899] ------------------------------------- Total time: 0.200 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:56:47 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfDual(C); Output: Magma V2.11-10 Thu Dec 8 2005 23:56:47 on modular [Seed = 131943090] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:56:22 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Thu Dec 8 2005 23:56:22 on modular [Seed = 47730510] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:55:57 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C) : Code -> BoolElt Output: Magma V2.11-10 Thu Dec 8 2005 23:55:56 on modular [Seed = 164840559] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } >> IsSelfOrthogonal(C) : Code -> BoolElt ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:52:06 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); Output: Magma V2.11-10 Thu Dec 8 2005 23:52:06 on modular [Seed = 298540893] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:50:33 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Output: Magma V2.11-10 Thu Dec 8 2005 23:50:33 on modular [Seed = 482759231] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:49:30 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); > FactoredOrder(aut); > CompositionFactors(aut); Output: Magma V2.11-10 Thu Dec 8 2005 23:49:30 on modular [Seed = 465655035] ------------------------------------- [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:43:43 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; AutomorphismGroup(C: parameters) : Code -> GrpPerm, PowMap, Map Output: Magma V2.11-10 Thu Dec 8 2005 23:43:42 on modular [Seed = 754345064] ------------------------------------- >> AutomorphismGroup(C: parameters) : Code -> GrpPerm, PowMap, Map ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:40:08 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; IsSelfOrthogonal(C) : Code -> BoolElt Output: Magma V2.11-10 Thu Dec 8 2005 23:40:08 on modular [Seed = 904887463] ------------------------------------- >> IsSelfOrthogonal(C) : Code -> BoolElt ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:39:47 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; IsSelfOrthogonal(C) : Code -> BoolElt; Output: Magma V2.11-10 Thu Dec 8 2005 23:39:47 on modular [Seed = 820672921] ------------------------------------- >> IsSelfOrthogonal(C) : Code -> BoolElt; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:39:10 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; Output: Magma V2.11-10 Thu Dec 8 2005 23:39:10 on modular [Seed = 1072002667] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:38:44 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]; Output: Magma V2.11-10 Thu Dec 8 2005 23:38:44 on modular [Seed = 3474294432] ------------------------------------- >> [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:35:40 2005 Input: C := BCHCode(GF(2),23,2); C; Output: Magma V2.11-10 Thu Dec 8 2005 23:35:40 on modular [Seed = 3607993712] ------------------------------------- [23, 12, 7] "BCH code (d = 2, b = 1)" Cyclic Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1 0 0] [0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0 1] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:34:17 2005 Input: > C := BCHCode(GF(2),23,2); > C; Output: Magma V2.11-10 Thu Dec 8 2005 23:34:17 on modular [Seed = 3691419310] ------------------------------------- [23, 12, 7] "BCH code (d = 2, b = 1)" Cyclic Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1 0 0] [0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0 1] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:10:54 2005 Input: 2+3 Output: Magma V2.11-10 Thu Dec 8 2005 23:10:54 on modular [Seed = 3812205334] ------------------------------------- 5 Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:13:05 2005 Input: Factorization(1111111111111111111111111111111111112222222222222222222222222222222222222222222222222222222222222222223); Output: Magma V2.11-10 Thu Dec 8 2005 22:13:04 on modular [Seed = 1569122342] ------------------------------------- [ <3, 2>, <41, 1>, <191, 1>, <6246943, 1>, <252365633729939524730246496421381170326644302337345845070768633297938261556952\ 8675426211359, 1> ] Total time: 0.210 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:12:31 2005 Input: Factorization(11111111111111111111111111111111111122222222222222222222222222222222222222222222222222222222222222222233333333333333333333333333333333333333333333333333333333333333333333333333333333333333); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Thu Dec 8 2005 22:12:11 on modular [Seed = 1518464874] ------------------------------------- Errors: /bin/sh: line 1: 30817 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:10:50 2005 Input: Factorization(1111111111111111111111111111111111112222222222222222222222222222222222222222222222222222222222222222223333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Thu Dec 8 2005 22:10:29 on modular [Seed = 1740427769] ------------------------------------- Errors: /bin/sh: line 1: 30812 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:09:25 2005 Input: Factorization(1111111111111111111111111111111111112222222222222222222222222222222222222222222222222222222222222222223333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444455555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555566666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Thu Dec 8 2005 22:09:05 on modular [Seed = 1823988230] ------------------------------------- Errors: /bin/sh: line 1: 30807 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:08:02 2005 Input: Factorization(11111111111111111111111111111111111122222222222222222222222222222222222222222222222222222222222222222233333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333334444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666667777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999990); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Thu Dec 8 2005 22:07:41 on modular [Seed = 2024001470] ------------------------------------- Errors: /bin/sh: line 1: 30787 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:06:11 2005 Input: Factorization(111111111111111111111111111111111111222222222222222222222222222222222222222222222222222222222222222222333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333344444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444445555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555556666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666677777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777778888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888889999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Thu Dec 8 2005 22:05:50 on modular [Seed = 115192243] ------------------------------------- Errors: /bin/sh: line 1: 30781 Alarm clock nice -n 19 /usr/local/bin/magma '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:52:57 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2*a-1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:52:57 on modular [Seed = 432184689] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:51:58 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:51:58 on modular [Seed = 315201085] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (a^3 - 4*a^2*b*c + a^2*c - a^2 + 4*a*b^2*c - a*b^2 + a*b - b^2*c)/(a^4 + 2*a^3*b + a^3*c - 3*a^3 + a^2*b^2 + 2*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + a*b^2*c - a*b^2 - 2*a*b*c + 2*a*b + a*c - a) (-a^4 - 2*a^3*b - a^3*c + a^3 - a^2*b^2 + 6*a^2*b*c + 2*a^2*b - a*b^2*c + a*b^2 - 4*a*b*c - a*b + b*c)/(a^4 + 2*a^3*b + a^3*c - 3*a^3 + a^2*b^2 + 2*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + a*b^2*c - a*b^2 - 2*a*b*c + 2*a*b + a*c - a) (4*a^2*b - 4*a*b + b)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a) (-a^4 - 2*a^3*b + a^3*c + 4*a^3 - a^2*b^2 - 6*a^2*b*c + 4*a^2*b - 4*a^2 + a*b^2*c + 4*a*b*c - a*b + a - b*c)/(a^4 + 2*a^3*b + a^3*c - 3*a^3 + a^2*b^2 + 2*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + a*b^2*c - a*b^2 - 2*a*b*c + 2*a*b + a*c - a) (-4*a^3*c - 2*a^3 + 4*a^2*b*c - 2*a^2*b + 6*a^2*c + 3*a^2 - 2*a*b*c + a*b - 4*a*c - a + c)/(a^4 + 2*a^3*b + a^3*c - 3*a^3 + a^2*b^2 + 2*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + a*b^2*c - a*b^2 - 2*a*b*c + 2*a*b + a*c - a) (-2*a^3 + 2*a^2*b + 5*a^2 - a*b - 4*a + 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a) (2*a^2*c - 2*a*b*c - a*c + b*c)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a) (-4*a^2*c + 4*a*c - c)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a) (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:49:04 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(2*(a-b)*(a+b+c-1)*(a+b-1)*(b+c-1)*(c+a-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1))); Output: Magma V2.11-10 Thu Dec 8 2005 18:49:04 on modular [Seed = 720482155] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.370 seconds, Total memory usage: 3.72MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:48:41 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(2*a*b*(a+b+c-1)*(a+b-1)*(b+c-1)*(c+a-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1))); Output: Magma V2.11-10 Thu Dec 8 2005 18:48:41 on modular [Seed = 670868047] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.370 seconds, Total memory usage: 3.82MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:48:11 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(2*a*b*c*(a+b+c-1)*(a+b-1)*(b+c-1)*(c+a-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1))); Output: Magma V2.11-10 Thu Dec 8 2005 18:48:10 on modular [Seed = 1055243061] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.380 seconds, Total memory usage: 3.82MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:47:45 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator((a+b+c-1)*(a+b-1)*(b+c-1)*(c+a-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1))); Output: Magma V2.11-10 Thu Dec 8 2005 18:47:45 on modular [Seed = 1004584491] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:39:28 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print (2*(a+b-1)*(b+c-1)*(a+c-1)-4*a*b*c+a+b+c-1); print Factorization(Numerator(2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - 6*a*c + 5*a + 2*b^2*c - 2*b^2 + 2*b*c^2 - 6*b*c + 5*b - 2*c^2 + 5*c - 3 )); Output: Magma V2.11-10 Thu Dec 8 2005 18:39:28 on modular [Seed = 3474213069] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - 6*a*c + 5*a + 2*b^2*c - 2*b^2 + 2*b*c^2 - 6*b*c + 5*b - 2*c^2 + 5*c - 3 [ ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:39:09 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print (2*(a+b-1)*(b+c-1)*(a+c-1)-4*a*b*c+a+b+c-1); print Factorization(2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - 6*a*c + 5*a + 2*b^2*c - 2*b^2 + 2*b*c^2 - 6*b*c + 5*b - 2*c^2 + 5*c - 3 ); Output: Magma V2.11-10 Thu Dec 8 2005 18:39:08 on modular [Seed = 3307094763] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - 6*a*c + 5*a + 2*b^2*c - 2*b^2 + 2*b*c^2 - 6*b*c + 5*b - 2*c^2 + 5*c - 3 >> print Factorization(2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - ^ Runtime error in 'Factorization': Bad argument types Argument types given: FldFunRatMElt Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:38:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print (2*(a+b-1)*(b+c-1)*(a+c-1)-4*a*b*c+a+b+c-1); Output: Magma V2.11-10 Thu Dec 8 2005 18:38:30 on modular [Seed = 3607910163] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - 6*a*c + 5*a + 2*b^2*c - 2*b^2 + 2*b*c^2 - 6*b*c + 5*b - 2*c^2 + 5*c - 3 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:37:25 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print ((a+b-1)*(b+c-1)*(a+c-1)-4*a*b*c+a+b+c-1); print Factorization(Numerator(a^2*b + a^2*c - a^2 + a*b^2 - 2*a*b*c - 3*a*b + a*c^2 - 3*a*c + 3*a + b^2*c - b^2 + b*c^2 - 3*b*c + 3*b - c^2 + 3*c - 2)); Output: Magma V2.11-10 Thu Dec 8 2005 18:37:25 on modular [Seed = 3996479081] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 a^2*b + a^2*c - a^2 + a*b^2 - 2*a*b*c - 3*a*b + a*c^2 - 3*a*c + 3*a + b^2*c - b^2 + b*c^2 - 3*b*c + 3*b - c^2 + 3*c - 2 [ ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:36:58 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print ((a+b-1)*(b+c-1)*(a+c-1)-4*a*b*c+a+b+c-1); Output: Magma V2.11-10 Thu Dec 8 2005 18:36:57 on modular [Seed = 3846071630] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 a^2*b + a^2*c - a^2 + a*b^2 - 2*a*b*c - 3*a*b + a*c^2 - 3*a*c + 3*a + b^2*c - b^2 + b*c^2 - 3*b*c + 3*b - c^2 + 3*c - 2 Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:34:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:34:33 on modular [Seed = 3812123608] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:34:18 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:34:17 on modular [Seed = 4263870261] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:34:01 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(q,-D/2+(0)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:34:01 on modular [Seed = 4113462318] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:33:47 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(q,-D/2+(0)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:33:46 on modular [Seed = 4029378840] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*a*b^2*c + a*b*c + 1/2*a*b + 1/4*a - 2*b^2*c + 1/2*b^2 - 1/2*b*c - 1/4*b + 1/4*c - 1/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (a^2*b*c - 1/4*a^2 - a*b^2*c - 4*a*b*c - a*b - 1/4*a*c + 1/4*a - 3/4*b^2 + 9/4*b*c + 3/4*b)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b + 3*b)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (-a^2*b*c - 3/4*a^2 + a*b^2*c + 4*a*b*c - a*b + 1/4*a*c + 7/4*a - 1/4*b^2 - 9/4*b*c + 5/4*b - 1)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (2*a^2*b*c + 2*a^2*c + 1/2*a^2 - 3*a*b*c + 1/2*a*b - 9/2*a*c - 5/4*a - 3/4*b + 9/4*c + 3/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b - 3*a + 3)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*b*c + c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*a*c - 3*c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (a*b*c - 9/4*a - 9/4*b + 3/4*c + 9/4)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:32:23 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(0)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:32:23 on modular [Seed = 2320567103] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [] (2*a*b^2*c + a*b*c + 1/2*a*b + 1/4*a - 2*b^2*c + 1/2*b^2 - 1/2*b*c - 1/4*b + 1/4*c - 1/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (a^2*b*c - 1/4*a^2 - a*b^2*c - 4*a*b*c - a*b - 1/4*a*c + 1/4*a - 3/4*b^2 + 9/4*b*c + 3/4*b)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b + 3*b)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (-a^2*b*c - 3/4*a^2 + a*b^2*c + 4*a*b*c - a*b + 1/4*a*c + 7/4*a - 1/4*b^2 - 9/4*b*c + 5/4*b - 1)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (2*a^2*b*c + 2*a^2*c + 1/2*a^2 - 3*a*b*c + 1/2*a*b - 9/2*a*c - 5/4*a - 3/4*b + 9/4*c + 3/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b - 3*a + 3)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*b*c + c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*a*c - 3*c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (a*b*c - 9/4*a - 9/4*b + 3/4*c + 9/4)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:31:56 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(2)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:31:55 on modular [Seed = 2170159320] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ ] (2*a*b^2*c + a*b*c + 1/2*a*b + 1/4*a - 2*b^2*c + 1/2*b^2 - 1/2*b*c - 1/4*b + 1/4*c - 1/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (a^2*b*c - 1/4*a^2 - a*b^2*c - 4*a*b*c - a*b - 1/4*a*c + 1/4*a - 3/4*b^2 + 9/4*b*c + 3/4*b)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b + 3*b)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (-a^2*b*c - 3/4*a^2 + a*b^2*c + 4*a*b*c - a*b + 1/4*a*c + 7/4*a - 1/4*b^2 - 9/4*b*c + 5/4*b - 1)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (2*a^2*b*c + 2*a^2*c + 1/2*a^2 - 3*a*b*c + 1/2*a*b - 9/2*a*c - 5/4*a - 3/4*b + 9/4*c + 3/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b - 3*a + 3)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*b*c + c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*a*c - 3*c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (a*b*c - 9/4*a - 9/4*b + 3/4*c + 9/4)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:31:46 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(2)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:31:45 on modular [Seed = 2571245770] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ ] (2*a*b - b)/(a^2 + a*b - a) (a^2 - a*b - 2*a + 1)/(a^2 + a*b - a) (-a + 1)/(a*c) (-a^2 + a*b + 2*a - 1)/(a^2 + a*b - a) (2*a^2*b + a^2 - 2*a*b - 2*a + 1)/(a^2*b + a*b^2 - a*b) (-a*b - a + 1)/(a*b*c) 1/a (a - 1)/(a*b) (a*b*c - a - b + 1)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:30:11 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(1)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:30:10 on modular [Seed = 2488210852] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*a*b - b)/(a^2 + a*b - a) (a^2 - a*b - 2*a + 1)/(a^2 + a*b - a) (-a + 1)/(a*c) (-a^2 + a*b + 2*a - 1)/(a^2 + a*b - a) (2*a^2*b + a^2 - 2*a*b - 2*a + 1)/(a^2*b + a*b^2 - a*b) (-a*b - a + 1)/(a*b*c) 1/a (a - 1)/(a*b) (a*b*c - a - b + 1)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:29:33 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(1)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2*a-1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:29:33 on modular [Seed = 2826119751] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:28:31 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(a^2+a*b-b)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2*a-1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:28:31 on modular [Seed = 2709136140] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ ] 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:27:49 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(a^2+a*b-b)*(a+b+c-1)/(2*a*b*c))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2*a-1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:27:49 on modular [Seed = 3211017720] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> ^2+a*b-b)*(a+b+c-1)/(2*a*b*c))); ^ User error: bad syntax 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:27:19 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Evaluate(Denominator(Q[3,3])),-D/2+(a^2+a*b-b)*(a+b+c-1)/(2*a*b*c)); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2*a-1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:27:18 on modular [Seed = 3126933726] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print Factorization(Evaluate(Denominator(Q[3,3])),-D/2+(a^2+a*b-b)*(a+b+c-1 ^ Runtime error in 'Evaluate': Bad argument types Argument types given: RngUPolElt[FldFunRat] 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:26:12 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a^2+a*b-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:26:12 on modular [Seed = 2976527255] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^3 - 2*a^2*b*c + a^2*b + a^2*c + a^2 + 2*a*c - a + 2*b*c - b + c - 1)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-a^4 - 2*a^3*b - a^3*c - a^3 - a^2*b^2 + a^2*b*c + a^2*b - 2*a^2*c + a^2 + 2*a*b^2 - 2*a*b*c + 2*a*b - a*c + a - b^2 + b*c - b)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (2*a^3*b + 2*a^2*b^2 - 2*a^2*b - 4*a*b^2 + 2*b^2)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-a^4 - 2*a^3*b + a^3*c + a^3 - a^2*b^2 - a^2*b*c + 3*a^2*b + 2*a^2*c + a^2 + 2*a*b^2 + 2*a*b*c + a*c - a - b^2 - b*c - b)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-2*a^3*c - a^3 - a^2*b + a^2*c + a^2 + 2*a*b + a - b + c - 1)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-2*a^4 - 2*a^3*b + 2*a^2*b - 2*a^2 - 2*a*b + 2*b)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (2*a^3*c + 2*a^2*b*c + 2*a^2*c - 2*b*c)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-2*a^3*c - 2*a^2*b*c + 2*a^2*c + 4*a*b*c - 2*b*c)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-a^4 - 2*a^3*b + a^3*c - a^2*b^2 + a^2*b*c + 2*a^2*b + a^2*c - 2*a^2 + 2*a*b^2 - 2*a*b*c - 2*a*b + a*c - b^2 + b*c + 2*b + c - 1)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:25:18 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:25:18 on modular [Seed = 1285017504] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a + 2*b*c - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b - a*c + a - b^2 + b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a*b + 2*b^2)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c + a - b^2 - b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c + a - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a^2 + 2*a*b - 2*a + 2*b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c - 2*a - b^2 + b*c + 2*b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:25:02 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c))); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:25:02 on modular [Seed = 1267779767] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c))); ^ User error: bad syntax >> end for; ^ User error: bad syntax >> end for; ^ User error: bad syntax Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:24:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(2*b-1)*(a+b+c-1)/((2*a*b*c)*(1-2*a))); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:24:29 on modular [Seed = 1100661470] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (4*a*b^2*c - a*b^2 - 2*a*b*c + 1/4*a - 2*b^3*c + b^3 + b^2*c - b^2 - 1/2*b*c + 3/4*b + 1/4*c - 1/4)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (-a^2*b^2 + 4*a^2*b*c - a^2*b - 1/4*a^2 + 2*a*b^3 - 5*a*b^2*c + a*b^2 - a*b*c + 1/2*a*b - 1/4*a*c + 1/4*a - b^4 + b^3*c + b^2*c - 1/4*b^2 + 1/4*b*c - 1/4*b)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (4*a^2*b^2 - 2*a^2*b - 6*a*b^3 + 2*a*b^2 + 1/2*a*b + 2*b^4 - 1/2*b^2)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (-a^2*b^2 - 4*a^2*b*c + a^2*b + 3/4*a^2 + 2*a*b^3 + 5*a*b^2*c - 3*a*b^2 + a*b*c + 1/2*a*b + 1/4*a*c - 3/4*a - b^4 - b^3*c + 2*b^3 - b^2*c - 5/4*b^2 - 1/4*b*c + 3/4*b)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (4*a^2*b*c - 2*a^2*b + 2*a^2*c - a^2 - 2*a*b^2*c + 3*a*b^2 - 4*a*b*c - 3/2*a*c + 5/4*a - b^3 + b^2*c + b*c - 1/4*b + 1/4*c - 1/4)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (2*a^2*b^2 + 2*a^2*b - 3/2*a^2 - 2*a*b^3 - 4*a*b^2 + 3/2*a*b + 1/2*a + 2*b^3 - 1/2*b)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (-2*a*b^2*c + 2*a*b*c - 1/2*a*c + 2*b^3*c - 2*b^2*c + 1/2*b*c)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (-4*a^2*b*c + 2*a^2*c + 6*a*b^2*c - 2*a*b*c - 1/2*a*c - 2*b^3*c + 1/2*b*c)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (-a^2*b^2 + 4*a^2*b*c + 3*a^2*b - 9/4*a^2 + 2*a*b^3 - 3*a*b^2*c - 4*a*b^2 - a*b*c + 1/2*a*b - 3/4*a*c + 3/2*a - b^4 + b^3*c + b^3 + 3/4*b^2 + 1/4*b*c - 1/2*b + 1/4*c - 1/4)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:23:16 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a+3*b-2)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:23:16 on modular [Seed = 1518459148] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^3 + 2*a^2*b*c + 9*a^2*b + a^2*c - 7*a^2 + 4*a*b^2*c + 23*a*b^2 + 6*a*b*c - 38*a*b - 6*a*c + 15*a - 30*b^3*c + 15*b^3 + 53*b^2*c - 39*b^2 - 36*b*c + 33*b + 9*c - 9)/(a^4 + 8*a^3*b + a^3*c - 6*a^3 + 22*a^2*b^2 + 11*a^2*b*c - 34*a^2*b - 5*a^2*c + 12*a^2 + 24*a*b^3 + 19*a*b^2*c - 58*a*b^2 - 26*a*b*c + 44*a*b + 7*a*c - 10*a + 9*b^4 + 9*b^3*c - 30*b^3 - 21*b^2*c + 36*b^2 + 15*b*c - 18*b - 3*c + 3) (-a^4 - 8*a^3*b - a^3*c + 7*a^3 - 22*a^2*b^2 - 9*a^2*b*c + 35*a^2*b + 6*a^2*c - 15*a^2 - 24*a*b^3 - 31*a*b^2*c + 49*a*b^2 + 32*a*b*c - 34*a*b - 9*a*c + 9*a - 9*b^4 + 9*b^3*c + 21*b^3 - 6*b^2*c - 15*b^2 + b*c + 3*b)/(a^4 + 8*a^3*b + a^3*c - 6*a^3 + 22*a^2*b^2 + 11*a^2*b*c - 34*a^2*b - 5*a^2*c + 12*a^2 + 24*a*b^3 + 19*a*b^2*c - 58*a*b^2 - 26*a*b*c + 44*a*b + 7*a*c - 10*a + 9*b^4 + 9*b^3*c - 30*b^3 - 21*b^2*c + 36*b^2 + 15*b*c - 18*b - 3*c + 3) (-2*a^2*b + 2*a*b + 18*b^3 - 18*b^2 + 4*b)/(a^3 + 7*a^2*b + a^2*c - 5*a^2 + 15*a*b^2 + 10*a*b*c - 22*a*b - 4*a*c + 7*a + 9*b^3 + 9*b^2*c - 21*b^2 - 12*b*c + 15*b + 3*c - 3) (-a^4 - 8*a^3*b + a^3*c + 7*a^3 - 22*a^2*b^2 + 9*a^2*b*c + 43*a^2*b - 6*a^2*c - 19*a^2 - 24*a*b^3 + 31*a*b^2*c + 81*a*b^2 - 32*a*b*c - 82*a*b + 9*a*c + 25*a - 9*b^4 - 9*b^3*c + 45*b^3 + 6*b^2*c - 75*b^2 - b*c + 51*b - 12)/(a^4 + 8*a^3*b + a^3*c - 6*a^3 + 22*a^2*b^2 + 11*a^2*b*c - 34*a^2*b - 5*a^2*c + 12*a^2 + 24*a*b^3 + 19*a*b^2*c - 58*a*b^2 - 26*a*b*c + 44*a*b + 7*a*c - 10*a + 9*b^4 + 9*b^3*c - 30*b^3 - 21*b^2*c + 36*b^2 + 15*b*c - 18*b - 3*c + 3) (2*a^3*c + a^3 + 4*a^2*b*c + a^2*b - 3*a^2*c - 3*a^2 - 30*a*b^2*c - 9*a*b^2 + 22*a*b*c + 10*a*b - 4*a*c - a - 9*b^3 + 9*b^2*c + 21*b^2 - 6*b*c - 15*b + c + 3)/(a^4 + 8*a^3*b + a^3*c - 6*a^3 + 22*a^2*b^2 + 11*a^2*b*c - 34*a^2*b - 5*a^2*c + 12*a^2 + 24*a*b^3 + 19*a*b^2*c - 58*a*b^2 - 26*a*b*c + 44*a*b + 7*a*c - 10*a + 9*b^4 + 9*b^3*c - 30*b^3 - 21*b^2*c + 36*b^2 + 15*b*c - 18*b - 3*c + 3) (-2*a^3 - 16*a^2*b + 8*a^2 - 30*a*b^2 + 38*a*b - 10*a + 18*b^2 - 18*b + 4)/(a^3 + 7*a^2*b + a^2*c - 5*a^2 + 15*a*b^2 + 10*a*b*c - 22*a*b - 4*a*c + 7*a + 9*b^3 + 9*b^2*c - 21*b^2 - 12*b*c + 15*b + 3*c - 3) (2*a^2*c + 16*a*b*c - 10*a*c + 30*b^2*c - 38*b*c + 12*c)/(a^3 + 7*a^2*b + a^2*c - 5*a^2 + 15*a*b^2 + 10*a*b*c - 22*a*b - 4*a*c + 7*a + 9*b^3 + 9*b^2*c - 21*b^2 - 12*b*c + 15*b + 3*c - 3) (2*a^2*c - 2*a*c - 18*b^2*c + 18*b*c - 4*c)/(a^3 + 7*a^2*b + a^2*c - 5*a^2 + 15*a*b^2 + 10*a*b*c - 22*a*b - 4*a*c + 7*a + 9*b^3 + 9*b^2*c - 21*b^2 - 12*b*c + 15*b + 3*c - 3) (-a^2 - 6*a*b + a*c + 2*a - 9*b^2 + 9*b*c + 6*b - 3*c - 1)/(a^2 + 6*a*b + a*c - 4*a + 9*b^2 + 9*b*c - 12*b - 3*c + 3) Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:17:06 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(-1+2*a)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:17:06 on modular [Seed = 2057406394] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:16:00 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(1-2*b)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:15:59 on modular [Seed = 115211163] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] 0 (-b + c)/(b + c - 1) (2*b - 1)/(b + c - 1) -1 (-b + c)/(b^2 + b*c - b) (2*b - 1)/(b^2 + b*c - b) 0 (-2*b*c + c)/(b^2 + b*c - b) (-b^2 + b*c + 2*b - 1)/(b^2 + b*c - b) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:15:21 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(1-2*b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)); Output: Magma V2.11-10 Thu Dec 8 2005 18:15:21 on modular [Seed = 349170095] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) [ , , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:14:46 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(1-2*b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)); Output: Magma V2.11-10 Thu Dec 8 2005 18:14:45 on modular [Seed = 331935300] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) [ , , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:12:48 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)); Output: Magma V2.11-10 Thu Dec 8 2005 18:12:48 on modular [Seed = 703780843] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) [ , , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:12:25 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1)); Output: Magma V2.11-10 Thu Dec 8 2005 18:12:24 on modular [Seed = 620747470] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) [ ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:11:51 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1)); Output: Magma V2.11-10 Thu Dec 8 2005 18:11:51 on modular [Seed = 603512230] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a + 2*b*c - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b - a*c + a - b^2 + b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a*b + 2*b^2)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c + a - b^2 - b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c + a - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a^2 + 2*a*b - 2*a + 2*b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c - 2*a - b^2 + b*c + 2*b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) [ ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:11:06 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2-(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1)); Output: Magma V2.11-10 Thu Dec 8 2005 18:11:06 on modular [Seed = 1055244422] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^3 - 6*a^2*b*c - 3*a^2*b + a^2*c + a^2 + 20*a*b^2*c - a*b^2 - 10*a*b*c + 2*a*b + 2*a*c - a - 6*b^3*c + 3*b^3 + 5*b^2*c - 7*b^2 - 4*b*c + 5*b + c - 1)/(a^4 + a^3*c - 2*a^3 - 2*a^2*b^2 + 3*a^2*b*c + 2*a^2*b - a^2*c + 3*a*b^2*c + 2*a*b^2 - 2*a*b*c - 4*a*b - a*c + 2*a + b^4 + b^3*c - 2*b^3 - b^2*c - b*c + 2*b + c - 1) (-a^4 - a^3*c - a^3 + 2*a^2*b^2 + 15*a^2*b*c - a^2*b - 2*a^2*c + a^2 - 15*a*b^2*c + a*b^2 - 2*a*b - a*c + a - b^4 + b^3*c + b^3 + 2*b^2*c + b^2 + b*c - b)/(a^4 + a^3*c - 2*a^3 - 2*a^2*b^2 + 3*a^2*b*c + 2*a^2*b - a^2*c + 3*a*b^2*c + 2*a*b^2 - 2*a*b*c - 4*a*b - a*c + 2*a + b^4 + b^3*c - 2*b^3 - b^2*c - b*c + 2*b + c - 1) (6*a^2*b - 8*a*b^2 - 2*a*b + 2*b^3 + 2*b^2)/(a^3 - a^2*b + a^2*c - a^2 - a*b^2 + 2*a*b*c + 2*a*b - a + b^3 + b^2*c - b^2 - b - c + 1) (-a^4 + a^3*c + 3*a^3 + 2*a^2*b^2 - 15*a^2*b*c - 5*a^2*b + 2*a^2*c + a^2 + 15*a*b^2*c - 3*a*b^2 + 6*a*b + a*c - 3*a - b^4 - b^3*c + 5*b^3 - 2*b^2*c - 7*b^2 - b*c + 3*b)/(a^4 + a^3*c - 2*a^3 - 2*a^2*b^2 + 3*a^2*b*c + 2*a^2*b - a^2*c + 3*a*b^2*c + 2*a*b^2 - 2*a*b*c - 4*a*b - a*c + 2*a + b^4 + b^3*c - 2*b^3 - b^2*c - b*c + 2*b + c - 1) (-6*a^3*c - 3*a^3 + 20*a^2*b*c + a^2*b + 5*a^2*c + a^2 - 6*a*b^2*c + 3*a*b^2 - 10*a*b*c - 6*a*b - 4*a*c + 3*a - b^3 + b^2*c + b^2 + 2*b*c + b + c - 1)/(a^4 + a^3*c - 2*a^3 - 2*a^2*b^2 + 3*a^2*b*c + 2*a^2*b - a^2*c + 3*a*b^2*c + 2*a*b^2 - 2*a*b*c - 4*a*b - a*c + 2*a + b^4 + b^3*c - 2*b^3 - b^2*c - b*c + 2*b + c - 1) (-2*a^3 + 8*a^2*b + 4*a^2 - 6*a*b^2 - 6*a*b - 2*a + 2*b^2 + 2*b)/(a^3 - a^2*b + a^2*c - a^2 - a*b^2 + 2*a*b*c + 2*a*b - a + b^3 + b^2*c - b^2 - b - c + 1) (2*a^2*c - 8*a*b*c + 2*a*c + 6*b^2*c - 2*b*c)/(a^3 - a^2*b + a^2*c - a^2 - a*b^2 + 2*a*b*c + 2*a*b - a + b^3 + b^2*c - b^2 - b - c + 1) (-6*a^2*c + 8*a*b*c + 2*a*c - 2*b^2*c - 2*b*c)/(a^3 - a^2*b + a^2*c - a^2 - a*b^2 + 2*a*b*c + 2*a*b - a + b^3 + b^2*c - b^2 - b - c + 1) (-a^2 + 2*a*b + a*c + 2*a - b^2 + b*c - 2*b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) [ ] Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:10:33 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1)); Output: Magma V2.11-10 Thu Dec 8 2005 18:10:32 on modular [Seed = 904842095] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a + 2*b*c - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b - a*c + a - b^2 + b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a*b + 2*b^2)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c + a - b^2 - b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c + a - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a^2 + 2*a*b - 2*a + 2*b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c - 2*a - b^2 + b*c + 2*b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) [ ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:09:23 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:09:22 on modular [Seed = 820760096] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a + 2*b*c - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b - a*c + a - b^2 + b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a*b + 2*b^2)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c + a - b^2 - b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c + a - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a^2 + 2*a*b - 2*a + 2*b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c - 2*a - b^2 + b*c + 2*b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:09:07 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j]),-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:09:07 on modular [Seed = 3457495278] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print Evaluate(Q[i,j]),-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); ^ User error: bad syntax >> end for; ^ User error: bad syntax >> end for; ^ User error: bad syntax Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:05:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(a-b)/(2*a*b)); end for; end for; print Factorization(Numerator(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)); Output: Magma V2.11-10 Thu Dec 8 2005 18:05:29 on modular [Seed = 3657508304] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) [ , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:04:49 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(a-b)/(2*a*b)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:04:49 on modular [Seed = 3607898357] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:04:16 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Factorization(Evaluate(Denominator(Q[i,j]),-D/2+(a-b)/(2*a*b))); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:04:15 on modular [Seed = 3963032624] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print Factorization(Evaluate(Denominator(Q[i,j]),-D/2+(a-b)/(2*a*b))); ^ Runtime error in 'Factorization': Bad argument types Argument types given: FldFunRatMElt Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:03:59 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(a-b)/(2*a*b)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:03:58 on modular [Seed = 3912372484] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:03:31 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2-(a-b)/(2*a*b)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:03:30 on modular [Seed = 3862765079] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*a^3*b^2*c - a^3*b*c^2 - 3/2*a^3*b*c - 1/2*a^3*b + 1/4*a^3*c^2 + 1/2*a^3*c + 1/4*a^3 + 4*a^2*b^3*c + 5*a^2*b^2*c^2 - 13/2*a^2*b^2*c - 3/2*a^2*b^2 - 3/2*a^2*b*c^3 - 11/4*a^2*b*c^2 + 2*a^2*b*c + 9/4*a^2*b + 1/4*a^2*c^3 + 3/4*a^2*c^2 - 1/4*a^2*c - 3/4*a^2 + 2*a*b^4*c + 5*a*b^3*c^2 - 9/2*a*b^3*c - 3/2*a*b^3 + 5*a*b^2*c^3 - 33/4*a*b^2*c^2 + 1/2*a*b^2*c + 15/4*a*b^2 - 5/2*a*b*c^3 + 5/2*a*b*c^2 + 3*a*b*c - 3*a*b + 1/2*a*c^3 - 1/4*a*c^2 - a*c + 3/4*a - b^4*c^2 + 1/2*b^4*c - 1/2*b^4 - 3/2*b^3*c^3 + 11/4*b^3*c^2 - 3*b^3*c + 7/4*b^3 + 5/4*b^2*c^3 - 17/4*b^2*c^2 + 21/4*b^2*c - 9/4*b^2 - b*c^3 + 13/4*b*c^2 - 7/2*b*c + 5/4*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^4*b*c + 1/4*a^4*c^2 - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b + 1/4*a^3*c^3 - 1/2*a^3*c^2 - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 7/2*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + 3/4*a^2*b*c^3 - 7/2*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/4*a^2*c^3 - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + 3/4*a*b^2*c^3 - 7/2*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - 1/2*a*b*c^3 + 7/2*a*b*c - 3*a*b - 1/4*a*c^3 + 3/2*a*c^2 - 9/4*a*c + a + 1/4*b^4*c^2 - 1/4*b^4 + 1/4*b^3*c^3 - 1/2*b^3*c^2 - 3/4*b^3*c + b^3 - 1/4*b^2*c^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 - 1/4*b*c^3 + 3/2*b*c^2 - 9/4*b*c + b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^4*b*c - 1/4*a^4*c^2 - 1/2*a^4*c - 1/4*a^4 + a^3*b^2*c + 4*a^3*b*c^2 - 5/2*a^3*b*c - 1/2*a^3*b - 1/4*a^3*c^3 - 3/4*a^3*c^2 + 1/4*a^3*c + 3/4*a^3 - a^2*b^3*c + 1/2*a^2*b^2*c^2 + 1/2*a^2*b^2*c + 15/4*a^2*b*c^3 - 19/4*a^2*b*c^2 + 1/4*a^2*b*c + 3/4*a^2*b - 1/2*a^2*c^3 + 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - a*b^4*c - 4*a*b^3*c^2 + 5/2*a*b^3*c + 1/2*a*b^3 - 15/4*a*b^2*c^3 + 19/4*a*b^2*c^2 - 5/4*a*b^2*c - 3/4*a*b^2 - 1/2*a*b*c^2 + 1/2*a*b*c - 1/4*a*c^3 + 3/4*a*c^2 - 3/4*a*c + 1/4*a - 1/4*b^4*c^2 + 1/4*b^4 + 1/4*b^3*c^3 + 3/4*b^3*c^2 + 3/4*b^3*c - 3/4*b^3 + 1/2*b^2*c^3 + 1/4*b^2*c^2 - 3/2*b^2*c + 3/4*b^2 + 1/4*b*c^3 - 3/4*b*c^2 + 3/4*b*c - 1/4*b)/(a^4*b*c + 1/4*a^4*c^2 - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b + 1/4*a^3*c^3 - 1/2*a^3*c^2 - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 7/2*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + 3/4*a^2*b*c^3 - 7/2*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/4*a^2*c^3 - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + 3/4*a*b^2*c^3 - 7/2*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - 1/2*a*b*c^3 + 7/2*a*b*c - 3*a*b - 1/4*a*c^3 + 3/2*a*c^2 - 9/4*a*c + a + 1/4*b^4*c^2 - 1/4*b^4 + 1/4*b^3*c^3 - 1/2*b^3*c^2 - 3/4*b^3*c + b^3 - 1/4*b^2*c^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 - 1/4*b*c^3 + 3/2*b*c^2 - 9/4*b*c + b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^3*b*c + 3/2*a^2*b*c^2 - 3/2*a^2*b*c - a*b^3*c - 2*a*b^2*c^2 + a*b^2*c - 1/2*a*b*c^2 + 1/2*a*b*c + 1/2*b^3*c^2 + 1/2*b^3*c + 1/2*b^2*c^2 - 1/2*b^2*c)/(a^3*b*c + 1/4*a^3*c^2 - 1/4*a^3 + 2*a^2*b^2*c + 7/4*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b + 1/4*a^2*c^3 - 1/4*a^2*c^2 - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 7/4*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + 1/2*a*b*c^3 - 3/2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 3/4*a*c^2 + 3/2*a*c - 3/4*a + 1/4*b^3*c^2 - 1/4*b^3 + 1/4*b^2*c^3 - 1/4*b^2*c^2 - 3/4*b^2*c + 3/4*b^2 - 3/4*b*c^2 + 3/2*b*c - 3/4*b - 1/4*c^3 + 3/4*c^2 - 3/4*c + 1/4) (-a^4*b*c - 1/4*a^4*c^2 + 1/4*a^4 - a^3*b^2*c - 4*a^3*b*c^2 + 3/2*a^3*b*c + 1/2*a^3*b + 1/4*a^3*c^3 + 5/4*a^3*c^2 + 5/4*a^3*c - 3/4*a^3 + a^2*b^3*c + 1/2*a^2*b^2*c^2 - 1/2*a^2*b^2*c - 15/4*a^2*b*c^3 + 13/4*a^2*b*c^2 + 5/4*a^2*b*c - 3/4*a^2*b + 1/2*a^2*c^3 + 1/4*a^2*c^2 - 5/2*a^2*c + 3/4*a^2 + a*b^4*c + 4*a*b^3*c^2 - 3/2*a*b^3*c - 1/2*a*b^3 + 15/4*a*b^2*c^3 - 21/4*a*b^2*c^2 - 1/4*a*b^2*c + 3/4*a*b^2 + 3/2*a*b*c^2 - 1/2*a*b*c + 1/4*a*c^3 - 5/4*a*c^2 + 5/4*a*c - 1/4*a - 1/4*b^4*c^2 + 1/2*b^4*c - 1/4*b^4 - 1/4*b^3*c^3 + 3/4*b^3*c^2 - 9/4*b^3*c + 3/4*b^3 - 1/2*b^2*c^3 - 7/4*b^2*c^2 + 3*b^2*c - 3/4*b^2 - 1/4*b*c^3 + 5/4*b*c^2 - 5/4*b*c + 1/4*b)/(a^4*b*c + 1/4*a^4*c^2 - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b + 1/4*a^3*c^3 - 1/2*a^3*c^2 - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 7/2*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + 3/4*a^2*b*c^3 - 7/2*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/4*a^2*c^3 - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + 3/4*a*b^2*c^3 - 7/2*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - 1/2*a*b*c^3 + 7/2*a*b*c - 3*a*b - 1/4*a*c^3 + 3/2*a*c^2 - 9/4*a*c + a + 1/4*b^4*c^2 - 1/4*b^4 + 1/4*b^3*c^3 - 1/2*b^3*c^2 - 3/4*b^3*c + b^3 - 1/4*b^2*c^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 - 1/4*b*c^3 + 3/2*b*c^2 - 9/4*b*c + b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (2*a^4*b*c - a^4*c^2 - 1/2*a^4*c - 1/2*a^4 + 4*a^3*b^2*c + 5*a^3*b*c^2 - 11/2*a^3*b*c - 3/2*a^3*b - 3/2*a^3*c^3 + 5/4*a^3*c^2 - 1/2*a^3*c + 7/4*a^3 + 2*a^2*b^3*c + 5*a^2*b^2*c^2 - 11/2*a^2*b^2*c - 3/2*a^2*b^2 + 5*a^2*b*c^3 - 31/4*a^2*b*c^2 + a^2*b*c + 15/4*a^2*b + 5/4*a^2*c^3 - 9/4*a^2*c^2 + 13/4*a^2*c - 9/4*a^2 - a*b^3*c^2 - 1/2*a*b^3*c - 1/2*a*b^3 - 3/2*a*b^2*c^3 - 5/4*a*b^2*c^2 - 1/2*a*b^2*c + 9/4*a*b^2 - 5/2*a*b*c^3 + 1/2*a*b*c^2 + 4*a*b*c - 3*a*b - a*c^3 + 11/4*a*c^2 - 3*a*c + 5/4*a - 1/4*b^3*c^2 + 1/4*b^3 + 1/4*b^2*c^3 + 3/4*b^2*c^2 + 3/4*b^2*c - 3/4*b^2 + 1/2*b*c^3 + 1/4*b*c^2 - 3/2*b*c + 3/4*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^4*b*c + 1/4*a^4*c^2 - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b + 1/4*a^3*c^3 - 1/2*a^3*c^2 - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 7/2*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + 3/4*a^2*b*c^3 - 7/2*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/4*a^2*c^3 - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + 3/4*a*b^2*c^3 - 7/2*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - 1/2*a*b*c^3 + 7/2*a*b*c - 3*a*b - 1/4*a*c^3 + 3/2*a*c^2 - 9/4*a*c + a + 1/4*b^4*c^2 - 1/4*b^4 + 1/4*b^3*c^3 - 1/2*b^3*c^2 - 3/4*b^3*c + b^3 - 1/4*b^2*c^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 - 1/4*b*c^3 + 3/2*b*c^2 - 9/4*b*c + b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^3*b*c - 1/2*a^3*c^2 + 1/2*a^3*c + 2*a^2*b*c^2 - a^2*b*c + a^2*c^2 - a^2*c - a*b^3*c - 3/2*a*b^2*c^2 + 1/2*a*b^2*c - 3/2*a*b*c^2 + 1/2*a*b*c - 1/2*a*c^2 + 1/2*a*c + 1/2*b^2*c^2 + 1/2*b^2*c + 1/2*b*c^2 - 1/2*b*c)/(a^3*b*c + 1/4*a^3*c^2 - 1/4*a^3 + 2*a^2*b^2*c + 7/4*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b + 1/4*a^2*c^3 - 1/4*a^2*c^2 - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 7/4*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + 1/2*a*b*c^3 - 3/2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 3/4*a*c^2 + 3/2*a*c - 3/4*a + 1/4*b^3*c^2 - 1/4*b^3 + 1/4*b^2*c^3 - 1/4*b^2*c^2 - 3/4*b^2*c + 3/4*b^2 - 3/4*b*c^2 + 3/2*b*c - 3/4*b - 1/4*c^3 + 3/4*c^2 - 3/4*c + 1/4) (-a^2*b*c^2 + 1/2*a^2*c^3 + 1/2*a^2*c^2 - 2*a*b*c^3 + a*b*c^2 + 1/2*a*c^3 - 1/2*a*c^2 + b^3*c^2 + 3/2*b^2*c^3 - 3/2*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^3*b*c + 1/4*a^3*c^2 - 1/4*a^3 + 2*a^2*b^2*c + 7/4*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b + 1/4*a^2*c^3 - 1/4*a^2*c^2 - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 7/4*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + 1/2*a*b*c^3 - 3/2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 3/4*a*c^2 + 3/2*a*c - 3/4*a + 1/4*b^3*c^2 - 1/4*b^3 + 1/4*b^2*c^3 - 1/4*b^2*c^2 - 3/4*b^2*c + 3/4*b^2 - 3/4*b*c^2 + 3/2*b*c - 3/4*b - 1/4*c^3 + 3/4*c^2 - 3/4*c + 1/4) (-a^3*c^2 - 3/2*a^2*c^3 + 3/2*a^2*c^2 + a*b^2*c^2 + 2*a*b*c^3 - a*b*c^2 + 1/2*a*c^3 - 1/2*a*c^2 - 1/2*b^2*c^3 - 1/2*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^3*b*c + 1/4*a^3*c^2 - 1/4*a^3 + 2*a^2*b^2*c + 7/4*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b + 1/4*a^2*c^3 - 1/4*a^2*c^2 - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 7/4*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + 1/2*a*b*c^3 - 3/2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 3/4*a*c^2 + 3/2*a*c - 3/4*a + 1/4*b^3*c^2 - 1/4*b^3 + 1/4*b^2*c^3 - 1/4*b^2*c^2 - 3/4*b^2*c + 3/4*b^2 - 3/4*b*c^2 + 3/2*b*c - 3/4*b - 1/4*c^3 + 3/4*c^2 - 3/4*c + 1/4) (a^2*b*c - 1/4*a^2*c^2 + 1/2*a^2*c - 1/4*a^2 + a*b^2*c + 5/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 + 1/2*a*c^2 - 5/4*a*c + 1/2*a - 1/4*b^2*c^2 - 1/2*b^2*c - 1/4*b^2 + 1/4*b*c^3 - 1/2*b*c^2 - 1/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:01:41 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)/(2*a*b)); end for; end for; print (1-D/2), (1+D/2); Output: Magma V2.11-10 Thu Dec 8 2005 18:01:41 on modular [Seed = 4230415514] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*a*b^2*c + a*b*c^2 - 1/2*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a + b^2*c^2 - 1/2*b^2*c - 1/2*b^2 + 1/2*b*c^3 - 1/4*b*c^2 - b*c + 3/4*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^2*b*c - 1/4*a^2*c^2 + 1/2*a^2*c - 1/4*a^2 - a*b^2*c + 1/2*a*b*c^2 - 1/2*a*b*c - 1/4*a*c^3 + 3/4*a*c^2 - 3/4*a*c + 1/4*a - 1/4*b^2*c^2 + 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c^2 + 3/4*b*c - 1/4*b)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (-a^2*b*c + a*b^2*c - 1/2*a*b*c^2 + 1/2*a*b*c + 1/2*b^2*c^2 - 1/2*b^2*c)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (-a^2*b*c - 1/4*a^2*c^2 + 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 + 1/2*a*b*c + 1/4*a*c^3 - 1/4*a*c^2 + 1/4*a*c - 1/4*a - 1/4*b^2*c^2 - 1/2*b^2*c - 1/4*b^2 - 1/4*b*c^3 + 1/4*b*c^2 - 1/4*b*c + 1/4*b)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (2*a^2*b*c + a^2*c^2 + 1/2*a^2*c - 1/2*a^2 + a*b*c^2 - 3/2*a*b*c - 1/2*a*b + 1/2*a*c^3 + 1/4*a*c^2 - 3/2*a*c + 3/4*a - 1/4*b*c^2 + 1/4*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (-a^2*b*c - 1/2*a^2*c^2 - 1/2*a^2*c + a*b^2*c + 1/2*a*b*c^2 + 1/2*a*b*c - 1/2*a*c^2 + 1/2*a*c + 1/2*b*c^2 - 1/2*b*c)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a*b*c^2 + 1/2*a*c^3 - 1/2*a*c^2 - b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^2*c^2 - a*b*c^2 + 1/2*a*c^3 - 1/2*a*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^2*b*c - 1/4*a^2*c^2 - 1/2*a^2*c - 1/4*a^2 + a*b^2*c + 5/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 1/2*a*c^2 - 1/4*a*c + 1/2*a - 1/4*b^2*c^2 + 1/2*b^2*c - 1/4*b^2 + 1/4*b*c^3 + 1/2*b*c^2 - 5/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (1/2*a + 1/2*b + 1/2*c - 1/2)/(a*b*c) (2*a*b*c - 1/2*a - 1/2*b - 1/2*c + 1/2)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:50:10 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; print (1-D/2), (1+D/2); Output: Magma V2.11-10 Thu Dec 8 2005 17:50:10 on modular [Seed = 1117370835] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 (1/2*a + 1/2*b + 1/2*c - 1/2)/(a*b*c) (2*a*b*c - 1/2*a - 1/2*b - 1/2*c + 1/2)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:46:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:46:33 on modular [Seed = 1501736355] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^3*c + a^3 - 2*a^2*b*c^2 + a^2*b*c + 2*a^2*b - a^2*c^2 + 4*a^2*c - 3*a^2 + 2*a*b^2*c + a*b^2 + 2*a*b*c^2 + a*b*c - 4*a*b + 2*a*c^2 - 5*a*c + 3*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^4*c + 2*a^3*b*c + a^3*c^2 - 2*a^3*c + a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c - a^2*c^2 + a^2*c) (-a^4*b*c + a^4 - 2*a^3*b^2*c - a^3*b*c^2 + 5*a^3*b*c + 2*a^3*b + 2*a^3*c - 4*a^3 - a^2*b^3*c + a^2*b^2*c^2 + 5*a^2*b^2*c + a^2*b^2 + 4*a^2*b*c^2 - 5*a^2*b*c - 6*a^2*b + a^2*c^2 - 6*a^2*c + 6*a^2 - 3*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 - a*b*c + 6*a*b - 2*a*c^2 + 6*a*c - 4*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^4*b*c + 2*a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c + a^2*b^3*c + a^2*b^2*c^2 - 2*a^2*b^2*c - a^2*b*c^2 + a^2*b*c) (2*a^3*b*c - a^3 + 2*a^2*b^2*c^2 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 5*a^2*b*c - 2*a^2*b - 2*a^2*c + 3*a^2 - 3*a*b^2*c - a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b - a*c^2 + 4*a*c - 3*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^3*b*c^2 + a^2*b^2*c^2 + a^2*b*c^3 - a^2*b*c^2) (-a^4*b*c - 2*a^3*b^2*c + a^3*b*c^2 - a^3*b*c + a^3*b - a^3*c + a^3 - a^2*b^3*c - a^2*b^2*c^2 - a^2*b^2*c + 2*a^2*b^2 - 4*a^2*b*c^2 + 5*a^2*b*c - a^2*c^2 + 4*a^2*c - 3*a^2 + a*b^3 + 4*a*b^2*c - a*b^2 + 3*a*b*c^2 - a*b*c - 3*a*b + 2*a*c^2 - 5*a*c + 3*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^4*b*c + 2*a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c + a^2*b^3*c + a^2*b^2*c^2 - 2*a^2*b^2*c - a^2*b*c^2 + a^2*b*c) (-2*a^4*b*c + a^4*b + a^4 - 2*a^3*b^2*c^2 - a^3*b^2*c + 2*a^3*b^2 - 2*a^3*b*c^2 + 9*a^3*b*c - a^3*b + 2*a^3*c - 4*a^3 + a^2*b^3*c + a^2*b^3 + 3*a^2*b^2*c^2 + 6*a^2*b^2*c - 3*a^2*b^2 + 6*a^2*b*c^2 - 10*a^2*b*c - 3*a^2*b + a^2*c^2 - 6*a^2*c + 6*a^2 - a*b^3 - 5*a*b^2*c - 4*a*b*c^2 + a*b*c + 5*a*b - 2*a*c^2 + 6*a*c - 4*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^4*b^2*c + 2*a^3*b^3*c + a^3*b^2*c^2 - 2*a^3*b^2*c + a^2*b^4*c + a^2*b^3*c^2 - 2*a^2*b^3*c - a^2*b^2*c^2 + a^2*b^2*c) (-2*a^3*b^2*c^2 + 2*a^3*b^2*c + 3*a^3*b*c - a^3*b - a^3 + 2*a^2*b^3*c + 4*a^2*b^2*c^2 + a^2*b^2*c - 2*a^2*b^2 + 3*a^2*b*c^2 - 8*a^2*b*c - 2*a^2*c + 3*a^2 - a*b^3 - 5*a*b^2*c + a*b^2 - 4*a*b*c^2 + 3*a*b*c + 3*a*b - a*c^2 + 4*a*c - 3*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2) (2*a^2*b*c^2 - 2*a^2*b*c - a^2*c + a^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b - a*c^2 + 3*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^3*b*c + a^2*b^2*c + a^2*b*c^2 - a^2*b*c) (-2*a^3*b*c + a^3 - 2*a^2*b^2*c^2 - 2*a^2*b^2*c - 2*a^2*b*c^2 + 5*a^2*b*c + 2*a^2*b + 2*a^2*c - 3*a^2 + 3*a*b^2*c + a*b^2 + 3*a*b*c^2 - a*b*c - 4*a*b + a*c^2 - 4*a*c + 3*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - a^2*b^3*c^2 + a^2*b^2*c^3 + a^2*b^2*c^2 + 6*a^2*b^2*c + 3*a^2*b*c^2 - 6*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 3*a*b^3*c + 3*a*b^2*c^2 - 6*a*b^2*c - 3*a*b^2 - 3*a*b*c^2 - a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:19:27 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; print Factorization(Numerator((1-(D/2))^2-(a+b+c-1)/(a*b*c))); Output: Magma V2.11-10 Thu Dec 8 2005 17:19:26 on modular [Seed = 2124778531] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.360 seconds, Total memory usage: 3.72MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:19:02 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; print (1-(D/2))^2-(a+b+c-1)/(a*b*c); Output: Magma V2.11-10 Thu Dec 8 2005 17:19:02 on modular [Seed = 2074118430] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 (-a^2*b*c + 1/4*a^2 - a*b^2*c - a*b*c^2 + a*b*c + 1/2*a*b + 1/2*a*c - 1/2*a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^2*b^2*c^2) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 17:18:17 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; print (1-(D/2))^2; Output: Magma V2.11-10 Thu Dec 8 2005 17:18:16 on modular [Seed = 1907000170] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 (1/4*a^2 + 1/2*a*b + 1/2*a*c - 1/2*a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^2*b^2*c^2) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:10:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-(2*a*b*c-a-b+1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:10:33 on modular [Seed = 131918701] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*a^3*b^2*c - a^3*b*c - 1/2*a^3*b + 1/4*a^3 + 4*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 - 3*a^2*b*c^2 + 9/4*a^2*b + 5/4*a^2*c - 3/4*a^2 + 2*a*b^4*c + 4*a*b^3*c^2 - 5*a*b^3*c - 3/2*a*b^3 + 2*a*b^2*c^3 - 6*a*b^2*c^2 + 15/4*a*b^2 - 2*a*b*c^3 + 11/2*a*b*c - 3*a*b + 2*a*c^2 - 5/2*a*c + 3/4*a - 1/2*b^4 + b^3*c^2 - 2*b^3*c + 7/4*b^3 + b^2*c^3 - 4*b^2*c^2 + 21/4*b^2*c - 9/4*b^2 - 2*b*c^3 + 5*b*c^2 - 9/2*b*c + 5/4*b + c^3 - 2*c^2 + 5/4*c - 1/4)/(a^4*b*c - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + a^2*b*c^3 - 4*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + a*b^2*c^3 - 4*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - a*b*c^3 + a*b*c^2 + 7/2*a*b*c - 3*a*b + a*c^2 - 9/4*a*c + a - 1/4*b^4 - 3/4*b^3*c + b^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 + b*c^2 - 9/4*b*c + b - 1/2*c^2 + 3/4*c - 1/4) (a^4*b*c - 1/4*a^4 + a^3*b^2*c + 2*a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 5/4*a^3*c + 3/4*a^3 - a^2*b^3*c + a^2*b*c^3 - 3/4*a^2*b*c + 3/4*a^2*b - 2*a^2*c^2 + 5/2*a^2*c - 3/4*a^2 - a*b^4*c - 2*a*b^3*c^2 + 2*a*b^3*c + 1/2*a*b^3 - a*b^2*c^3 + 4*a*b^2*c^2 - 3/4*a*b^2*c - 3/4*a*b^2 + 2*a*b*c^3 - 4*a*b*c^2 + a*b*c - a*c^3 + 2*a*c^2 - 5/4*a*c + 1/4*a + 1/4*b^4 + 3/4*b^3*c - 3/4*b^3 - 3/2*b^2*c + 3/4*b^2 + 3/4*b*c - 1/4*b)/(a^4*b*c - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + a^2*b*c^3 - 4*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + a*b^2*c^3 - 4*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - a*b*c^3 + a*b*c^2 + 7/2*a*b*c - 3*a*b + a*c^2 - 9/4*a*c + a - 1/4*b^4 - 3/4*b^3*c + b^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 + b*c^2 - 9/4*b*c + b - 1/2*c^2 + 3/4*c - 1/4) (a^2*b*c + a*b^2*c + a*b*c^2 - 3/2*a*b*c - 1/2*b^2*c + 1/2*b*c)/(a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - 3/4*a - 1/4*b^3 - 3/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 3/2*b*c - 3/4*b + 1/2*c^2 - 3/4*c + 1/4) (-a^4*b*c + 1/4*a^4 - a^3*b^2*c - 2*a^3*b*c^2 + 2*a^3*b*c + 1/2*a^3*b + 3/4*a^3*c - 3/4*a^3 + a^2*b^3*c - a^2*b*c^3 - 3/4*a^2*b*c - 3/4*a^2*b + a^2*c^2 - a^2*c + 3/4*a^2 + a*b^4*c + 2*a*b^3*c^2 - 2*a*b^3*c - 1/2*a*b^3 + a*b^2*c^3 - 4*a*b^2*c^2 - 3/4*a*b^2*c + 3/4*a*b^2 - 2*a*b*c^3 + 2*a*b*c^2 + 2*a*b*c + a*c^3 - 1/4*a*c - 1/4*a - 1/4*b^4 - 5/4*b^3*c + 3/4*b^3 - b^2*c^2 + 3*b^2*c - 3/4*b^2 + 2*b*c^2 - 9/4*b*c + 1/4*b - c^2 + 1/2*c)/(a^4*b*c - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + a^2*b*c^3 - 4*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + a*b^2*c^3 - 4*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - a*b*c^3 + a*b*c^2 + 7/2*a*b*c - 3*a*b + a*c^2 - 9/4*a*c + a - 1/4*b^4 - 3/4*b^3*c + b^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 + b*c^2 - 9/4*b*c + b - 1/2*c^2 + 3/4*c - 1/4) (2*a^4*b*c - 1/2*a^4 + 4*a^3*b^2*c + 4*a^3*b*c^2 - 5*a^3*b*c - 3/2*a^3*b - a^3*c^2 - 2*a^3*c + 7/4*a^3 + 2*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + 2*a^2*b*c^3 - 6*a^2*b*c^2 + 15/4*a^2*b - a^2*c^3 + 19/4*a^2*c - 9/4*a^2 - a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 + 9/4*a*b^2 + 9/2*a*b*c - 3*a*b + a*c^2 - 7/2*a*c + 5/4*a + 1/4*b^3 + 3/4*b^2*c - 3/4*b^2 - 3/2*b*c + 3/4*b + 3/4*c - 1/4)/(a^4*b*c - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + a^2*b*c^3 - 4*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + a*b^2*c^3 - 4*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - a*b*c^3 + a*b*c^2 + 7/2*a*b*c - 3*a*b + a*c^2 - 9/4*a*c + a - 1/4*b^4 - 3/4*b^3*c + b^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 + b*c^2 - 9/4*b*c + b - 1/2*c^2 + 3/4*c - 1/4) (a^2*b*c + 1/2*a^2*c + a*b^2*c + a*b*c^2 - 1/2*a*b*c - a*c - 1/2*b*c + 1/2*c)/(a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - 3/4*a - 1/4*b^3 - 3/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 3/2*b*c - 3/4*b + 1/2*c^2 - 3/4*c + 1/4) (-a*b*c^2 + 1/2*a*c^2 - b^2*c^2 - b*c^3 + 3/2*b*c^2 + c^3 - 1/2*c^2)/(a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - 3/4*a - 1/4*b^3 - 3/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 3/2*b*c - 3/4*b + 1/2*c^2 - 3/4*c + 1/4) (-a^2*c^2 - a*b*c^2 - a*c^3 + 3/2*a*c^2 + 1/2*b*c^2 - 1/2*c^2)/(a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - 3/4*a - 1/4*b^3 - 3/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 3/2*b*c - 3/4*b + 1/2*c^2 - 3/4*c + 1/4) (a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 1/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 + 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 1/2*a*c - 3/4*a - 1/4*b^3 - 1/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 1/2*b*c - 3/4*b + 1/2*c^2 - 1/4*c + 1/4)/(a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - 3/4*a - 1/4*b^3 - 3/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 3/2*b*c - 3/4*b + 1/2*c^2 - 3/4*c + 1/4) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 17:09:51 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],c); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:09:51 on modular [Seed = 499577089] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^3*b^2*c^4 + 4*a^3*b^2*c^3 + 4*a^3*b^2*c^2 + 3*a^3*b^2*c - 2*a^3*b*c^2 - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^5 - 3*a^2*b^3*c^4 + 3*a^2*b^3*c^2 + 6*a^2*b^3*c + a^2*b^2*c^5 + 3*a^2*b^2*c^4 + 4*a^2*b^2*c^3 - 13*a^2*b^2*c - 3*a^2*b^2 - 4*a^2*b*c^3 - 4*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^3 - a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^4 + 5*a*b^3*c^3 + 7*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 4*a*b^2*c^4 - 3*a*b^2*c^3 - 10*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 2*a*b*c^4 + a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^4 + 2*a^4*b^2*c^3 + a^4*b^2*c^2 - a^4*b*c^2 + 2*a^3*b^3*c^4 + 4*a^3*b^3*c^3 + 2*a^3*b^3*c^2 + a^3*b^2*c^5 - 3*a^3*b^2*c^3 - 5*a^3*b^2*c^2 - 2*a^3*b*c^3 + 3*a^3*b*c^2 + a^2*b^4*c^4 + 2*a^2*b^4*c^3 + a^2*b^4*c^2 + a^2*b^3*c^5 - 3*a^2*b^3*c^3 - 5*a^2*b^3*c^2 - a^2*b^2*c^5 - a^2*b^2*c^4 - 3*a^2*b^2*c^3 + 7*a^2*b^2*c^2 - a^2*b*c^4 + 4*a^2*b*c^3 - 3*a^2*b*c^2 - a*b^4*c^2 - 2*a*b^3*c^3 + 3*a*b^3*c^2 - a*b^2*c^4 + 4*a*b^2*c^3 - 3*a*b^2*c^2 + a*b*c^4 - 2*a*b*c^3 + a*b*c^2) (-a^3*b^2*c^4 - 2*a^3*b^2*c^3 - a^3*b^2*c^2 + 2*a^3*b*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^4 - 4*a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^5 - a^2*b^2*c^4 - 3*a^2*b^2*c^3 + 2*a^2*b^2*c^2 + 6*a^2*b^2*c + 4*a^2*b*c^3 + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^4 - 2*a*b^4*c^3 - a*b^4*c^2 + a*b^3*c^5 + 3*a*b^3*c^4 - a*b^3*c^3 + a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^4 + 5*a*b^2*c^3 + 3*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 2*a*b*c^4 - a*b*c^3 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a + b^4*c^2 + b^3*c^3 - 3*b^3*c^2 + 2*b^3*c - b^3 - 2*b^2*c^3 + 6*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^4 + 2*a^3*b^2*c^3 + a^3*b^2*c^2 - a^3*b*c^2 + 2*a^2*b^3*c^4 + 4*a^2*b^3*c^3 + 2*a^2*b^3*c^2 + a^2*b^2*c^5 - 3*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 2*a^2*b*c^3 + 3*a^2*b*c^2 + a*b^4*c^4 + 2*a*b^4*c^3 + a*b^4*c^2 + a*b^3*c^5 - 3*a*b^3*c^3 - 5*a*b^3*c^2 - a*b^2*c^5 - a*b^2*c^4 - 3*a*b^2*c^3 + 7*a*b^2*c^2 - a*b*c^4 + 4*a*b*c^3 - 3*a*b*c^2 - b^4*c^2 - 2*b^3*c^3 + 3*b^3*c^2 - b^2*c^4 + 4*b^2*c^3 - 3*b^2*c^2 + b*c^4 - 2*b*c^3 + b*c^2) (-2*a^2*b*c^2 - 2*a^2*b*c + a^2 + 2*a*b^2*c^4 + 4*a*b^2*c^3 - 2*a*b^2*c - 2*a*b*c^3 - a*b*c^2 + a*b*c + 2*a*b + 2*a*c - 2*a - b^2*c^2 - b^2*c + b^2 - b*c^3 + 3*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^4 + 2*a^2*b*c^3 + a^2*b*c^2 - a^2*c^2 + a*b^2*c^4 + 2*a*b^2*c^3 + a*b^2*c^2 + a*b*c^5 + a*b*c^4 - a*b*c^3 - 3*a*b*c^2 - 2*a*c^3 + 2*a*c^2 - b^2*c^2 - 2*b*c^3 + 2*b*c^2 - c^4 + 2*c^3 - c^2) (-a^4*b^2*c^4 - 2*a^4*b^2*c^3 - a^4*b^2*c^2 + a^4*b*c^2 - 2*a^3*b^3*c^4 - 4*a^3*b^3*c^3 - 2*a^3*b^3*c^2 + a^3*b^2*c^5 + 5*a^3*b^2*c^4 + 11*a^3*b^2*c^3 + 11*a^3*b^2*c^2 + 3*a^3*b^2*c - a^3*b*c^3 - 8*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^4 - 2*a^2*b^4*c^3 - a^2*b^4*c^2 - a^2*b^3*c^5 + a^2*b^3*c^4 + 9*a^2*b^3*c^3 + 12*a^2*b^3*c^2 + 6*a^2*b^3*c + 2*a^2*b^2*c^4 - 3*a^2*b^2*c^3 - 17*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 2*a^2*b*c^4 - 5*a^2*b*c^3 + 3*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 2*a*b^4*c^2 + 3*a*b^4*c + 2*a*b^3*c^3 - 3*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 - 4*a*b^2*c^3 - 3*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 + a*b*c^3 + 5*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^4 + 2*a^4*b^2*c^3 + a^4*b^2*c^2 - a^4*b*c^2 + 2*a^3*b^3*c^4 + 4*a^3*b^3*c^3 + 2*a^3*b^3*c^2 + a^3*b^2*c^5 - 3*a^3*b^2*c^3 - 5*a^3*b^2*c^2 - 2*a^3*b*c^3 + 3*a^3*b*c^2 + a^2*b^4*c^4 + 2*a^2*b^4*c^3 + a^2*b^4*c^2 + a^2*b^3*c^5 - 3*a^2*b^3*c^3 - 5*a^2*b^3*c^2 - a^2*b^2*c^5 - a^2*b^2*c^4 - 3*a^2*b^2*c^3 + 7*a^2*b^2*c^2 - a^2*b*c^4 + 4*a^2*b*c^3 - 3*a^2*b*c^2 - a*b^4*c^2 - 2*a*b^3*c^3 + 3*a*b^3*c^2 - a*b^2*c^4 + 4*a*b^2*c^3 - 3*a*b^2*c^2 + a*b*c^4 - 2*a*b*c^3 + a*b*c^2) (2*a^3*b*c^3 + 3*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^5 - 5*a^2*b^2*c^4 - 4*a^2*b^2*c^3 + a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^4 + a^2*b*c^3 + 2*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - a*b^3*c^4 - 4*a*b^3*c^3 - 2*a*b^3*c^2 + 3*a*b^3*c + a*b^2*c^5 + a*b^2*c^4 + 4*a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 7*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c^2 + b^3*c - b^3 + b^2*c^3 - b^2*c^2 - 4*b^2*c + 3*b^2 - b*c^3 - b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^4 + 2*a^3*b^2*c^3 + a^3*b^2*c^2 - a^3*b*c^2 + 2*a^2*b^3*c^4 + 4*a^2*b^3*c^3 + 2*a^2*b^3*c^2 + a^2*b^2*c^5 - 3*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 2*a^2*b*c^3 + 3*a^2*b*c^2 + a*b^4*c^4 + 2*a*b^4*c^3 + a*b^4*c^2 + a*b^3*c^5 - 3*a*b^3*c^3 - 5*a*b^3*c^2 - a*b^2*c^5 - a*b^2*c^4 - 3*a*b^2*c^3 + 7*a*b^2*c^2 - a*b*c^4 + 4*a*b*c^3 - 3*a*b*c^2 - b^4*c^2 - 2*b^3*c^3 + 3*b^3*c^2 - b^2*c^4 + 4*b^2*c^3 - 3*b^2*c^2 + b*c^4 - 2*b*c^3 + b*c^2) (-2*a^2*b*c^4 - 4*a^2*b*c^3 - 4*a^2*b*c^2 - 2*a^2*b*c + a^2*c^2 + a^2*c + a^2 - 2*a*b^2*c^2 - 2*a*b^2*c + 2*a*b*c^4 + 2*a*b*c^3 + 3*a*b*c^2 + 3*a*b*c + 2*a*b + a*c^3 - a*c^2 - 2*a + b^2 - b*c^2 + b*c - 2*b - c^3 + c^2 - c + 1)/(a^2*b*c^4 + 2*a^2*b*c^3 + a^2*b*c^2 - a^2*c^2 + a*b^2*c^4 + 2*a*b^2*c^3 + a*b^2*c^2 + a*b*c^5 + a*b*c^4 - a*b*c^3 - 3*a*b*c^2 - 2*a*c^3 + 2*a*c^2 - b^2*c^2 - 2*b*c^3 + 2*b*c^2 - c^4 + 2*c^3 - c^2) (2*a^2*b^2*c^4 + 4*a^2*b^2*c^3 + 4*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c^2 - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c^2 + 2*a*b^3*c + 2*a*b^2*c^3 - 3*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^3 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^3 + 2*a^3*b^2*c^2 + a^3*b^2*c - a^3*b*c + a^2*b^3*c^3 + 2*a^2*b^3*c^2 + a^2*b^3*c + a^2*b^2*c^4 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3*a^2*b^2*c - 2*a^2*b*c^2 + 2*a^2*b*c - a*b^3*c - 2*a*b^2*c^2 + 2*a*b^2*c - a*b*c^3 + 2*a*b*c^2 - a*b*c) (2*a^2*b*c^2 + 2*a^2*b*c - a^2 - 2*a*b^2*c^4 - 4*a*b^2*c^3 + 2*a*b^2*c + 2*a*b*c^3 + a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c^2 + b^2*c - b^2 + b*c^3 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c^3 + 2*a^2*b^2*c^2 + a^2*b^2*c - a^2*b*c + a*b^3*c^3 + 2*a*b^3*c^2 + a*b^3*c + a*b^2*c^4 + a*b^2*c^3 - a*b^2*c^2 - 3*a*b^2*c - 2*a*b*c^2 + 2*a*b*c - b^3*c - 2*b^2*c^2 + 2*b^2*c - b*c^3 + 2*b*c^2 - b*c) (-a^2*b*c^3 - 2*a^2*b*c^2 - a^2*b*c + a^2 - a*b^2*c^3 - 2*a*b^2*c^2 - a*b^2*c + a*b*c^4 + 3*a*b*c^3 + 3*a*b*c^2 + a*b*c + 2*a*b - a*c^2 + a*c - 2*a + b^2 - b*c^2 + b*c - 2*b - c^3 + c^2 - c + 1)/(a^2*b*c^3 + 2*a^2*b*c^2 + a^2*b*c - a^2*c + a*b^2*c^3 + 2*a*b^2*c^2 + a*b^2*c + a*b*c^4 + a*b*c^3 - a*b*c^2 - 3*a*b*c - 2*a*c^2 + 2*a*c - b^2*c - 2*b*c^2 + 2*b*c - c^3 + 2*c^2 - c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 17:03:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; print Factorization(Numerator(1-D^2/4)); Output: Magma V2.11-10 Thu Dec 8 2005 17:03:07 on modular [Seed = 1038537621] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 17:02:01 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:02:01 on modular [Seed = 987878480] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:01:41 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],+D/2); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:01:40 on modular [Seed = 938268992] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*a^3*b^2*c^2 + a^3*b^2*c - 9/4*a^3*b*c - 3/8*a^3*b + 9/16*a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 2*a^2*b^3*c + 2*a^2*b^2*c^3 + 5/2*a^2*b^2*c^2 - 6*a^2*b^2*c - 9/8*a^2*b^2 - 9/2*a^2*b*c^2 + 21/8*a^2*b*c + 45/16*a^2*b + 27/16*a^2*c - 27/16*a^2 - a*b^4*c^2 + a*b^4*c - a*b^3*c^3 + 11/2*a*b^3*c^2 - 13/4*a*b^3*c - 9/8*a*b^3 + 7/2*a*b^2*c^3 - 8*a*b^2*c^2 + 3/4*a*b^2*c + 63/16*a*b^2 - 9/4*a*b*c^3 + 15/8*a*b*c^2 + 39/8*a*b*c - 9/2*a*b + 27/16*a*c^2 - 27/8*a*c + 27/16*a + 1/2*b^4*c - 3/8*b^4 + b^3*c^2 - 23/8*b^3*c + 27/16*b^3 + 1/2*b^2*c^3 - 29/8*b^2*c^2 + 95/16*b^2*c - 45/16*b^2 - 9/8*b*c^3 + 69/16*b*c^2 - 21/4*b*c + 33/16*b + 9/16*c^3 - 27/16*c^2 + 27/16*c - 9/16)/(a^4*b^2*c^2 - 3/4*a^4*b*c + 3/16*a^4 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 - 9/4*a^3*b^2*c - 3/2*a^3*b*c^2 + 9/4*a^3*b*c + 3/4*a^3*b + 9/16*a^3*c - 3/4*a^3 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 9/4*a^2*b^3*c - a^2*b^2*c^3 - 2*a^2*b^2*c^2 + 9/2*a^2*b^2*c + 9/8*a^2*b^2 - 3/4*a^2*b*c^3 + 3*a^2*b*c^2 - 9/16*a^2*b*c - 9/4*a^2*b + 9/16*a^2*c^2 - 27/16*a^2*c + 9/8*a^2 - 3/4*a*b^4*c - 3/2*a*b^3*c^2 + 9/4*a*b^3*c + 3/4*a*b^3 - 3/4*a*b^2*c^3 + 3*a*b^2*c^2 - 9/16*a*b^2*c - 9/4*a*b^2 + 3/4*a*b*c^3 - 3/8*a*b*c^2 - 21/8*a*b*c + 9/4*a*b + 3/16*a*c^3 - 9/8*a*c^2 + 27/16*a*c - 3/4*a + 3/16*b^4 + 9/16*b^3*c - 3/4*b^3 + 9/16*b^2*c^2 - 27/16*b^2*c + 9/8*b^2 + 3/16*b*c^3 - 9/8*b*c^2 + 27/16*b*c - 3/4*b - 3/16*c^3 + 9/16*c^2 - 9/16*c + 3/16) (-a^4*b^2*c^2 + 7/4*a^4*b*c - 9/16*a^4 - 2*a^3*b^3*c^2 - a^3*b^2*c^3 - a^3*b^2*c^2 + 17/4*a^3*b^2*c + 7/2*a^3*b*c^2 - 5/2*a^3*b*c - 15/8*a^3*b - 27/16*a^3*c + 27/16*a^3 - a^2*b^4*c^2 + a^2*b^3*c^3 - a^2*b^3*c^2 + 13/4*a^2*b^3*c - 2*a^2*b^2*c^3 + 11/2*a^2*b^2*c^2 - 3*a^2*b^2*c - 9/4*a^2*b^2 + 7/4*a^2*b*c^3 - 3/2*a^2*b*c^2 - 63/16*a^2*b*c + 63/16*a^2*b - 27/16*a^2*c^2 + 27/8*a^2*c - 27/16*a^2 + 3/4*a*b^4*c - 1/2*a*b^3*c - 9/8*a*b^3 - 3/4*a*b^2*c^3 + 2*a*b^2*c^2 - 57/16*a*b^2*c + 45/16*a*b^2 + a*b*c^3 - 15/4*a*b*c^2 + 5*a*b*c - 9/4*a*b - 9/16*a*c^3 + 27/16*a*c^2 - 27/16*a*c + 9/16*a - 3/16*b^4 - 5/16*b^3*c + 9/16*b^3 - 1/16*b^2*c^2 + 5/8*b^2*c - 9/16*b^2 + 1/16*b*c^3 + 1/16*b*c^2 - 5/16*b*c + 3/16*b)/(a^4*b^2*c^2 - 3/4*a^4*b*c + 3/16*a^4 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 - 9/4*a^3*b^2*c - 3/2*a^3*b*c^2 + 9/4*a^3*b*c + 3/4*a^3*b + 9/16*a^3*c - 3/4*a^3 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 9/4*a^2*b^3*c - a^2*b^2*c^3 - 2*a^2*b^2*c^2 + 9/2*a^2*b^2*c + 9/8*a^2*b^2 - 3/4*a^2*b*c^3 + 3*a^2*b*c^2 - 9/16*a^2*b*c - 9/4*a^2*b + 9/16*a^2*c^2 - 27/16*a^2*c + 9/8*a^2 - 3/4*a*b^4*c - 3/2*a*b^3*c^2 + 9/4*a*b^3*c + 3/4*a*b^3 - 3/4*a*b^2*c^3 + 3*a*b^2*c^2 - 9/16*a*b^2*c - 9/4*a*b^2 + 3/4*a*b*c^3 - 3/8*a*b*c^2 - 21/8*a*b*c + 9/4*a*b + 3/16*a*c^3 - 9/8*a*c^2 + 27/16*a*c - 3/4*a + 3/16*b^4 + 9/16*b^3*c - 3/4*b^3 + 9/16*b^2*c^2 - 27/16*b^2*c + 9/8*b^2 + 3/16*b*c^3 - 9/8*b*c^2 + 27/16*b*c - 3/4*b - 3/16*c^3 + 9/16*c^2 - 9/16*c + 3/16) (-a^3*b^2*c + 1/2*a^3*b + 2*a^2*b^3*c^2 - a^2*b^3*c - a^2*b^2*c^2 - 1/2*a^2*b^2*c + a^2*b^2 + a^2*b*c - 3/4*a^2*b - 3/2*a*b^3*c + 1/2*a*b^3 - 3/2*a*b^2*c^2 + 5/2*a*b^2*c - 1/2*a*b^2 + 1/2*a*b*c^2 - 1/2*a*b*c + 1/4*b^3 + 1/2*b^2*c - 1/2*b^2 + 1/4*b*c^2 - 1/2*b*c + 1/4*b)/(a^3*b^2*c^2 - 3/4*a^3*b*c + 3/16*a^3 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3/2*a^2*b^2*c - 3/2*a^2*b*c^2 + 3/2*a^2*b*c + 9/16*a^2*b + 9/16*a^2*c - 9/16*a^2 - 3/4*a*b^3*c - 3/2*a*b^2*c^2 + 3/2*a*b^2*c + 9/16*a*b^2 - 3/4*a*b*c^3 + 3/2*a*b*c^2 + 3/8*a*b*c - 9/8*a*b + 9/16*a*c^2 - 9/8*a*c + 9/16*a + 3/16*b^3 + 9/16*b^2*c - 9/16*b^2 + 9/16*b*c^2 - 9/8*b*c + 9/16*b + 3/16*c^3 - 9/16*c^2 + 9/16*c - 3/16) (-a^4*b^2*c^2 + 3/4*a^4*b*c - 3/16*a^4 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 5*a^3*b^2*c^2 + 13/4*a^3*b^2*c - a^3*b*c^2 - 5*a^3*b*c - 9/8*a^3*b + 3/16*a^3*c + 21/16*a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 5*a^2*b^3*c^2 + 17/4*a^2*b^3*c + 2*a^2*b^2*c^3 - 5/2*a^2*b^2*c^2 - 12*a^2*b^2*c - 9/4*a^2*b^2 - 7/4*a^2*b*c^3 - 7/2*a^2*b*c^2 + 111/16*a^2*b*c + 81/16*a^2*b + 15/16*a^2*c^2 + 9/8*a^2*c - 45/16*a^2 + 7/4*a*b^4*c + 5/2*a*b^3*c^2 - 7*a*b^3*c - 15/8*a*b^3 + 3/4*a*b^2*c^3 - 7*a*b^2*c^2 + 105/16*a*b^2*c + 99/16*a*b^2 - a*b*c^3 + 19/4*a*b*c^2 + 3/2*a*b*c - 27/4*a*b + 9/16*a*c^3 - 3/16*a*c^2 - 45/16*a*c + 39/16*a - 9/16*b^4 - 19/16*b^3*c + 39/16*b^3 - 11/16*b^2*c^2 + 31/8*b^2*c - 63/16*b^2 - 1/16*b*c^3 + 23/16*b*c^2 - 67/16*b*c + 45/16*b - 3/4*c^2 + 3/2*c - 3/4)/(a^4*b^2*c^2 - 3/4*a^4*b*c + 3/16*a^4 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 - 9/4*a^3*b^2*c - 3/2*a^3*b*c^2 + 9/4*a^3*b*c + 3/4*a^3*b + 9/16*a^3*c - 3/4*a^3 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 9/4*a^2*b^3*c - a^2*b^2*c^3 - 2*a^2*b^2*c^2 + 9/2*a^2*b^2*c + 9/8*a^2*b^2 - 3/4*a^2*b*c^3 + 3*a^2*b*c^2 - 9/16*a^2*b*c - 9/4*a^2*b + 9/16*a^2*c^2 - 27/16*a^2*c + 9/8*a^2 - 3/4*a*b^4*c - 3/2*a*b^3*c^2 + 9/4*a*b^3*c + 3/4*a*b^3 - 3/4*a*b^2*c^3 + 3*a*b^2*c^2 - 9/16*a*b^2*c - 9/4*a*b^2 + 3/4*a*b*c^3 - 3/8*a*b*c^2 - 21/8*a*b*c + 9/4*a*b + 3/16*a*c^3 - 9/8*a*c^2 + 27/16*a*c - 3/4*a + 3/16*b^4 + 9/16*b^3*c - 3/4*b^3 + 9/16*b^2*c^2 - 27/16*b^2*c + 9/8*b^2 + 3/16*b*c^3 - 9/8*b*c^2 + 27/16*b*c - 3/4*b - 3/16*c^3 + 9/16*c^2 - 9/16*c + 3/16) (a^4*b*c^2 + a^4*b*c - 1/2*a^4*c - 3/8*a^4 - 2*a^3*b^2*c^3 - a^3*b^2*c^2 + 2*a^3*b^2*c + a^3*b*c^3 + 3/2*a^3*b*c^2 - 7/4*a^3*b*c - 9/8*a^3*b - a^3*c^2 + 1/8*a^3*c + 15/16*a^3 - 2*a^2*b^3*c^2 + a^2*b^3*c + 9/2*a^2*b^2*c^2 - 9/8*a^2*b^2 + 3/2*a^2*b*c^3 - 5/2*a^2*b*c^2 - 9/4*a^2*b*c + 27/16*a^2*b - 1/2*a^2*c^3 + 3/8*a^2*c^2 + 15/16*a^2*c - 9/16*a^2 + 5/4*a*b^3*c - 3/8*a*b^3 + a*b^2*c^2 - 27/8*a*b^2*c + 9/16*a*b^2 - 1/4*a*b*c^3 - 13/8*a*b*c^2 + 19/8*a*b*c - 1/8*a*c^3 + 9/16*a*c^2 - 1/4*a*c - 3/16*a - 3/16*b^3 - 5/16*b^2*c + 9/16*b^2 - 1/16*b*c^2 + 5/8*b*c - 9/16*b + 1/16*c^3 + 1/16*c^2 - 5/16*c + 3/16)/(a^4*b^2*c^2 - 3/4*a^4*b*c + 3/16*a^4 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 - 9/4*a^3*b^2*c - 3/2*a^3*b*c^2 + 9/4*a^3*b*c + 3/4*a^3*b + 9/16*a^3*c - 3/4*a^3 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 9/4*a^2*b^3*c - a^2*b^2*c^3 - 2*a^2*b^2*c^2 + 9/2*a^2*b^2*c + 9/8*a^2*b^2 - 3/4*a^2*b*c^3 + 3*a^2*b*c^2 - 9/16*a^2*b*c - 9/4*a^2*b + 9/16*a^2*c^2 - 27/16*a^2*c + 9/8*a^2 - 3/4*a*b^4*c - 3/2*a*b^3*c^2 + 9/4*a*b^3*c + 3/4*a*b^3 - 3/4*a*b^2*c^3 + 3*a*b^2*c^2 - 9/16*a*b^2*c - 9/4*a*b^2 + 3/4*a*b*c^3 - 3/8*a*b*c^2 - 21/8*a*b*c + 9/4*a*b + 3/16*a*c^3 - 9/8*a*c^2 + 27/16*a*c - 3/4*a + 3/16*b^4 + 9/16*b^3*c - 3/4*b^3 + 9/16*b^2*c^2 - 27/16*b^2*c + 9/8*b^2 + 3/16*b*c^3 - 9/8*b*c^2 + 27/16*b*c - 3/4*b - 3/16*c^3 + 9/16*c^2 - 9/16*c + 3/16) (-2*a^3*b^2*c^2 - a^3*b^2*c + 3/2*a^3*b*c + 1/2*a^3*b - 1/4*a^3 - a^2*b^3*c + a^2*b^2*c^2 + 5/2*a^2*b^2*c + a^2*b^2 + 3/2*a^2*b*c^2 - 2*a^2*b*c - 3/2*a^2*b - 1/2*a^2*c + 3/4*a^2 + 1/2*a*b^3 - 1/2*a*b^2*c - 5/4*a*b^2 - a*b*c^2 + 3/2*a*b - 1/4*a*c^2 + a*c - 3/4*a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^3*b^2*c^2 - 3/4*a^3*b*c + 3/16*a^3 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3/2*a^2*b^2*c - 3/2*a^2*b*c^2 + 3/2*a^2*b*c + 9/16*a^2*b + 9/16*a^2*c - 9/16*a^2 - 3/4*a*b^3*c - 3/2*a*b^2*c^2 + 3/2*a*b^2*c + 9/16*a*b^2 - 3/4*a*b*c^3 + 3/2*a*b*c^2 + 3/8*a*b*c - 9/8*a*b + 9/16*a*c^2 - 9/8*a*c + 9/16*a + 3/16*b^3 + 9/16*b^2*c - 9/16*b^2 + 9/16*b*c^2 - 9/8*b*c + 9/16*b + 3/16*c^3 - 9/16*c^2 + 9/16*c - 3/16) (2*a^2*b^2*c^3 + a^2*b^2*c^2 - 5/2*a^2*b*c^2 - 1/2*a^2*b*c + 3/4*a^2*c + a*b^3*c^2 + a*b^2*c^3 - 7/2*a*b^2*c^2 - a*b^2*c - 5/2*a*b*c^3 + 3/2*a*b*c^2 + 5/2*a*b*c + 3/2*a*c^2 - 3/2*a*c - 1/2*b^3*c - b^2*c^2 + 7/4*b^2*c - 1/2*b*c^3 + 5/2*b*c^2 - 2*b*c + 3/4*c^3 - 3/2*c^2 + 3/4*c)/(a^3*b^2*c^2 - 3/4*a^3*b*c + 3/16*a^3 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3/2*a^2*b^2*c - 3/2*a^2*b*c^2 + 3/2*a^2*b*c + 9/16*a^2*b + 9/16*a^2*c - 9/16*a^2 - 3/4*a*b^3*c - 3/2*a*b^2*c^2 + 3/2*a*b^2*c + 9/16*a*b^2 - 3/4*a*b*c^3 + 3/2*a*b*c^2 + 3/8*a*b*c - 9/8*a*b + 9/16*a*c^2 - 9/8*a*c + 9/16*a + 3/16*b^3 + 9/16*b^2*c - 9/16*b^2 + 9/16*b*c^2 - 9/8*b*c + 9/16*b + 3/16*c^3 - 9/16*c^2 + 9/16*c - 3/16) (a^3*b*c^2 - 1/2*a^3*c - 2*a^2*b^2*c^3 + a^2*b^2*c^2 + a^2*b*c^3 + 1/2*a^2*b*c^2 - a^2*b*c - a^2*c^2 + 3/4*a^2*c + 3/2*a*b^2*c^2 - 1/2*a*b^2*c + 3/2*a*b*c^3 - 5/2*a*b*c^2 + 1/2*a*b*c - 1/2*a*c^3 + 1/2*a*c^2 - 1/4*b^2*c - 1/2*b*c^2 + 1/2*b*c - 1/4*c^3 + 1/2*c^2 - 1/4*c)/(a^3*b^2*c^2 - 3/4*a^3*b*c + 3/16*a^3 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3/2*a^2*b^2*c - 3/2*a^2*b*c^2 + 3/2*a^2*b*c + 9/16*a^2*b + 9/16*a^2*c - 9/16*a^2 - 3/4*a*b^3*c - 3/2*a*b^2*c^2 + 3/2*a*b^2*c + 9/16*a*b^2 - 3/4*a*b*c^3 + 3/2*a*b*c^2 + 3/8*a*b*c - 9/8*a*b + 9/16*a*c^2 - 9/8*a*c + 9/16*a + 3/16*b^3 + 9/16*b^2*c - 9/16*b^2 + 9/16*b*c^2 - 9/8*b*c + 9/16*b + 3/16*c^3 - 9/16*c^2 + 9/16*c - 3/16) (-a^3*b^2*c^2 + 3/4*a^3*b*c - 1/16*a^3 - a^2*b^3*c^2 + a^2*b^2*c^3 + a^2*b^2*c^2 + 3/2*a^2*b^2*c - 3/2*a^2*b*c - 3/16*a^2*b + 1/16*a^2*c + 3/16*a^2 + 3/4*a*b^3*c - 3/2*a*b^2*c - 3/16*a*b^2 - 3/4*a*b*c^3 + 7/8*a*b*c + 3/8*a*b + 5/16*a*c^2 - 1/8*a*c - 3/16*a - 1/16*b^3 + 1/16*b^2*c + 3/16*b^2 + 5/16*b*c^2 - 1/8*b*c - 3/16*b + 3/16*c^3 - 5/16*c^2 + 1/16*c + 1/16)/(a^3*b^2*c^2 - 3/4*a^3*b*c + 3/16*a^3 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3/2*a^2*b^2*c - 3/2*a^2*b*c^2 + 3/2*a^2*b*c + 9/16*a^2*b + 9/16*a^2*c - 9/16*a^2 - 3/4*a*b^3*c - 3/2*a*b^2*c^2 + 3/2*a*b^2*c + 9/16*a*b^2 - 3/4*a*b*c^3 + 3/2*a*b*c^2 + 3/8*a*b*c - 9/8*a*b + 9/16*a*c^2 - 9/8*a*c + 9/16*a + 3/16*b^3 + 9/16*b^2*c - 9/16*b^2 + 9/16*b*c^2 - 9/8*b*c + 9/16*b + 3/16*c^3 - 9/16*c^2 + 9/16*c - 3/16) T ** WARNING: Output too long, hence truncated. '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:01:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:01:30 on modular [Seed = 3540532677] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:01:20 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j]),-D/2); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:01:19 on modular [Seed = 3591190741] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print Evaluate(Q[i,j]),-D/2); ^ User error: bad syntax >> end for; ^ User error: bad syntax >> end for; ^ User error: bad syntax Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:58:15 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Factorization(Denominator(Evaluate(Numerator(Q[i,j]),0))); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:58:14 on modular [Seed = 3624089474] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , , , ] [ , , , , ] [ , , , ] [ , , , , ] [ , , , , ] [ , , , ] [ , , , ] [ , , , ] [ , , , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:57:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Numerator(Q[i,j]),0); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:57:34 on modular [Seed = 3256429984] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (-2*a^2*b*c + a^2 + 2*a*b^2*c^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c - 2*a - b^2*c + b^2 - b*c^2 + 3*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (-a^4*b^2*c^2 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 3*a^3*b^2*c - 3*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 7*a^2*b^3*c^2 + 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^3 - 4*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 3*a*b^4*c + 4*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 + a*b^2*c^3 - 11*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 - 2*a*b*c^3 + 8*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (2*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^3 - a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - 3*a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 6*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (-2*a^2*b*c^2 - 2*a^2*b*c + a^2*c + a^2 - 2*a*b^2*c + 3*a*b*c + 2*a*b + a*c^2 - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (2*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) (2*a^2*b*c - a^2 - 2*a*b^2*c^2 + 2*a*b^2*c + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c - b^2 + b*c^2 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c) (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 16:55:47 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Denominator(Q[i,j])*(a+b+c-1)*(a+b-1); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:55:47 on modular [Seed = 3307089162] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 16:52:11 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..1] do for j in [1..3] do print Numerator(Q[i,j])*(a+b+c-1)*(a+b-1); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:52:11 on modular [Seed = 3424068843] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a - 2*b*c + b + c - 1)*$.1^2 + (4*a^2*b*c + a^2*b - 2*a^2 - 4*a*b^2*c^2 + 2*a*b^2*c + 2*a*b^2 + 4*a*b*c^2 + a*b*c - 6*a*b - 4*a*c + 4*a - 2*b^3*c + b^3 - 2*b^2*c^2 + 7*b^2*c - 4*b^2 + 4*b*c^2 - 9*b*c + 5*b - 2*c^2 + 4*c - 2)/(a*b*c)*$.1 + (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^2*b^2*c^2) (-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)*$.1^2 + (-2*a^3*b*c + 2*a^3 - 4*a^2*b^2*c - 2*a^2*b*c^2 - 2*a^2*b*c + 5*a^2*b + 4*a^2*c - 4*a^2 - 2*a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + 4*a*b^2 - 4*a*b*c^2 + 9*a*b*c - 6*a*b + 2*a*c^2 - 4*a*c + 2*a + b^3 + b^2*c - 2*b^2 - b*c + b)/(a*b*c)*$.1 + (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a*b^2*c^2) (2*a*b + 2*b^2 - 2*b)*$.1^2 + (-2*a^3 + 4*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + 4*a*b^2*c - 2*a*b^2 - 6*a*b*c + 2*a*b + a*c - b^2 - b*c + 2*b + c - 1)/(a*c)*$.1 + (-2*a^3*b*c + a^3 + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 2*a^2*b*c^2 + 3*a^2*b*c + 3*a^2*b + 2*a^2*c - 3*a^2 + 2*a*b^3*c^2 - 2*a*b^3*c - 4*a*b^2*c^2 + 2*a*b^2*c + 3*a*b^2 + a*b*c^2 + 4*a*b*c - 6*a*b + a*c^2 - 4*a*c + 3*a - b^3*c + b^3 - b^2*c^2 + 4*b^2*c - 3*b^2 + 2*b*c^2 - 5*b*c + 3*b - c^2 + 2*c - 1)/(a*b*c^2) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:51:18 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..1] do for j in [1..3] do print Evaluate(Numerator(Q[i,j]),(a+b+c-1)*(a+b-1)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:51:17 on modular [Seed = 4146863630] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^7*b^2*c^2 - 2*a^6*b^3*c^3 + 5*a^6*b^3*c^2 + 3*a^6*b^2*c^3 - 5*a^6*b^2*c^2 - 8*a^5*b^4*c^3 + 10*a^5*b^4*c^2 - 4*a^5*b^3*c^4 + 20*a^5*b^3*c^3 - 20*a^5*b^3*c^2 + 3*a^5*b^2*c^4 - 12*a^5*b^2*c^3 + 14*a^5*b^2*c^2 + a^5*b^2*c - 2*a^5*b*c - 12*a^4*b^5*c^3 + 10*a^4*b^5*c^2 - 12*a^4*b^4*c^4 + 42*a^4*b^4*c^3 - 30*a^4*b^4*c^2 - 2*a^4*b^3*c^5 + 21*a^4*b^3*c^4 - 52*a^4*b^3*c^3 + 40*a^4*b^3*c^2 + 4*a^4*b^3*c + a^4*b^2*c^5 - 9*a^4*b^2*c^4 + 26*a^4*b^2*c^3 - 16*a^4*b^2*c^2 - 12*a^4*b^2*c - 6*a^4*b*c^2 + 8*a^4*b*c - 8*a^3*b^6*c^3 + 5*a^3*b^6*c^2 - 12*a^3*b^5*c^4 + 36*a^3*b^5*c^3 - 20*a^3*b^5*c^2 - 4*a^3*b^4*c^5 + 33*a^3*b^4*c^4 - 68*a^3*b^4*c^3 + 36*a^3*b^4*c^2 + 6*a^3*b^4*c + 6*a^3*b^3*c^5 - 34*a^3*b^3*c^4 + 64*a^3*b^3*c^3 - 20*a^3*b^3*c^2 - 24*a^3*b^3*c - 2*a^3*b^2*c^5 + 13*a^3*b^2*c^4 - 19*a^3*b^2*c^3 - 16*a^3*b^2*c^2 + 33*a^3*b^2*c - 6*a^3*b*c^3 + 18*a^3*b*c^2 - 16*a^3*b*c - a^3*b + a^3 - 2*a^2*b^7*c^3 + a^2*b^7*c^2 - 4*a^2*b^6*c^4 + 11*a^2*b^6*c^3 - 5*a^2*b^6*c^2 - 2*a^2*b^5*c^5 + 15*a^2*b^5*c^4 - 28*a^2*b^5*c^3 + 8*a^2*b^5*c^2 + 4*a^2*b^5*c + 5*a^2*b^4*c^5 - 25*a^2*b^4*c^4 + 34*a^2*b^4*c^3 + 8*a^2*b^4*c^2 - 20*a^2*b^4*c - 4*a^2*b^3*c^5 + 19*a^2*b^3*c^4 - 10*a^2*b^3*c^3 - 42*a^2*b^3*c^2 + 42*a^2*b^3*c + a^2*b^2*c^5 - 3*a^2*b^2*c^4 - 16*a^2*b^2*c^3 + 55*a^2*b^2*c^2 - 41*a^2*b^2*c - 3*a^2*b^2 - 2*a^2*b*c^4 + 12*a^2*b*c^3 - 26*a^2*b*c^2 + 12*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^6*c^2 + a*b^6*c - 4*a*b^5*c^3 + 12*a*b^5*c^2 - 6*a*b^5*c - 2*a*b^4*c^4 + 17*a*b^4*c^3 - 32*a*b^4*c^2 + 17*a*b^4*c + 6*a*b^3*c^4 - 30*a*b^3*c^3 + 49*a*b^3*c^2 - 24*a*b^3*c - 3*a*b^3 - 6*a*b^2*c^4 + 27*a*b^2*c^3 - 39*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 + 2*a*b*c^4 - 10*a*b*c^3 + 9*a*b*c^2 + 8*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-a^7*b^2*c^2 - 6*a^6*b^3*c^2 - 3*a^6*b^2*c^3 + 5*a^6*b^2*c^2 - 15*a^5*b^4*c^2 - 13*a^5*b^3*c^3 + 25*a^5*b^3*c^2 - 3*a^5*b^2*c^4 + 12*a^5*b^2*c^3 - 12*a^5*b^2*c^2 + 2*a^5*b*c - 20*a^4*b^5*c^2 - 22*a^4*b^4*c^3 + 50*a^4*b^4*c^2 - 8*a^4*b^3*c^4 + 40*a^4*b^3*c^3 - 48*a^4*b^3*c^2 - a^4*b^2*c^5 + 9*a^4*b^2*c^4 - 22*a^4*b^2*c^3 + 12*a^4*b^2*c^2 + 9*a^4*b^2*c + 6*a^4*b*c^2 - 8*a^4*b*c - 15*a^3*b^6*c^2 - 18*a^3*b^5*c^3 + 50*a^3*b^5*c^2 - 6*a^3*b^4*c^4 + 48*a^3*b^4*c^3 - 72*a^3*b^4*c^2 - a^3*b^3*c^5 + 15*a^3*b^3*c^4 - 50*a^3*b^3*c^3 + 36*a^3*b^3*c^2 + 16*a^3*b^3*c + 2*a^3*b^2*c^5 - 11*a^3*b^2*c^4 + 12*a^3*b^2*c^3 + 20*a^3*b^2*c^2 - 28*a^3*b^2*c + 6*a^3*b*c^3 - 18*a^3*b*c^2 + 15*a^3*b*c - a^3 - 6*a^2*b^7*c^2 - 7*a^2*b^6*c^3 + 25*a^2*b^6*c^2 + 24*a^2*b^5*c^3 - 48*a^2*b^5*c^2 + a^2*b^4*c^5 + 3*a^2*b^4*c^4 - 34*a^2*b^4*c^3 + 36*a^2*b^4*c^2 + 14*a^2*b^4*c - 6*a^2*b^3*c^4 + 8*a^2*b^3*c^3 + 24*a^2*b^3*c^2 - 36*a^2*b^3*c - a^2*b^2*c^5 + a^2*b^2*c^4 + 21*a^2*b^2*c^3 - 54*a^2*b^2*c^2 + 36*a^2*b^2*c + 2*a^2*b*c^4 - 12*a^2*b*c^3 + 24*a^2*b*c^2 - 12*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^8*c^2 - a*b^7*c^3 + 5*a*b^7*c^2 + a*b^6*c^4 + 4*a*b^6*c^3 - 12*a*b^6*c^2 + a*b^5*c^5 - 3*a*b^5*c^4 - 6*a*b^5*c^3 + 12*a*b^5*c^2 + 6*a*b^5*c - 2*a*b^4*c^5 + 5*a*b^4*c^4 - 4*a*b^4*c^3 + 12*a*b^4*c^2 - 20*a*b^4*c + a*b^3*c^5 - 7*a*b^3*c^4 + 24*a*b^3*c^3 - 42*a*b^3*c^2 + 27*a*b^3*c + 6*a*b^2*c^4 - 26*a*b^2*c^3 + 37*a*b^2*c^2 - 14*a*b^2*c - 3*a*b^2 - 2*a*b*c^4 + 9*a*b*c^3 - 8*a*b*c^2 - 5*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a + b^6*c + 2*b^5*c^2 - 4*b^5*c + b^4*c^3 - 6*b^4*c^2 + 6*b^4*c - 2*b^3*c^3 + 5*b^3*c^2 - 2*b^3*c - b^3 + 3*b^2*c^2 - 6*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (2*a^5*b^2*c^2 + 8*a^4*b^3*c^2 + 4*a^4*b^2*c^3 - 8*a^4*b^2*c^2 - 2*a^4*b*c + 12*a^3*b^4*c^2 + 12*a^3*b^3*c^3 - 24*a^3*b^3*c^2 + 2*a^3*b^2*c^4 - 12*a^3*b^2*c^3 + 16*a^3*b^2*c^2 - 6*a^3*b^2*c - 4*a^3*b*c^2 + 5*a^3*b*c + 8*a^2*b^5*c^2 + 12*a^2*b^4*c^3 - 24*a^2*b^4*c^2 + 4*a^2*b^3*c^4 - 24*a^2*b^3*c^3 + 32*a^2*b^3*c^2 - 6*a^2*b^3*c - 4*a^2*b^2*c^4 + 16*a^2*b^2*c^3 - 24*a^2*b^2*c^2 + 9*a^2*b^2*c - 2*a^2*b*c^3 + 6*a^2*b*c^2 - 5*a^2*b*c + a^2 + 2*a*b^6*c^2 + 4*a*b^5*c^3 - 8*a*b^5*c^2 + 2*a*b^4*c^4 - 12*a*b^4*c^3 + 16*a*b^4*c^2 - 2*a*b^4*c - 4*a*b^3*c^4 + 16*a*b^3*c^3 - 20*a*b^3*c^2 + 3*a*b^3*c + 2*a*b^2*c^4 - 10*a*b^2*c^3 + 12*a*b^2*c^2 - 2*a*b^2*c + a*b*c^3 - 2*a*b*c^2 + 2*a*b + 2*a*c - 2*a - b^4*c - 2*b^3*c^2 + 3*b^3*c - b^2*c^3 + 4*b^2*c^2 - 4*b^2*c + b^2 + b*c^3 - 3*b*c^2 + 4*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 16:47:53 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Numerator(Q[i,j])*(a+b+c-1)*(a+b-1); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:47:52 on modular [Seed = 2538372787] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a - 2*b*c + b + c - 1)*$.1^2 + (4*a^2*b*c + a^2*b - 2*a^2 - 4*a*b^2*c^2 + 2*a*b^2*c + 2*a*b^2 + 4*a*b*c^2 + a*b*c - 6*a*b - 4*a*c + 4*a - 2*b^3*c + b^3 - 2*b^2*c^2 + 7*b^2*c - 4*b^2 + 4*b*c^2 - 9*b*c + 5*b - 2*c^2 + 4*c - 2)/(a*b*c)*$.1 + (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^2*b^2*c^2) (-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)*$.1^2 + (-2*a^3*b*c + 2*a^3 - 4*a^2*b^2*c - 2*a^2*b*c^2 - 2*a^2*b*c + 5*a^2*b + 4*a^2*c - 4*a^2 - 2*a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + 4*a*b^2 - 4*a*b*c^2 + 9*a*b*c - 6*a*b + 2*a*c^2 - 4*a*c + 2*a + b^3 + b^2*c - 2*b^2 - b*c + b)/(a*b*c)*$.1 + (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a*b^2*c^2) (2*a*b + 2*b^2 - 2*b)*$.1^2 + (-2*a^3 + 4*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + 4*a*b^2*c - 2*a*b^2 - 6*a*b*c + 2*a*b + a*c - b^2 - b*c + 2*b + c - 1)/(a*c)*$.1 + (-2*a^3*b*c + a^3 + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 2*a^2*b*c^2 + 3*a^2*b*c + 3*a^2*b + 2*a^2*c - 3*a^2 + 2*a*b^3*c^2 - 2*a*b^3*c - 4*a*b^2*c^2 + 2*a*b^2*c + 3*a*b^2 + a*b*c^2 + 4*a*b*c - 6*a*b + a*c^2 - 4*a*c + 3*a - b^3*c + b^3 - b^2*c^2 + 4*b^2*c - 3*b^2 + 2*b*c^2 - 5*b*c + 3*b - c^2 + 2*c - 1)/(a*b*c^2) (-a^2 - 2*a*b + a*c + 3*a - b^2 - b*c + 3*b - 2)*$.1^2 + (-2*a^3*b*c + a^3 - 4*a^2*b^2*c + 2*a^2*b*c^2 + 10*a^2*b*c + 4*a^2*b - a^2*c - 5*a^2 - 2*a*b^3*c - 2*a*b^2*c^2 + 10*a*b^2*c + 5*a*b^2 + 4*a*b*c^2 - 7*a*b*c - 12*a*b - 2*a*c^2 - 2*a*c + 7*a + 2*b^3 + 2*b^2*c - 7*b^2 - 5*b*c + 8*b + 3*c - 3)/(a*b*c)*$.1 + (-a^4*b^2*c^2 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 3*a^3*b^2*c - 3*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 7*a^2*b^3*c^2 + 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^3 - 4*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 3*a*b^4*c + 4*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 + a*b^2*c^3 - 11*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 - 2*a*b*c^3 + 8*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^2*b^2*c^2) (-2*a*c - a - b + c + 1)*$.1^2 + (2*a^3*c + a^3 - 4*a^2*b*c^2 - 2*a^2*b*c + 2*a^2*b + 2*a^2*c^2 - a^2*c - a^2 - 4*a*b^2*c + a*b^2 + 5*a*b*c - a + b^2 + b*c - 2*b - c + 1)/(a*b*c)*$.1 + (2*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^3 - a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - 3*a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 6*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a*b^2*c^2) (-2*a^2 - 2*a*b + 4*a + 2*b - 2)*$.1^2 + (-4*a^3*b*c - 2*a^3*b + a^3 - 4*a^2*b^2*c - 4*a^2*b^2 + 6*a^2*b*c + 6*a^2*b + a^2*c - 3*a^2 - 2*a*b^3 + 2*a*b^2*c + 5*a*b^2 - a*b*c - 6*a*b - 2*a*c + 3*a - b^2 - b*c + 2*b + c - 1)/(a*b*c)*$.1 + (-2*a^3*b*c^2 - 2*a^3*b*c + a^3*c + a^3 - 2*a^2*b^2*c^2 - 4*a^2*b^2*c + 2*a^2*b*c^2 + 6*a^2*b*c + 3*a^2*b + a^2*c^2 - a^2*c - 3*a^2 - 2*a*b^3*c + 5*a*b^2*c + 3*a*b^2 + a*b*c^2 - 2*a*b*c - 6*a*b - a*c^2 - a*c + 3*a + b^3 + b^2*c - 3*b^2 - 2*b*c + 3*b + c - 1)/(a*b*c^2) (2*a*c + 2*b*c - 2*c)*$.1^2 + (4*a^2*b*c + 2*a^2*b - 3*a^2 + 4*a*b^2*c + 4*a*b^2 - 2*a*b*c - 10*a*b - 3*a*c + 6*a + 2*b^3 + 2*b^2*c - 7*b^2 - 5*b*c + 8*b + 3*c - 3)/(a*b)*$.1 + (2*a^3*b^2*c^2 + 2*a^3*b^2*c - 3*a^3*b*c - a^3*b + a^3 + 2*a^2*b^3*c^2 + 4*a^2*b^3*c - 10*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 2*a^2*c - 3*a^2 + 2*a*b^4*c + 2*a*b^3*c^2 - 7*a*b^3*c - 3*a*b^3 - 5*a*b^2*c^2 + 4*a*b^2*c + 9*a*b^2 + 2*a*b*c^2 + 5*a*b*c - 9*a*b + a*c^2 - 4*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^2*b^2*c) (-2*a*c - 2*b*c + 2*c)*$.1^2 + (2*a^3 - 4*a^2*b*c + 4*a^2*b + 2*a^2*c - 3*a^2 - 4*a*b^2*c + 2*a*b^2 + 6*a*b*c - 2*a*b - a*c + b^2 + b*c - 2*b - c + 1)/(a*b)*$.1 + (2*a^3*b*c - a^3 - 2*a^2*b^2*c^2 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 3*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 - 2*a*b^3*c^2 + 2*a*b^3*c + 4*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 - a*b*c^2 - 4*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a*b^2*c) (-a^2 - 2*a*b + a*c + 2*a - b^2 + b*c + 2*b - c - 1)*$.1^2 + (-2*a^3*b - 4*a^2*b^2 + 2*a^2*b*c + 4*a^2*b - a^2 - 2*a*b^3 + 2*a*b^2*c + 4*a*b^2 - 2*a*b*c - 4*a*b - a*c + 2*a - b^2 - b*c + 2*b + c - 1)/(a*b)*$.1 + (-a^3*b*c + a^3 - 2*a^2*b^2*c + a^2*b*c^2 + 2*a^2*b*c + 3*a^2*b + a^2*c - 3*a^2 - a*b^3*c + a*b^2*c^2 + 2*a*b^2*c + 3*a*b^2 - a*b*c^2 + a*b*c - 6*a*b - 2*a*c + 3*a + b^3 + b^2*c - 3*b^2 - 2*b*c + 3*b + c - 1)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:46:52 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Numerator(Q[i,j])*(a+b+c-1)^2*(a+b-1); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:46:52 on modular [Seed = 2587979898] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2 - 2*a*b*c + 2*a*b + 2*a*c - 2*a - 2*b^2*c + b^2 - 2*b*c^2 + 4*b*c - 2*b + c^2 - 2*c + 1)*$.1^2 + (4*a^3*b*c + a^3*b - 2*a^3 - 4*a^2*b^2*c^2 + 6*a^2*b^2*c + 3*a^2*b^2 + 8*a^2*b*c^2 - 2*a^2*b*c - 9*a^2*b - 6*a^2*c + 6*a^2 - 4*a*b^3*c^2 + 3*a*b^3 - 4*a*b^2*c^3 + 8*a*b^2*c^2 + 8*a*b^2*c - 12*a*b^2 + 4*a*b*c^3 + a*b*c^2 - 20*a*b*c + 15*a*b - 6*a*c^2 + 12*a*c - 6*a - 2*b^4*c + b^4 - 4*b^3*c^2 + 10*b^3*c - 5*b^3 - 2*b^2*c^3 + 13*b^2*c^2 - 20*b^2*c + 9*b^2 + 4*b*c^3 - 15*b*c^2 + 18*b*c - 7*b - 2*c^3 + 6*c^2 - 6*c + 2)/(a*b*c)*$.1 + (3*a^4*b^2*c^2 + 3*a^4*b^2*c - 4*a^4*b*c - a^4*b + a^4 - 2*a^3*b^3*c^3 + 4*a^3*b^3*c^2 + 9*a^3*b^3*c + 6*a^3*b^2*c^3 + 6*a^3*b^2*c^2 - 20*a^3*b^2*c - 4*a^3*b^2 - 12*a^3*b*c^2 + 7*a^3*b*c + 8*a^3*b + 4*a^3*c - 4*a^3 - 2*a^2*b^4*c^3 - a^2*b^4*c^2 + 9*a^2*b^4*c - 2*a^2*b^3*c^4 + 4*a^2*b^3*c^3 + 22*a^2*b^3*c^2 - 27*a^2*b^3*c - 6*a^2*b^3 + 3*a^2*b^2*c^4 + 9*a^2*b^2*c^3 - 42*a^2*b^2*c^2 + 15*a^2*b^2*c + 18*a^2*b^2 - 12*a^2*b*c^3 + 15*a^2*b*c^2 + 15*a^2*b*c - 18*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 2*a*b^5*c^2 + 3*a*b^5*c - 4*a*b^4*c^3 + 16*a*b^4*c^2 - 10*a*b^4*c - 4*a*b^4 - 2*a*b^3*c^4 + 19*a*b^3*c^3 - 32*a*b^3*c^2 + 16*a*b^3 + 6*a*b^2*c^4 - 24*a*b^2*c^3 + 12*a*b^2*c^2 + 30*a*b^2*c - 24*a*b^2 - 4*a*b*c^4 + 5*a*b*c^3 + 18*a*b*c^2 - 35*a*b*c + 16*a*b + 4*a*c^3 - 12*a*c^2 + 12*a*c - 4*a + b^5*c - b^5 + 3*b^4*c^2 - 8*b^4*c + 5*b^4 + 3*b^3*c^3 - 15*b^3*c^2 + 22*b^3*c - 10*b^3 + b^2*c^4 - 10*b^2*c^3 + 27*b^2*c^2 - 28*b^2*c + 10*b^2 - 2*b*c^4 + 11*b*c^3 - 21*b*c^2 + 17*b*c - 5*b + c^4 - 4*c^3 + 6*c^2 - 4*c + 1)/(a^2*b^2*c^2) (-a^3 - 3*a^2*b - 2*a^2*c + 2*a^2 - 3*a*b^2 - 2*a*b*c + 4*a*b - a*c^2 + 2*a*c - a - b^3 + 2*b^2 + b*c^2 - b)*$.1^2 + (-2*a^4*b*c + 2*a^4 - 6*a^3*b^2*c - 4*a^3*b*c^2 + 7*a^3*b + 6*a^3*c - 6*a^3 - 6*a^2*b^3*c - 4*a^2*b^2*c^2 + 9*a^2*b^2 - 2*a^2*b*c^3 - 4*a^2*b*c^2 + 20*a^2*b*c - 15*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 2*a*b^4*c + 5*a*b^3 + 2*a*b^2*c^3 - 8*a*b^2*c^2 + 16*a*b^2*c - 12*a*b^2 - 4*a*b*c^3 + 15*a*b*c^2 - 20*a*b*c + 9*a*b + 2*a*c^3 - 6*a*c^2 + 6*a*c - 2*a + b^4 + 2*b^3*c - 3*b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - b*c^2 + 2*b*c - b)/(a*b*c)*$.1 + (-a^4*b^2*c^2 + 3*a^4*b*c - a^4 - 3*a^3*b^3*c^2 - 2*a^3*b^2*c^3 - 2*a^3*b^2*c^2 + 9*a^3*b^2*c + 9*a^3*b*c^2 - 7*a^3*b*c - 4*a^3*b - 4*a^3*c + 4*a^3 - 3*a^2*b^4*c^2 - 2*a^2*b^3*c^3 - 4*a^2*b^3*c^2 + 9*a^2*b^3*c - a^2*b^2*c^4 - 6*a^2*b^2*c^3 + 24*a^2*b^2*c^2 - 12*a^2*b^2*c - 6*a^2*b^2 + 9*a^2*b*c^3 - 12*a^2*b*c^2 - 9*a^2*b*c + 12*a^2*b - 6*a^2*c^2 + 12*a^2*c - 6*a^2 - a*b^5*c^2 - 2*a*b^4*c^2 + 3*a*b^4*c + a*b^3*c^4 - 8*a*b^3*c^3 + 14*a*b^3*c^2 - 3*a*b^3*c - 4*a*b^3 - 4*a*b^2*c^4 + 15*a*b^2*c^3 - 8*a*b^2*c^2 - 15*a*b^2*c + 12*a*b^2 + 3*a*b*c^4 - 3*a*b*c^3 - 15*a*b*c^2 + 27*a*b*c - 12*a*b - 4*a*c^3 + 12*a*c^2 - 12*a*c + 4*a - b^4*c^2 + 2*b^4*c - b^4 - 2*b^3*c^3 + 8*b^3*c^2 - 10*b^3*c + 4*b^3 - b^2*c^4 + 8*b^2*c^3 - 19*b^2*c^2 + 18*b^2*c - 6*b^2 + 2*b*c^4 - 10*b*c^3 + 18*b*c^2 - 14*b*c + 4*b - c^4 + 4*c^3 - 6*c^2 + 4*c - 1)/(a*b^2*c^2) (2*a^2*b + 4*a*b^2 + 2*a*b*c - 4*a*b + 2*b^3 + 2*b^2*c - 4*b^2 - 2*b*c + 2*b)*$.1^2 + (-2*a^4 + 4*a^3*b*c - 6*a^3*b - 4*a^3*c + 5*a^3 + 8*a^2*b^2*c - 6*a^2*b^2 + 4*a^2*b*c^2 - 16*a^2*b*c + 9*a^2*b - 2*a^2*c^2 + 6*a^2*c - 3*a^2 + 4*a*b^3*c - 2*a*b^3 + 4*a*b^2*c^2 - 12*a*b^2*c + 3*a*b^2 - 6*a*b*c^2 + 8*a*b*c + a*c^2 - a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*c)*$.1 + (-2*a^4*b*c + a^4 + 2*a^3*b^2*c^2 - 6*a^3*b^2*c - 4*a^3*b*c^2 + 5*a^3*b*c + 4*a^3*b + 3*a^3*c - 4*a^3 + 4*a^2*b^3*c^2 - 6*a^2*b^3*c + 2*a^2*b^2*c^3 - 12*a^2*b^2*c^2 + 9*a^2*b^2*c + 6*a^2*b^2 - 2*a^2*b*c^3 + 6*a^2*b*c^2 + 6*a^2*b*c - 12*a^2*b + 3*a^2*c^2 - 9*a^2*c + 6*a^2 + 2*a*b^4*c^2 - 2*a*b^4*c + 2*a*b^3*c^3 - 8*a*b^3*c^2 + 3*a*b^3*c + 4*a*b^3 - 4*a*b^2*c^3 + 6*a*b^2*c^2 + 9*a*b^2*c - 12*a*b^2 + a*b*c^3 + 6*a*b*c^2 - 19*a*b*c + 12*a*b + a*c^3 - 6*a*c^2 + 9*a*c - 4*a - b^4*c + b^4 - 2*b^3*c^2 + 6*b^3*c - 4*b^3 - b^2*c^3 + 7*b^2*c^2 - 12*b^2*c + 6*b^2 + 2*b*c^3 - 8*b*c^2 + 10*b*c - 4*b - c^3 + 3*c^2 - 3*c + 1)/(a*b*c^2) (-a^3 - 3*a^2*b + 4*a^2 - 3*a*b^2 - 2*a*b*c + 8*a*b + a*c^2 + 2*a*c - 5*a - b^3 - 2*b^2*c + 4*b^2 - b*c^2 + 4*b*c - 5*b - 2*c + 2)*$.1^2 + (-2*a^4*b*c + a^4 - 6*a^3*b^2*c + 12*a^3*b*c + 5*a^3*b - 6*a^3 - 6*a^2*b^3*c - 4*a^2*b^2*c^2 + 24*a^2*b^2*c + 9*a^2*b^2 + 2*a^2*b*c^3 + 12*a^2*b*c^2 - 14*a^2*b*c - 21*a^2*b - 3*a^2*c^2 - 6*a^2*c + 12*a^2 - 2*a*b^4*c - 4*a*b^3*c^2 + 12*a*b^3*c + 7*a*b^3 - 2*a*b^2*c^3 + 16*a*b^2*c^2 - 10*a*b^2*c - 24*a*b^2 + 4*a*b*c^3 - 13*a*b*c^2 - 12*a*b*c + 27*a*b - 2*a*c^3 + 12*a*c - 10*a + 2*b^4 + 4*b^3*c - 9*b^3 + 2*b^2*c^2 - 14*b^2*c + 15*b^2 - 5*b*c^2 + 16*b*c - 11*b + 3*c^2 - 6*c + 3)/(a*b*c)*$.1 + (-a^5*b^2*c^2 - 3*a^4*b^3*c^2 + 8*a^4*b^2*c^2 + 3*a^4*b^2*c - 3*a^4*b*c^2 - 5*a^4*b*c - a^4*b + a^4*c + a^4 - 3*a^3*b^4*c^2 - 2*a^3*b^3*c^3 + 16*a^3*b^3*c^2 + 9*a^3*b^3*c + a^3*b^2*c^4 + 10*a^3*b^2*c^3 - 12*a^3*b^2*c^2 - 24*a^3*b^2*c - 4*a^3*b^2 - 6*a^3*b*c^3 - 6*a^3*b*c^2 + 15*a^3*b*c + 8*a^3*b + 3*a^3*c^2 - 4*a^3 - a^2*b^5*c^2 - 2*a^2*b^4*c^3 + 8*a^2*b^4*c^2 + 9*a^2*b^4*c - a^2*b^3*c^4 + 12*a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 33*a^2*b^3*c - 6*a^2*b^3 + 4*a^2*b^2*c^4 - 11*a^2*b^2*c^3 - 26*a^2*b^2*c^2 + 33*a^2*b^2*c + 18*a^2*b^2 - 3*a^2*b*c^4 - 3*a^2*b*c^3 + 24*a^2*b*c^2 - 3*a^2*b*c - 18*a^2*b + 3*a^2*c^3 - 3*a^2*c^2 - 6*a^2*c + 6*a^2 + 3*a*b^5*c + 7*a*b^4*c^2 - 14*a*b^4*c - 4*a*b^4 + 5*a*b^3*c^3 - 26*a*b^3*c^2 + 16*a*b^3*c + 16*a*b^3 + a*b^2*c^4 - 14*a*b^2*c^3 + 28*a*b^2*c^2 + 6*a*b^2*c - 24*a*b^2 - 2*a*b*c^4 + 11*a*b*c^3 - 6*a*b*c^2 - 19*a*b*c + 16*a*b + a*c^4 - 2*a*c^3 - 3*a*c^2 + 8*a*c - 4*a - b^5 - 3*b^4*c + 5*b^4 - 3*b^3*c^2 + 12*b^3*c - 10*b^3 - b^2*c^3 + 9*b^2*c^2 - 18*b^2*c + 10*b^2 + 2*b*c^3 - 9*b*c^2 + 12*b*c - 5*b - c^3 + 3*c^2 - 3*c + 1)/(a^2*b^2*c^2) (-2*a^2*c - a^2 - 2*a*b*c - 2*a*b - 2*a*c^2 + 2*a*c + 2*a - b^2 + 2*b + c^2 - 1)*$.1^2 + (2*a^4*c + a^4 - 4*a^3*b*c^2 + 3*a^3*b + 4*a^3*c^2 - 2*a^3*c - 2*a^3 - 4*a^2*b^2*c^2 - 6*a^2*b^2*c + 3*a^2*b^2 - 4*a^2*b*c^3 + 4*a^2*b*c^2 + 8*a^2*b*c - 3*a^2*b + 2*a^2*c^3 - 3*a^2*c^2 - 4*a*b^3*c + a*b^3 - 4*a*b^2*c^2 + 10*a*b^2*c + 5*a*b*c^2 - 4*a*b*c - 3*a*b - 2*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a*b*c)*$.1 + (2*a^4*b*c^2 + 3*a^4*b*c - a^4*c - a^4 - 2*a^3*b^2*c^3 + a^3*b^2*c^2 + 9*a^3*b^2*c + 4*a^3*b*c^3 + 4*a^3*b*c^2 - 11*a^3*b*c - 4*a^3*b - 3*a^3*c^2 + 4*a^3 - 2*a^2*b^3*c^3 - 4*a^2*b^3*c^2 + 9*a^2*b^3*c - 2*a^2*b^2*c^4 + 2*a^2*b^2*c^3 + 18*a^2*b^2*c^2 - 18*a^2*b^2*c - 6*a^2*b^2 + 2*a^2*b*c^4 + 3*a^2*b*c^3 - 18*a^2*b*c^2 + 3*a^2*b*c + 12*a^2*b - 3*a^2*c^3 + 3*a^2*c^2 + 6*a^2*c - 6*a^2 - 3*a*b^4*c^2 + 3*a*b^4*c - 4*a*b^3*c^3 + 14*a*b^3*c^2 - 7*a*b^3*c - 4*a*b^3 - a*b^2*c^4 + 11*a*b^2*c^3 - 18*a*b^2*c^2 - 3*a*b^2*c + 12*a*b^2 + 2*a*b*c^4 - 9*a*b*c^3 + 4*a*b*c^2 + 15*a*b*c - 12*a*b - a*c^4 + 2*a*c^3 + 3*a*c^2 - 8*a*c + 4*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a*b^2*c^2) (-2*a^3 - 4*a^2*b - 2*a^2*c + 6*a^2 - 2*a*b^2 - 2*a*b*c + 8*a*b + 4*a*c - 6*a + 2*b^2 + 2*b*c - 4*b - 2*c + 2)*$.1^2 + (-4*a^4*b*c - 2*a^4*b + a^4 - 8*a^3*b^2*c - 6*a^3*b^2 - 4*a^3*b*c^2 + 8*a^3*b*c + 9*a^3*b + 2*a^3*c - 4*a^3 - 4*a^2*b^3*c - 6*a^2*b^3 - 4*a^2*b^2*c^2 + 8*a^2*b^2*c + 15*a^2*b^2 + 6*a^2*b*c^2 - 15*a^2*b + a^2*c^2 - 6*a^2*c + 6*a^2 - 2*a*b^4 + 7*a*b^3 + 2*a*b^2*c^2 + 2*a*b^2*c - 12*a*b^2 - a*b*c^2 - 8*a*b*c + 11*a*b - 2*a*c^2 + 6*a*c - 4*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + (-2*a^4*b*c^2 - 2*a^4*b*c + a^4*c + a^4 - 4*a^3*b^2*c^2 - 6*a^3*b^2*c - 2*a^3*b*c^3 + 2*a^3*b*c^2 + 9*a^3*b*c + 4*a^3*b + 2*a^3*c^2 - a^3*c - 4*a^3 - 2*a^2*b^3*c^2 - 6*a^2*b^3*c - 2*a^2*b^2*c^3 + 15*a^2*b^2*c + 6*a^2*b^2 + 2*a^2*b*c^3 + 6*a^2*b*c^2 - 6*a^2*b*c - 12*a^2*b + a^2*c^3 - 3*a^2*c^2 - 3*a^2*c + 6*a^2 - 2*a*b^4*c - 2*a*b^3*c^2 + 7*a*b^3*c + 4*a*b^3 + 6*a*b^2*c^2 - 3*a*b^2*c - 12*a*b^2 + a*b*c^3 - 4*a*b*c^2 - 7*a*b*c + 12*a*b - a*c^3 + 5*a*c - 4*a + b^4 + 2*b^3*c - 4*b^3 + b^2*c^2 - 6*b^2*c + 6*b^2 - 2*b*c^2 + 6*b*c - 4*b + c^2 - 2*c + 1)/(a*b*c^2) (2*a^2*c + 4*a*b*c + 2*a*c^2 - 4*a*c + 2*b^2*c + 2*b*c^2 - 4*b*c - 2*c^2 + 2*c)*$.1^2 + (4*a^3*b*c + 2*a^3*b - 3*a^3 + 8*a^2*b^2*c + 6*a^2*b^2 + 4*a^2*b*c^2 - 4*a^2*b*c - 15*a^2*b - 6*a^2*c + 9*a^2 + 4*a*b^3*c + 6*a*b^3 + 4*a*b^2*c^2 - 21*a*b^2 - 2*a*b*c^2 - 16*a*b*c + 24*a*b - 3*a*c^2 + 12*a*c - 9*a + 2*b^4 + 4*b^3*c - 9*b^3 + 2*b^2*c^2 - 14*b^2*c + 15*b^2 - 5*b*c^2 + 16*b*c - 11*b + 3*c^2 - 6*c + 3)/(a*b)*$.1 + (2*a^4*b^2*c^2 + 2*a^4*b^2*c - 3*a^4*b*c - a^4*b + a^4 + 4*a^3*b^3*c^2 + 6*a^3*b^3*c + 2*a^3*b^2*c^3 - 15*a^3*b^2*c - 4*a^3*b^2 - 6*a^3*b*c^2 + 6*a^3*b*c + 8*a^3*b + 3*a^3*c - 4*a^3 + 2*a^2*b^4*c^2 + 6*a^2*b^4*c + 2*a^2*b^3*c^3 + 4*a^2*b^3*c^2 - 21*a^2*b^3*c - 6*a^2*b^3 - 18*a^2*b^2*c^2 + 15*a^2*b^2*c + 18*a^2*b^2 - 3*a^2*b*c^3 + 9*a^2*b*c^2 + 9*a^2*b*c - 18*a^2*b + 3*a^2*c^2 - 9*a^2*c + 6*a^2 + 2*a*b^5*c + 4*a*b^4*c^2 - 9*a*b^4*c - 4*a*b^4 + 2*a*b^3*c^3 - 14*a*b^3*c^2 + 6*a*b^3*c + 16*a*b^3 - 5*a*b^2*c^3 + 10*a*b^2*c^2 + 16*a*b^2*c - 24*a*b^2 + 2*a*b*c^3 + 6*a*b*c^2 - 24*a*b*c + 16*a*b + a*c^3 - 6*a*c^2 + 9*a*c - 4*a - b^5 - 3*b^4*c + 5*b^4 - 3*b^3*c^2 + 12*b^3*c - 10*b^3 - b^2*c^3 + 9*b^2*c^2 - 18*b^2*c + 10*b^2 + 2*b*c^3 - 9*b*c^2 + 12*b*c - 5*b - c^3 + 3*c^2 - 3*c + 1)/(a^2*b^2*c) (-2*a^2*c - 4*a*b*c - 2*a*c^2 + 4*a*c - 2*b ** WARNING: Output too long, hence truncated. '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:46:13 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Numerator(Q[i,j])*(a+b+c-1)^2; end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:46:12 on modular [Seed = 2638640917] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2 - 2*a*b*c + 2*a*b + 2*a*c - 2*a - 2*b^2*c + b^2 - 2*b*c^2 + 4*b*c - 2*b + c^2 - 2*c + 1)/(a + b - 1)*$.1^2 + (4*a^3*b*c + a^3*b - 2*a^3 - 4*a^2*b^2*c^2 + 6*a^2*b^2*c + 3*a^2*b^2 + 8*a^2*b*c^2 - 2*a^2*b*c - 9*a^2*b - 6*a^2*c + 6*a^2 - 4*a*b^3*c^2 + 3*a*b^3 - 4*a*b^2*c^3 + 8*a*b^2*c^2 + 8*a*b^2*c - 12*a*b^2 + 4*a*b*c^3 + a*b*c^2 - 20*a*b*c + 15*a*b - 6*a*c^2 + 12*a*c - 6*a - 2*b^4*c + b^4 - 4*b^3*c^2 + 10*b^3*c - 5*b^3 - 2*b^2*c^3 + 13*b^2*c^2 - 20*b^2*c + 9*b^2 + 4*b*c^3 - 15*b*c^2 + 18*b*c - 7*b - 2*c^3 + 6*c^2 - 6*c + 2)/(a^2*b*c + a*b^2*c - a*b*c)*$.1 + (3*a^4*b^2*c^2 + 3*a^4*b^2*c - 4*a^4*b*c - a^4*b + a^4 - 2*a^3*b^3*c^3 + 4*a^3*b^3*c^2 + 9*a^3*b^3*c + 6*a^3*b^2*c^3 + 6*a^3*b^2*c^2 - 20*a^3*b^2*c - 4*a^3*b^2 - 12*a^3*b*c^2 + 7*a^3*b*c + 8*a^3*b + 4*a^3*c - 4*a^3 - 2*a^2*b^4*c^3 - a^2*b^4*c^2 + 9*a^2*b^4*c - 2*a^2*b^3*c^4 + 4*a^2*b^3*c^3 + 22*a^2*b^3*c^2 - 27*a^2*b^3*c - 6*a^2*b^3 + 3*a^2*b^2*c^4 + 9*a^2*b^2*c^3 - 42*a^2*b^2*c^2 + 15*a^2*b^2*c + 18*a^2*b^2 - 12*a^2*b*c^3 + 15*a^2*b*c^2 + 15*a^2*b*c - 18*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 2*a*b^5*c^2 + 3*a*b^5*c - 4*a*b^4*c^3 + 16*a*b^4*c^2 - 10*a*b^4*c - 4*a*b^4 - 2*a*b^3*c^4 + 19*a*b^3*c^3 - 32*a*b^3*c^2 + 16*a*b^3 + 6*a*b^2*c^4 - 24*a*b^2*c^3 + 12*a*b^2*c^2 + 30*a*b^2*c - 24*a*b^2 - 4*a*b*c^4 + 5*a*b*c^3 + 18*a*b*c^2 - 35*a*b*c + 16*a*b + 4*a*c^3 - 12*a*c^2 + 12*a*c - 4*a + b^5*c - b^5 + 3*b^4*c^2 - 8*b^4*c + 5*b^4 + 3*b^3*c^3 - 15*b^3*c^2 + 22*b^3*c - 10*b^3 + b^2*c^4 - 10*b^2*c^3 + 27*b^2*c^2 - 28*b^2*c + 10*b^2 - 2*b*c^4 + 11*b*c^3 - 21*b*c^2 + 17*b*c - 5*b + c^4 - 4*c^3 + 6*c^2 - 4*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2) (-a^3 - 3*a^2*b - 2*a^2*c + 2*a^2 - 3*a*b^2 - 2*a*b*c + 4*a*b - a*c^2 + 2*a*c - a - b^3 + 2*b^2 + b*c^2 - b)/(a + b - 1)*$.1^2 + (-2*a^4*b*c + 2*a^4 - 6*a^3*b^2*c - 4*a^3*b*c^2 + 7*a^3*b + 6*a^3*c - 6*a^3 - 6*a^2*b^3*c - 4*a^2*b^2*c^2 + 9*a^2*b^2 - 2*a^2*b*c^3 - 4*a^2*b*c^2 + 20*a^2*b*c - 15*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 2*a*b^4*c + 5*a*b^3 + 2*a*b^2*c^3 - 8*a*b^2*c^2 + 16*a*b^2*c - 12*a*b^2 - 4*a*b*c^3 + 15*a*b*c^2 - 20*a*b*c + 9*a*b + 2*a*c^3 - 6*a*c^2 + 6*a*c - 2*a + b^4 + 2*b^3*c - 3*b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - b*c^2 + 2*b*c - b)/(a^2*b*c + a*b^2*c - a*b*c)*$.1 + (-a^4*b^2*c^2 + 3*a^4*b*c - a^4 - 3*a^3*b^3*c^2 - 2*a^3*b^2*c^3 - 2*a^3*b^2*c^2 + 9*a^3*b^2*c + 9*a^3*b*c^2 - 7*a^3*b*c - 4*a^3*b - 4*a^3*c + 4*a^3 - 3*a^2*b^4*c^2 - 2*a^2*b^3*c^3 - 4*a^2*b^3*c^2 + 9*a^2*b^3*c - a^2*b^2*c^4 - 6*a^2*b^2*c^3 + 24*a^2*b^2*c^2 - 12*a^2*b^2*c - 6*a^2*b^2 + 9*a^2*b*c^3 - 12*a^2*b*c^2 - 9*a^2*b*c + 12*a^2*b - 6*a^2*c^2 + 12*a^2*c - 6*a^2 - a*b^5*c^2 - 2*a*b^4*c^2 + 3*a*b^4*c + a*b^3*c^4 - 8*a*b^3*c^3 + 14*a*b^3*c^2 - 3*a*b^3*c - 4*a*b^3 - 4*a*b^2*c^4 + 15*a*b^2*c^3 - 8*a*b^2*c^2 - 15*a*b^2*c + 12*a*b^2 + 3*a*b*c^4 - 3*a*b*c^3 - 15*a*b*c^2 + 27*a*b*c - 12*a*b - 4*a*c^3 + 12*a*c^2 - 12*a*c + 4*a - b^4*c^2 + 2*b^4*c - b^4 - 2*b^3*c^3 + 8*b^3*c^2 - 10*b^3*c + 4*b^3 - b^2*c^4 + 8*b^2*c^3 - 19*b^2*c^2 + 18*b^2*c - 6*b^2 + 2*b*c^4 - 10*b*c^3 + 18*b*c^2 - 14*b*c + 4*b - c^4 + 4*c^3 - 6*c^2 + 4*c - 1)/(a^2*b^2*c^2 + a*b^3*c^2 - a*b^2*c^2) (2*a*b + 2*b^2 + 2*b*c - 2*b)*$.1^2 + (-2*a^3 + 4*a^2*b*c - 4*a^2*b - 4*a^2*c + 3*a^2 + 4*a*b^2*c - 2*a*b^2 + 4*a*b*c^2 - 8*a*b*c + 2*a*b - 2*a*c^2 + 2*a*c - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*c)*$.1 + (-2*a^3*b*c + a^3 + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 4*a^2*b*c^2 + 3*a^2*b*c + 3*a^2*b + 3*a^2*c - 3*a^2 + 2*a*b^3*c^2 - 2*a*b^3*c + 2*a*b^2*c^3 - 6*a*b^2*c^2 + 2*a*b^2*c + 3*a*b^2 - 2*a*b*c^3 + 2*a*b*c^2 + 6*a*b*c - 6*a*b + 3*a*c^2 - 6*a*c + 3*a - b^3*c + b^3 - 2*b^2*c^2 + 5*b^2*c - 3*b^2 - b*c^3 + 5*b*c^2 - 7*b*c + 3*b + c^3 - 3*c^2 + 3*c - 1)/(a*b*c^2) (-a^3 - 3*a^2*b + 4*a^2 - 3*a*b^2 - 2*a*b*c + 8*a*b + a*c^2 + 2*a*c - 5*a - b^3 - 2*b^2*c + 4*b^2 - b*c^2 + 4*b*c - 5*b - 2*c + 2)/(a + b - 1)*$.1^2 + (-2*a^4*b*c + a^4 - 6*a^3*b^2*c + 12*a^3*b*c + 5*a^3*b - 6*a^3 - 6*a^2*b^3*c - 4*a^2*b^2*c^2 + 24*a^2*b^2*c + 9*a^2*b^2 + 2*a^2*b*c^3 + 12*a^2*b*c^2 - 14*a^2*b*c - 21*a^2*b - 3*a^2*c^2 - 6*a^2*c + 12*a^2 - 2*a*b^4*c - 4*a*b^3*c^2 + 12*a*b^3*c + 7*a*b^3 - 2*a*b^2*c^3 + 16*a*b^2*c^2 - 10*a*b^2*c - 24*a*b^2 + 4*a*b*c^3 - 13*a*b*c^2 - 12*a*b*c + 27*a*b - 2*a*c^3 + 12*a*c - 10*a + 2*b^4 + 4*b^3*c - 9*b^3 + 2*b^2*c^2 - 14*b^2*c + 15*b^2 - 5*b*c^2 + 16*b*c - 11*b + 3*c^2 - 6*c + 3)/(a^2*b*c + a*b^2*c - a*b*c)*$.1 + (-a^5*b^2*c^2 - 3*a^4*b^3*c^2 + 8*a^4*b^2*c^2 + 3*a^4*b^2*c - 3*a^4*b*c^2 - 5*a^4*b*c - a^4*b + a^4*c + a^4 - 3*a^3*b^4*c^2 - 2*a^3*b^3*c^3 + 16*a^3*b^3*c^2 + 9*a^3*b^3*c + a^3*b^2*c^4 + 10*a^3*b^2*c^3 - 12*a^3*b^2*c^2 - 24*a^3*b^2*c - 4*a^3*b^2 - 6*a^3*b*c^3 - 6*a^3*b*c^2 + 15*a^3*b*c + 8*a^3*b + 3*a^3*c^2 - 4*a^3 - a^2*b^5*c^2 - 2*a^2*b^4*c^3 + 8*a^2*b^4*c^2 + 9*a^2*b^4*c - a^2*b^3*c^4 + 12*a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 33*a^2*b^3*c - 6*a^2*b^3 + 4*a^2*b^2*c^4 - 11*a^2*b^2*c^3 - 26*a^2*b^2*c^2 + 33*a^2*b^2*c + 18*a^2*b^2 - 3*a^2*b*c^4 - 3*a^2*b*c^3 + 24*a^2*b*c^2 - 3*a^2*b*c - 18*a^2*b + 3*a^2*c^3 - 3*a^2*c^2 - 6*a^2*c + 6*a^2 + 3*a*b^5*c + 7*a*b^4*c^2 - 14*a*b^4*c - 4*a*b^4 + 5*a*b^3*c^3 - 26*a*b^3*c^2 + 16*a*b^3*c + 16*a*b^3 + a*b^2*c^4 - 14*a*b^2*c^3 + 28*a*b^2*c^2 + 6*a*b^2*c - 24*a*b^2 - 2*a*b*c^4 + 11*a*b*c^3 - 6*a*b*c^2 - 19*a*b*c + 16*a*b + a*c^4 - 2*a*c^3 - 3*a*c^2 + 8*a*c - 4*a - b^5 - 3*b^4*c + 5*b^4 - 3*b^3*c^2 + 12*b^3*c - 10*b^3 - b^2*c^3 + 9*b^2*c^2 - 18*b^2*c + 10*b^2 + 2*b*c^3 - 9*b*c^2 + 12*b*c - 5*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2) (-2*a^2*c - a^2 - 2*a*b*c - 2*a*b - 2*a*c^2 + 2*a*c + 2*a - b^2 + 2*b + c^2 - 1)/(a + b - 1)*$.1^2 + (2*a^4*c + a^4 - 4*a^3*b*c^2 + 3*a^3*b + 4*a^3*c^2 - 2*a^3*c - 2*a^3 - 4*a^2*b^2*c^2 - 6*a^2*b^2*c + 3*a^2*b^2 - 4*a^2*b*c^3 + 4*a^2*b*c^2 + 8*a^2*b*c - 3*a^2*b + 2*a^2*c^3 - 3*a^2*c^2 - 4*a*b^3*c + a*b^3 - 4*a*b^2*c^2 + 10*a*b^2*c + 5*a*b*c^2 - 4*a*b*c - 3*a*b - 2*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c + a*b^2*c - a*b*c)*$.1 + (2*a^4*b*c^2 + 3*a^4*b*c - a^4*c - a^4 - 2*a^3*b^2*c^3 + a^3*b^2*c^2 + 9*a^3*b^2*c + 4*a^3*b*c^3 + 4*a^3*b*c^2 - 11*a^3*b*c - 4*a^3*b - 3*a^3*c^2 + 4*a^3 - 2*a^2*b^3*c^3 - 4*a^2*b^3*c^2 + 9*a^2*b^3*c - 2*a^2*b^2*c^4 + 2*a^2*b^2*c^3 + 18*a^2*b^2*c^2 - 18*a^2*b^2*c - 6*a^2*b^2 + 2*a^2*b*c^4 + 3*a^2*b*c^3 - 18*a^2*b*c^2 + 3*a^2*b*c + 12*a^2*b - 3*a^2*c^3 + 3*a^2*c^2 + 6*a^2*c - 6*a^2 - 3*a*b^4*c^2 + 3*a*b^4*c - 4*a*b^3*c^3 + 14*a*b^3*c^2 - 7*a*b^3*c - 4*a*b^3 - a*b^2*c^4 + 11*a*b^2*c^3 - 18*a*b^2*c^2 - 3*a*b^2*c + 12*a*b^2 + 2*a*b*c^4 - 9*a*b*c^3 + 4*a*b*c^2 + 15*a*b*c - 12*a*b - a*c^4 + 2*a*c^3 + 3*a*c^2 - 8*a*c + 4*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^2*b^2*c^2 + a*b^3*c^2 - a*b^2*c^2) (-2*a^2 - 2*a*b - 2*a*c + 4*a + 2*b + 2*c - 2)*$.1^2 + (-4*a^3*b*c - 2*a^3*b + a^3 - 4*a^2*b^2*c - 4*a^2*b^2 - 4*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 2*a^2*c - 3*a^2 - 2*a*b^3 + 5*a*b^2 + 2*a*b*c^2 + 2*a*b*c - 6*a*b + a*c^2 - 4*a*c + 3*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b*c)*$.1 + (-2*a^3*b*c^2 - 2*a^3*b*c + a^3*c + a^3 - 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 2*a^2*b*c^3 + 6*a^2*b*c + 3*a^2*b + 2*a^2*c^2 - 3*a^2 - 2*a*b^3*c - 2*a*b^2*c^2 + 5*a*b^2*c + 3*a*b^2 + 4*a*b*c^2 - 6*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a*b*c^2) (2*a*c + 2*b*c + 2*c^2 - 2*c)*$.1^2 + (4*a^2*b*c + 2*a^2*b - 3*a^2 + 4*a*b^2*c + 4*a*b^2 + 4*a*b*c^2 - 10*a*b - 6*a*c + 6*a + 2*b^3 + 4*b^2*c - 7*b^2 + 2*b*c^2 - 10*b*c + 8*b - 3*c^2 + 6*c - 3)/(a*b)*$.1 + (2*a^3*b^2*c^2 + 2*a^3*b^2*c - 3*a^3*b*c - a^3*b + a^3 + 2*a^2*b^3*c^2 + 4*a^2*b^3*c + 2*a^2*b^2*c^3 + 2*a^2*b^2*c^2 - 10*a^2*b^2*c - 3*a^2*b^2 - 6*a^2*b*c^2 + 3*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 + 2*a*b^4*c + 4*a*b^3*c^2 - 7*a*b^3*c - 3*a*b^3 + 2*a*b^2*c^3 - 10*a*b^2*c^2 + 2*a*b^2*c + 9*a*b^2 - 3*a*b*c^3 + 3*a*b*c^2 + 9*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a - b^4 - 3*b^3*c + 4*b^3 - 3*b^2*c^2 + 9*b^2*c - 6*b^2 - b*c^3 + 6*b*c^2 - 9*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^2*b^2*c) (-2*a*c - 2*b*c - 2*c^2 + 2*c)*$.1^2 + (2*a^3 - 4*a^2*b*c + 4*a^2*b + 4*a^2*c - 3*a^2 - 4*a*b^2*c + 2*a*b^2 - 4*a*b*c^2 + 8*a*b*c - 2*a*b + 2*a*c^2 - 2*a*c + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a*b)*$.1 + (2*a^3*b*c - a^3 - 2*a^2*b^2*c^2 + 4*a^2*b^2*c + 4*a^2*b*c^2 - 3*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - 2*a*b^3*c^2 + 2*a*b^3*c - 2*a*b^2*c^3 + 6*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 2*a*b*c^2 - 6*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a + b^3*c - b^3 + 2*b^2*c^2 - 5*b^2*c + 3*b^2 + b*c^3 - 5*b*c^2 + 7*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a*b^2*c) (-a^2 - 2*a*b + 2*a - b^2 + 2*b + c^2 - 1)*$.1^2 + (-2*a^3*b - 4*a^2*b^2 + 4*a^2*b - a^2 - 2*a*b^3 + 4*a*b^2 + 2*a*b*c^2 - 4*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b)*$.1 + (-a^3*b*c + a^3 - 2*a^2*b^2*c + 2*a^2*b*c + 3*a^2*b + 2*a^2*c - 3*a^2 - a*b^3*c + 2*a*b^2*c + 3*a*b^2 + a*b*c^3 + 3*a*b*c - 6*a*b + a*c^2 - 4*a*c + 3*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:45:31 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Numerator(Q[i,j]); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:45:30 on modular [Seed = 2153474774] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a - 2*b*c + b + c - 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (4*a^2*b*c + a^2*b - 2*a^2 - 4*a*b^2*c^2 + 2*a*b^2*c + 2*a*b^2 + 4*a*b*c^2 + a*b*c - 6*a*b - 4*a*c + 4*a - 2*b^3*c + b^3 - 2*b^2*c^2 + 7*b^2*c - 4*b^2 + 4*b*c^2 - 9*b*c + 5*b - 2*c^2 + 4*c - 2)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (-2*a^3*b*c + 2*a^3 - 4*a^2*b^2*c - 2*a^2*b*c^2 - 2*a^2*b*c + 5*a^2*b + 4*a^2*c - 4*a^2 - 2*a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + 4*a*b^2 - 4*a*b*c^2 + 9*a*b*c - 6*a*b + 2*a*c^2 - 4*a*c + 2*a + b^3 + b^2*c - 2*b^2 - b*c + b)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) 2*b/(a + b + c - 1)*$.1^2 + (-2*a^2 + 4*a*b*c - 2*a*b - 2*a*c + a - b - c + 1)/(a^2*c + a*b*c + a*c^2 - a*c)*$.1 + (-2*a^2*b*c + a^2 + 2*a*b^2*c^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c - 2*a - b^2*c + b^2 - b*c^2 + 3*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (-a^2 - 2*a*b + a*c + 3*a - b^2 - b*c + 3*b - 2)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (-2*a^3*b*c + a^3 - 4*a^2*b^2*c + 2*a^2*b*c^2 + 10*a^2*b*c + 4*a^2*b - a^2*c - 5*a^2 - 2*a*b^3*c - 2*a*b^2*c^2 + 10*a*b^2*c + 5*a*b^2 + 4*a*b*c^2 - 7*a*b*c - 12*a*b - 2*a*c^2 - 2*a*c + 7*a + 2*b^3 + 2*b^2*c - 7*b^2 - 5*b*c + 8*b + 3*c - 3)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (-a^4*b^2*c^2 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 3*a^3*b^2*c - 3*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 7*a^2*b^3*c^2 + 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^3 - 4*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 3*a*b^4*c + 4*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 + a*b^2*c^3 - 11*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 - 2*a*b*c^3 + 8*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-2*a*c - a - b + c + 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*c + a^3 - 4*a^2*b*c^2 - 2*a^2*b*c + 2*a^2*b + 2*a^2*c^2 - a^2*c - a^2 - 4*a*b^2*c + a*b^2 + 5*a*b*c - a + b^2 + b*c - 2*b - c + 1)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (2*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^3 - a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - 3*a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 6*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (-2*a + 2)/(a + b + c - 1)*$.1^2 + (-4*a^2*b*c - 2*a^2*b + a^2 - 2*a*b^2 + 2*a*b*c + 3*a*b + a*c - 2*a - b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*$.1 + (-2*a^2*b*c^2 - 2*a^2*b*c + a^2*c + a^2 - 2*a*b^2*c + 3*a*b*c + 2*a*b + a*c^2 - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) 2*c/(a + b + c - 1)*$.1^2 + (4*a*b*c + 2*a*b - 3*a + 2*b^2 + 2*b*c - 5*b - 3*c + 3)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (2*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) -2*c/(a + b + c - 1)*$.1^2 + (2*a^2 - 4*a*b*c + 2*a*b + 2*a*c - a + b + c - 1)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (2*a^2*b*c - a^2 - 2*a*b^2*c^2 + 2*a*b^2*c + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c - b^2 + b*c^2 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c) (-a - b + c + 1)/(a + b + c - 1)*$.1^2 + (-2*a^2*b - 2*a*b^2 + 2*a*b*c + 2*a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 16:45:05 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Numerator(Q[i,j]),1/(a+b+c-1)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:45:04 on modular [Seed = 2337300935] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (3*a^5*b^2*c^2 + 3*a^5*b^2*c - 4*a^5*b*c - a^5*b + a^5 - 2*a^4*b^3*c^3 + 7*a^4*b^3*c^2 + 12*a^4*b^3*c + 9*a^4*b^2*c^3 + 10*a^4*b^2*c^2 - 26*a^4*b^2*c - 5*a^4*b^2 - 16*a^4*b*c^2 + 8*a^4*b*c + 10*a^4*b + 5*a^4*c - 5*a^4 - 4*a^3*b^4*c^3 + 3*a^3*b^4*c^2 + 18*a^3*b^4*c - 4*a^3*b^3*c^4 + 12*a^3*b^3*c^3 + 39*a^3*b^3*c^2 - 53*a^3*b^3*c - 10*a^3*b^3 + 9*a^3*b^2*c^4 + 17*a^3*b^2*c^3 - 81*a^3*b^2*c^2 + 29*a^3*b^2*c + 30*a^3*b^2 - 24*a^3*b*c^3 + 28*a^3*b*c^2 + 26*a^3*b*c - 30*a^3*b + 10*a^3*c^2 - 20*a^3*c + 10*a^3 - 2*a^2*b^5*c^3 - 3*a^2*b^5*c^2 + 12*a^2*b^5*c - 4*a^2*b^4*c^4 - 3*a^2*b^4*c^3 + 48*a^2*b^4*c^2 - 43*a^2*b^4*c - 10*a^2*b^4 - 2*a^2*b^3*c^5 + 3*a^2*b^3*c^4 + 52*a^2*b^3*c^3 - 114*a^2*b^3*c^2 + 24*a^2*b^3*c + 40*a^2*b^3 + 3*a^2*b^2*c^5 + 16*a^2*b^2*c^4 - 85*a^2*b^2*c^3 + 63*a^2*b^2*c^2 + 63*a^2*b^2*c - 60*a^2*b^2 - 16*a^2*b*c^4 + 26*a^2*b*c^3 + 36*a^2*b*c^2 - 86*a^2*b*c + 40*a^2*b + 10*a^2*c^3 - 30*a^2*c^2 + 30*a^2*c - 10*a^2 - 2*a*b^6*c^2 + 3*a*b^6*c - 6*a*b^5*c^3 + 19*a*b^5*c^2 - 11*a*b^5*c - 5*a*b^5 - 6*a*b^4*c^4 + 35*a*b^4*c^3 - 45*a*b^4*c^2 - 7*a*b^4*c + 25*a*b^4 - 2*a*b^3*c^5 + 25*a*b^3*c^4 - 59*a*b^3*c^3 + 9*a*b^3*c^2 + 77*a*b^3*c - 50*a*b^3 + 6*a*b^2*c^5 - 29*a*b^2*c^4 + 16*a*b^2*c^3 + 81*a*b^2*c^2 - 124*a*b^2*c + 50*a*b^2 - 4*a*b*c^5 + 5*a*b*c^4 + 34*a*b*c^3 - 92*a*b*c^2 + 82*a*b*c - 25*a*b + 5*a*c^4 - 20*a*c^3 + 30*a*c^2 - 20*a*c + 5*a + b^6*c - b^6 + 4*b^5*c^2 - 10*b^5*c + 6*b^5 + 6*b^4*c^3 - 26*b^4*c^2 + 35*b^4*c - 15*b^4 + 4*b^3*c^4 - 28*b^3*c^3 + 64*b^3*c^2 - 60*b^3*c + 20*b^3 + b^2*c^5 - 13*b^2*c^4 + 48*b^2*c^3 - 76*b^2*c^2 + 55*b^2*c - 15*b^2 - 2*b*c^5 + 14*b*c^4 - 36*b*c^3 + 44*b*c^2 - 26*b*c + 6*b + c^5 - 5*c^4 + 10*c^3 - 10*c^2 + 5*c - 1)/(a^6*b^2*c^2 + 4*a^5*b^3*c^2 + 3*a^5*b^2*c^3 - 4*a^5*b^2*c^2 + 6*a^4*b^4*c^2 + 9*a^4*b^3*c^3 - 12*a^4*b^3*c^2 + 3*a^4*b^2*c^4 - 9*a^4*b^2*c^3 + 6*a^4*b^2*c^2 + 4*a^3*b^5*c^2 + 9*a^3*b^4*c^3 - 12*a^3*b^4*c^2 + 6*a^3*b^3*c^4 - 18*a^3*b^3*c^3 + 12*a^3*b^3*c^2 + a^3*b^2*c^5 - 6*a^3*b^2*c^4 + 9*a^3*b^2*c^3 - 4*a^3*b^2*c^2 + a^2*b^6*c^2 + 3*a^2*b^5*c^3 - 4*a^2*b^5*c^2 + 3*a^2*b^4*c^4 - 9*a^2*b^4*c^3 + 6*a^2*b^4*c^2 + a^2*b^3*c^5 - 6*a^2*b^3*c^4 + 9*a^2*b^3*c^3 - 4*a^2*b^3*c^2 - a^2*b^2*c^5 + 3*a^2*b^2*c^4 - 3*a^2*b^2*c^3 + a^2*b^2*c^2) (-a^5*b^2*c^2 + 3*a^5*b*c - a^5 - 4*a^4*b^3*c^2 - 3*a^4*b^2*c^3 - 3*a^4*b^2*c^2 + 12*a^4*b^2*c + 12*a^4*b*c^2 - 8*a^4*b*c - 5*a^4*b - 5*a^4*c + 5*a^4 - 6*a^3*b^4*c^2 - 7*a^3*b^3*c^3 - 9*a^3*b^3*c^2 + 18*a^3*b^3*c - 3*a^3*b^2*c^4 - 10*a^3*b^2*c^3 + 43*a^3*b^2*c^2 - 21*a^3*b^2*c - 10*a^3*b^2 + 18*a^3*b*c^3 - 22*a^3*b*c^2 - 16*a^3*b*c + 20*a^3*b - 10*a^3*c^2 + 20*a^3*c - 10*a^3 - 4*a^2*b^5*c^2 - 5*a^2*b^4*c^3 - 9*a^2*b^4*c^2 + 12*a^2*b^4*c - 2*a^2*b^3*c^4 - 20*a^2*b^3*c^3 + 49*a^2*b^3*c^2 - 15*a^2*b^3*c - 10*a^2*b^3 - a^2*b^2*c^5 - 11*a^2*b^2*c^4 + 49*a^2*b^2*c^3 - 35*a^2*b^2*c^2 - 33*a^2*b^2*c + 30*a^2*b^2 + 12*a^2*b*c^4 - 18*a^2*b*c^3 - 30*a^2*b*c^2 + 66*a^2*b*c - 30*a^2*b - 10*a^2*c^3 + 30*a^2*c^2 - 30*a^2*c + 10*a^2 - a*b^6*c^2 - a*b^5*c^3 - 3*a*b^5*c^2 + 3*a*b^5*c + a*b^4*c^4 - 10*a*b^4*c^3 + 17*a*b^4*c^2 + a*b^4*c - 5*a*b^4 + a*b^3*c^5 - 11*a*b^3*c^4 + 28*a*b^3*c^3 - 38*a*b^3*c + 20*a*b^3 - 4*a*b^2*c^5 + 17*a*b^2*c^4 - 3*a*b^2*c^3 - 61*a*b^2*c^2 + 81*a*b^2*c - 30*a*b^2 + 3*a*b*c^5 - 2*a*b*c^4 - 32*a*b*c^3 + 78*a*b*c^2 - 67*a*b*c + 20*a*b - 5*a*c^4 + 20*a*c^3 - 30*a*c^2 + 20*a*c - 5*a - b^5*c^2 + 3*b^5*c - b^5 - 3*b^4*c^3 + 13*b^4*c^2 - 16*b^4*c + 5*b^4 - 3*b^3*c^4 + 19*b^3*c^3 - 41*b^3*c^2 + 35*b^3*c - 10*b^3 - b^2*c^5 + 11*b^2*c^4 - 38*b^2*c^3 + 57*b^2*c^2 - 39*b^2*c + 10*b^2 + 2*b*c^5 - 13*b*c^4 + 32*b*c^3 - 38*b*c^2 + 22*b*c - 5*b - c^5 + 5*c^4 - 10*c^3 + 10*c^2 - 5*c + 1)/(a^5*b^2*c^2 + 4*a^4*b^3*c^2 + 3*a^4*b^2*c^3 - 4*a^4*b^2*c^2 + 6*a^3*b^4*c^2 + 9*a^3*b^3*c^3 - 12*a^3*b^3*c^2 + 3*a^3*b^2*c^4 - 9*a^3*b^2*c^3 + 6*a^3*b^2*c^2 + 4*a^2*b^5*c^2 + 9*a^2*b^4*c^3 - 12*a^2*b^4*c^2 + 6*a^2*b^3*c^4 - 18*a^2*b^3*c^3 + 12*a^2*b^3*c^2 + a^2*b^2*c^5 - 6*a^2*b^2*c^4 + 9*a^2*b^2*c^3 - 4*a^2*b^2*c^2 + a*b^6*c^2 + 3*a*b^5*c^3 - 4*a*b^5*c^2 + 3*a*b^4*c^4 - 9*a*b^4*c^3 + 6*a*b^4*c^2 + a*b^3*c^5 - 6*a*b^3*c^4 + 9*a*b^3*c^3 - 4*a*b^3*c^2 - a*b^2*c^5 + 3*a*b^2*c^4 - 3*a*b^2*c^3 + a*b^2*c^2) (-2*a^4*b*c + a^4 + 2*a^3*b^2*c^2 - 6*a^3*b^2*c - 6*a^3*b*c^2 + 3*a^3*b*c + 4*a^3*b + 4*a^3*c - 4*a^3 + 4*a^2*b^3*c^2 - 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 12*a^2*b^2*c^2 + 5*a^2*b^2*c + 6*a^2*b^2 - 6*a^2*b*c^3 + 5*a^2*b*c^2 + 12*a^2*b*c - 12*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 + 2*a*b^4*c^2 - 2*a*b^4*c + 4*a*b^3*c^3 - 6*a*b^3*c^2 + a*b^3*c + 4*a*b^3 + 2*a*b^2*c^4 - 6*a*b^2*c^3 + 2*a*b^2*c^2 + 14*a*b^2*c - 12*a*b^2 - 2*a*b*c^4 + a*b*c^3 + 14*a*b*c^2 - 25*a*b*c + 12*a*b + 4*a*c^3 - 12*a*c^2 + 12*a*c - 4*a - b^4*c + b^4 - 3*b^3*c^2 + 6*b^3*c - 4*b^3 - 3*b^2*c^3 + 10*b^2*c^2 - 13*b^2*c + 6*b^2 - b*c^4 + 6*b*c^3 - 13*b*c^2 + 12*b*c - 4*b + c^4 - 4*c^3 + 6*c^2 - 4*c + 1)/(a^4*b*c^2 + 3*a^3*b^2*c^2 + 3*a^3*b*c^3 - 3*a^3*b*c^2 + 3*a^2*b^3*c^2 + 6*a^2*b^2*c^3 - 6*a^2*b^2*c^2 + 3*a^2*b*c^4 - 6*a^2*b*c^3 + 3*a^2*b*c^2 + a*b^4*c^2 + 3*a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^2*c^4 - 6*a*b^2*c^3 + 3*a*b^2*c^2 + a*b*c^5 - 3*a*b*c^4 + 3*a*b*c^3 - a*b*c^2) (-a^6*b^2*c^2 - 4*a^5*b^3*c^2 - a^5*b^2*c^3 + 7*a^5*b^2*c^2 + 3*a^5*b^2*c - 3*a^5*b*c^2 - 4*a^5*b*c - a^5*b + a^5*c + a^5 - 6*a^4*b^4*c^2 - 5*a^4*b^3*c^3 + 21*a^4*b^3*c^2 + 12*a^4*b^3*c + a^4*b^2*c^4 + 18*a^4*b^2*c^3 - 9*a^4*b^2*c^2 - 27*a^4*b^2*c - 5*a^4*b^2 - 9*a^4*b*c^3 - 8*a^4*b*c^2 + 14*a^4*b*c + 10*a^4*b + 4*a^4*c^2 - 5*a^4 - 4*a^3*b^5*c^2 - 7*a^3*b^4*c^3 + 21*a^3*b^4*c^2 + 18*a^3*b^4*c - 2*a^3*b^3*c^4 + 36*a^3*b^3*c^3 + a^3*b^3*c^2 - 57*a^3*b^3*c - 10*a^3*b^3 + a^3*b^2*c^5 + 15*a^3*b^2*c^4 - 26*a^3*b^2*c^3 - 55*a^3*b^2*c^2 + 47*a^3*b^2*c + 30*a^3*b^2 - 9*a^3*b*c^4 - 6*a^3*b*c^3 + 42*a^3*b*c^2 + 2*a^3*b*c - 30*a^3*b + 6*a^3*c^3 - 6*a^3*c^2 - 10*a^3*c + 10*a^3 - a^2*b^6*c^2 - 3*a^2*b^5*c^3 + 7*a^2*b^5*c^2 + 12*a^2*b^5*c - 3*a^2*b^4*c^4 + 18*a^2*b^4*c^3 + 17*a^2*b^4*c^2 - 49*a^2*b^4*c - 10*a^2*b^4 - a^2*b^3*c^5 + 15*a^2*b^3*c^4 - 5*a^2*b^3*c^3 - 90*a^2*b^3*c^2 + 52*a^2*b^3*c + 40*a^2*b^3 + 4*a^2*b^2*c^5 - 13*a^2*b^2*c^4 - 45*a^2*b^2*c^3 + 97*a^2*b^2*c^2 + 15*a^2*b^2*c - 60*a^2*b^2 - 3*a^2*b*c^5 - 4*a^2*b*c^4 + 41*a^2*b*c^3 - 24*a^2*b*c^2 - 50*a^2*b*c + 40*a^2*b + 4*a^2*c^4 - 8*a^2*c^3 - 6*a^2*c^2 + 20*a^2*c - 10*a^2 + 3*a*b^6*c + 10*a*b^5*c^2 - 15*a*b^5*c - 5*a*b^5 + 12*a*b^4*c^3 - 43*a*b^4*c^2 + 14*a*b^4*c + 25*a*b^4 + 6*a*b^3*c^4 - 43*a*b^3*c^3 + 53*a*b^3*c^2 + 33*a*b^3*c - 50*a*b^3 + a*b^2*c^5 - 17*a*b^2*c^4 + 47*a*b^2*c^3 - 3*a*b^2*c^2 - 78*a*b^2*c + 50*a*b^2 - 2*a*b*c^5 + 14*a*b*c^4 - 14*a*b*c^3 - 31*a*b*c^2 + 58*a*b*c - 25*a*b + a*c^5 - 3*a*c^4 - 2*a*c^3 + 14*a*c^2 - 15*a*c + 5*a - b^6 - 4*b^5*c + 6*b^5 - 6*b^4*c^2 + 20*b^4*c - 15*b^4 - 4*b^3*c^3 + 24*b^3*c^2 - 40*b^3*c + 20*b^3 - b^2*c^4 + 12*b^2*c^3 - 36*b^2*c^2 + 40*b^2*c - 15*b^2 + 2*b*c^4 - 12*b*c^3 + 24*b*c^2 - 20*b*c + 6*b - c^4 + 4*c^3 - 6*c^2 + 4*c - 1)/(a^6*b^2*c^2 + 4*a^5*b^3*c^2 + 3*a^5*b^2*c^3 - 4*a^5*b^2*c^2 + 6*a^4*b^4*c^2 + 9*a^4*b^3*c^3 - 12*a^4*b^3*c^2 + 3*a^4*b^2*c^4 - 9*a^4*b^2*c^3 + 6*a^4*b^2*c^2 + 4*a^3*b^5*c^2 + 9*a^3*b^4*c^3 - 12*a^3*b^4*c^2 + 6*a^3*b^3*c^4 - 18*a^3*b^3*c^3 + 12*a^3*b^3*c^2 + a^3*b^2*c^5 - 6*a^3*b^2*c^4 + 9*a^3*b^2*c^3 - 4*a^3*b^2*c^2 + a^2*b^6*c^2 + 3*a^2*b^5*c^3 - 4*a^2*b^5*c^2 + 3*a^2*b^4*c^4 - 9*a^2*b^4*c^3 + 6*a^2*b^4*c^2 + a^2*b^3*c^5 - 6*a^2*b^3*c^4 + 9*a^2*b^3*c^3 - 4*a^2*b^3*c^2 - a^2*b^2*c^5 + 3*a^2*b^2*c^4 - 3*a^2*b^2*c^3 + a^2*b^2*c^2) (2*a^5*b*c^2 + 3*a^5*b*c - a^5*c - a^5 - 2*a^4*b^2*c^3 + 3*a^4*b^2*c^2 + 12*a^4*b^2*c + 6*a^4*b*c^3 + 7*a^4*b*c^2 - 14*a^4*b*c - 5*a^4*b - 4*a^4*c^2 + 5*a^4 - 4*a^3*b^3*c^3 - 3*a^3*b^3*c^2 + 18*a^3*b^3*c - 4*a^3*b^2*c^4 + 5*a^3*b^2*c^3 + 30*a^3*b^2*c^2 - 35*a^3*b^2*c - 10*a^3*b^2 + 6*a^3*b*c^4 + 7*a^3*b*c^3 - 38*a^3*b*c^2 + 8*a^3*b*c + 20*a^3*b - 6*a^3*c^3 + 6*a^3*c^2 + 10*a^3*c - 10*a^3 - 2*a^2*b^4*c^3 - 7*a^2*b^4*c^2 + 12*a^2*b^4*c - 4*a^2*b^3*c^4 - 8*a^2*b^3*c^3 + 39*a^2*b^3*c^2 - 31*a^2*b^3*c - 10*a^2*b^3 - 2*a^2*b^2*c^5 + a^2*b^2*c^4 + 32*a^2*b^2*c^3 - 65*a^2*b^2*c^2 + 9*a^2*b^2*c + 30*a^2*b^2 + 2*a^2*b*c^5 + 5*a^2*b*c^4 - 36*a^2*b*c^3 + 28*a^2*b*c^2 + 30*a^2*b*c - 30*a^2*b - 4*a^2*c^4 + 8*a^2*c^3 + 6*a^2*c^2 - 20*a^2*c + 10*a^2 - 3*a*b^5*c^2 + 3*a*b^5*c - 7*a*b^4*c^3 + 16*a*b^4*c^2 - 8*a*b^4*c - 5*a*b^4 - 5*a*b^3*c^4 + 25*a*b^3*c^3 - 28*a*b^3*c^2 - 6*a*b^3*c + 20*a*b^3 - a*b^2*c^5 + 14*a*b^2*c^4 - 31*a*b^2*c^3 + 9*a*b^2*c^2 + 39*a*b^2*c - 30*a*b^2 + 2*a*b*c^5 - 12*a*b*c^4 + 13*a*b*c^3 + 20*a*b*c^2 - 43*a*b*c + 20*a*b - a*c^5 + 3*a*c^4 + 2*a*c^3 - 14*a*c^2 + 15*a*c - 5*a + b^5*c - b^5 + 3*b^4*c^2 - 7*b^4*c + 5*b^4 + 3*b^3*c^3 - 13*b^3*c^2 + 19*b^3*c - 10*b^3 + b^2*c^4 - 9*b^2*c^3 + 23*b^2*c^2 - 25*b^2*c + 10*b^2 - 2*b*c^4 + 10*b*c^3 - 19*b*c^2 + 16*b*c - 5*b + c^4 - 4*c^3 + 6*c^2 - 4*c + 1)/(a^5*b^2*c^2 + 4*a^4*b^3*c^2 + 3*a^4*b^2*c^3 - 4*a^4*b^2*c^2 + 6*a^3*b^4*c^2 + 9*a^3*b^3*c^3 - 12*a^3*b^3*c^2 + 3*a^3*b^2*c^4 - 9*a^3*b^2*c^3 + 6*a^3*b^2*c^2 + 4*a^2*b^5*c^2 + 9*a^2*b^4*c^3 - 12*a^2*b^4*c^2 + 6*a^2*b^3*c^4 - 18*a^2*b^3*c^3 + 12*a^2*b^3*c^2 + a^2*b^2*c^5 - 6*a^2*b^2*c^4 + 9*a^2*b^2*c^3 - 4*a^2*b^2*c^2 + a*b^6*c^2 + 3*a*b^5*c^3 - 4*a*b^5*c^2 + 3*a*b^4*c^4 - 9*a*b^4*c^3 + 6*a*b^4*c^2 + a*b^3*c^5 - 6*a*b^3*c^4 + 9*a*b^3*c^3 - 4*a*b^3*c^2 - a*b^2*c^5 + 3*a*b^2*c^4 - 3*a*b^2*c^3 + a*b^2*c^2) (-2*a^4*b*c^2 - 2*a^4*b*c + a^4*c + a^4 - 4*a^3*b^2*c^2 - 6*a^3*b^2*c - ** WARNING: Output too long, hence truncated. '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:44:26 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Numerator(Q[i,j]),a+b+c-1); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:44:25 on modular [Seed = 2387960987] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^5*b^2*c^2 - 2*a^4*b^3*c^3 + 3*a^4*b^3*c^2 + 3*a^4*b^2*c^3 + a^4*b^2*c^2 + a^4*b^2*c - 2*a^4*b*c - 4*a^3*b^4*c^3 + 3*a^3*b^4*c^2 - 4*a^3*b^3*c^4 + 6*a^3*b^3*c^3 + 3*a^3*b^3*c + 3*a^3*b^2*c^4 + 2*a^3*b^2*c^3 + 4*a^3*b^2*c^2 - 6*a^3*b^2*c - 6*a^3*b*c^2 + 2*a^3*b*c - a^3*b + a^3 - 2*a^2*b^5*c^3 + a^2*b^5*c^2 - 4*a^2*b^4*c^4 + 3*a^2*b^4*c^3 - 3*a^2*b^4*c^2 + 3*a^2*b^4*c - 2*a^2*b^3*c^5 + 3*a^2*b^3*c^4 - 2*a^2*b^3*c^3 + 12*a^2*b^3*c^2 - 6*a^2*b^3*c + a^2*b^2*c^5 + a^2*b^2*c^4 + 7*a^2*b^2*c^3 - 15*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b^2 - 6*a^2*b*c^3 + 4*a^2*b*c^2 - 2*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^5*c^2 + a*b^5*c - 4*a*b^4*c^3 + 8*a*b^4*c^2 - 2*a*b^4*c - 2*a*b^3*c^4 + 11*a*b^3*c^3 - 9*a*b^3*c^2 + a*b^3*c - 3*a*b^3 + 4*a*b^2*c^4 - 9*a*b^2*c^3 + 3*a*b^2*c^2 - 6*a*b^2*c + 9*a*b^2 - 2*a*b*c^4 + 2*a*b*c^3 - 3*a*b*c^2 + 12*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-a^5*b^2*c^2 - 4*a^4*b^3*c^2 - 3*a^4*b^2*c^3 + a^4*b^2*c^2 + 2*a^4*b*c - 6*a^3*b^4*c^2 - 7*a^3*b^3*c^3 + 3*a^3*b^3*c^2 - 3*a^3*b^2*c^4 + 2*a^3*b^2*c^3 - 4*a^3*b^2*c^2 + 7*a^3*b^2*c + 6*a^3*b*c^2 - 3*a^3*b*c - a^3 - 4*a^2*b^5*c^2 - 5*a^2*b^4*c^3 + 3*a^2*b^4*c^2 - 2*a^2*b^3*c^4 + 4*a^2*b^3*c^3 - 8*a^2*b^3*c^2 + 9*a^2*b^3*c - a^2*b^2*c^5 + a^2*b^2*c^4 - 8*a^2*b^2*c^3 + 18*a^2*b^2*c^2 - 9*a^2*b^2*c + 6*a^2*b*c^3 - 6*a^2*b*c^2 + 2*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^6*c^2 - a*b^5*c^3 + a*b^5*c^2 + a*b^4*c^4 + 2*a*b^4*c^3 - 4*a*b^4*c^2 + 5*a*b^4*c + a*b^3*c^5 + a*b^3*c^4 - 8*a*b^3*c^3 + 14*a*b^3*c^2 - 9*a*b^3*c - 4*a*b^2*c^4 + 11*a*b^2*c^3 - 11*a*b^2*c^2 + 7*a*b^2*c - 3*a*b^2 + 2*a*b*c^4 - 3*a*b*c^3 + 4*a*b*c^2 - 9*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a + b^5*c + 2*b^4*c^2 - 3*b^4*c + b^3*c^3 - 5*b^3*c^2 + 5*b^3*c - b^3 - 2*b^2*c^3 + 7*b^2*c^2 - 8*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (2*a^3*b^2*c^2 - 2*a^3*b*c + 4*a^2*b^3*c^2 + 4*a^2*b^2*c^3 - 4*a^2*b^2*c - 4*a^2*b*c^2 + a^2*b*c + a^2 + 2*a*b^4*c^2 + 4*a*b^3*c^3 - 2*a*b^3*c + 2*a*b^2*c^4 - 4*a*b^2*c^2 - 2*a*b*c^3 + a*b*c + 2*a*b + 2*a*c - 2*a - b^3*c - 2*b^2*c^2 + b^2*c + b^2 - b*c^3 + b*c^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (-a^6*b^2*c^2 - 4*a^5*b^3*c^2 - a^5*b^2*c^3 + 3*a^5*b^2*c^2 + a^5*b*c - 6*a^4*b^4*c^2 - 5*a^4*b^3*c^3 + 9*a^4*b^3*c^2 + a^4*b^2*c^4 + 6*a^4*b^2*c^3 + 2*a^4*b^2*c^2 + 5*a^4*b^2*c - 6*a^4*b*c - 4*a^3*b^5*c^2 - 7*a^3*b^4*c^3 + 9*a^3*b^4*c^2 - 2*a^3*b^3*c^4 + 12*a^3*b^3*c^3 + 4*a^3*b^3*c^2 + 9*a^3*b^3*c + a^3*b^2*c^5 + 3*a^3*b^2*c^4 + 4*a^3*b^2*c^3 - 18*a^3*b^2*c - 3*a^3*b*c^3 - 9*a^3*b*c^2 + 7*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^6*c^2 - 3*a^2*b^5*c^3 + 3*a^2*b^5*c^2 - 3*a^2*b^4*c^4 + 6*a^2*b^4*c^3 + 2*a^2*b^4*c^2 + 7*a^2*b^4*c - a^2*b^3*c^5 + 3*a^2*b^3*c^4 + 4*a^2*b^3*c^3 + 4*a^2*b^3*c^2 - 18*a^2*b^3*c + 2*a^2*b^2*c^4 - 5*a^2*b^2*c^3 - 19*a^2*b^2*c^2 + 11*a^2*b^2*c - 3*a^2*b^2 - 2*a^2*b*c^4 - 3*a^2*b*c^3 + 8*a^2*b*c^2 + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 2*a*b^5*c + 4*a*b^4*c^2 - 6*a*b^4*c + 2*a*b^3*c^3 - 10*a*b^3*c^2 + 4*a*b^3*c - 3*a*b^3 - 4*a*b^2*c^3 + 5*a*b^2*c^2 - a*b^2*c + 9*a*b^2 + a*b*c^3 + 2*a*b*c^2 + 4*a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-2*a^4*b^2*c^3 - a^4*b^2*c^2 + 2*a^4*b*c^2 + a^4*b*c - 4*a^3*b^3*c^3 - 3*a^3*b^3*c^2 - 4*a^3*b^2*c^4 - a^3*b^2*c^3 + 3*a^3*b^2*c^2 + 3*a^3*b^2*c + 4*a^3*b*c^3 + a^3*b*c - a^3*c - a^3 - 2*a^2*b^4*c^3 - 3*a^2*b^4*c^2 - 4*a^2*b^3*c^4 - 2*a^2*b^3*c^3 + 3*a^2*b^3*c - 2*a^2*b^2*c^5 + a^2*b^2*c^4 + 2*a^2*b^2*c^3 + 4*a^2*b^2*c^2 + 3*a^2*b^2*c + 2*a^2*b*c^4 - a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - a*b^5*c^2 - a*b^4*c^3 - a*b^4*c^2 + a*b^4*c + a*b^3*c^4 - 2*a*b^3*c^3 + 4*a*b^3*c^2 + 3*a*b^3*c + a*b^2*c^5 - a*b^2*c^4 + 3*a*b^2*c^3 + 5*a*b^2*c^2 - 8*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 8*a*b*c^2 + a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^4*c + 2*b^3*c^2 - 2*b^3*c - b^3 + b^2*c^3 - 3*b^2*c^2 - b^2*c + 3*b^2 - b*c^3 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (-2*a^4*b*c^2 - 4*a^3*b^2*c^2 - 4*a^3*b*c^3 + 2*a^3*b*c^2 - 2*a^3*b*c + a^3*c - 2*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 4*a^2*b^2*c^2 - 4*a^2*b^2*c - 2*a^2*b*c^4 + 4*a^2*b*c^3 - 4*a^2*b*c^2 + 4*a^2*b*c + 2*a^2*c^2 - 2*a^2*c + a^2 + 2*a*b^3*c^2 - 2*a*b^3*c + 4*a*b^2*c^3 - 4*a*b^2*c^2 + 3*a*b^2*c + 2*a*b*c^4 - 2*a*b*c^3 + 4*a*b*c^2 - 3*a*b*c + 2*a*b + a*c^3 - 3*a*c^2 + 3*a*c - 2*a - b^2*c + b^2 - 2*b*c^2 + 3*b*c - 2*b - c^3 + 2*c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (2*a^4*b^2*c^2 + 4*a^3*b^3*c^2 + 4*a^3*b^2*c^3 + 2*a^3*b^2*c - 3*a^3*b*c + 2*a^2*b^4*c^2 + 4*a^2*b^3*c^3 + 4*a^2*b^3*c + 2*a^2*b^2*c^4 + 4*a^2*b^2*c^2 - 8*a^2*b^2*c - 6*a^2*b*c^2 + 3*a^2*b*c - a^2*b + a^2 + 2*a*b^4*c + 4*a*b^3*c^2 - 5*a*b^3*c + 2*a*b^2*c^3 - 8*a*b^2*c^2 + 3*a*b^2*c - 2*a*b^2 - 3*a*b*c^3 + 3*a*b*c^2 - 2*a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) (-2*a^3*b^2*c^2 + 2*a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 4*a^2*b^2*c + 4*a^2*b*c^2 - a^2*b*c - a^2 - 2*a*b^4*c^2 - 4*a*b^3*c^3 + 2*a*b^3*c - 2*a*b^2*c^4 + 4*a*b^2*c^2 + 2*a*b*c^3 - a*b*c - 2*a*b - 2*a*c + 2*a + b^3*c + 2*b^2*c^2 - b^2*c - b^2 + b*c^3 - b*c^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c) (-a^4*b*c - 3*a^3*b^2*c - a^3*b*c^2 + a^3*b*c - 3*a^2*b^3*c - 2*a^2*b^2*c^2 + 2*a^2*b^2*c + a^2*b*c^3 + 2*a^2*b*c^2 - a^2*c + a^2 - a*b^4*c - a*b^3*c^2 + a*b^3*c + a*b^2*c^3 + 2*a*b^2*c^2 + a*b*c^4 + a*b*c^3 - 2*a*b*c + 2*a*b - 2*a*c^2 + 3*a*c - 2*a - b^2*c + b^2 - 2*b*c^2 + 3*b*c - 2*b - c^3 + 2*c^2 - 2*c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 16:00:31 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Numerator(Q[i,j]); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:00:31 on modular [Seed = 1806743208] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a - 2*b*c + b + c - 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (4*a^2*b*c + a^2*b - 2*a^2 - 4*a*b^2*c^2 + 2*a*b^2*c + 2*a*b^2 + 4*a*b*c^2 + a*b*c - 6*a*b - 4*a*c + 4*a - 2*b^3*c + b^3 - 2*b^2*c^2 + 7*b^2*c - 4*b^2 + 4*b*c^2 - 9*b*c + 5*b - 2*c^2 + 4*c - 2)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (-2*a^3*b*c + 2*a^3 - 4*a^2*b^2*c - 2*a^2*b*c^2 - 2*a^2*b*c + 5*a^2*b + 4*a^2*c - 4*a^2 - 2*a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + 4*a*b^2 - 4*a*b*c^2 + 9*a*b*c - 6*a*b + 2*a*c^2 - 4*a*c + 2*a + b^3 + b^2*c - 2*b^2 - b*c + b)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) 2*b/(a + b + c - 1)*$.1^2 + (-2*a^2 + 4*a*b*c - 2*a*b - 2*a*c + a - b - c + 1)/(a^2*c + a*b*c + a*c^2 - a*c)*$.1 + (-2*a^2*b*c + a^2 + 2*a*b^2*c^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c - 2*a - b^2*c + b^2 - b*c^2 + 3*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (-a^2 - 2*a*b + a*c + 3*a - b^2 - b*c + 3*b - 2)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (-2*a^3*b*c + a^3 - 4*a^2*b^2*c + 2*a^2*b*c^2 + 10*a^2*b*c + 4*a^2*b - a^2*c - 5*a^2 - 2*a*b^3*c - 2*a*b^2*c^2 + 10*a*b^2*c + 5*a*b^2 + 4*a*b*c^2 - 7*a*b*c - 12*a*b - 2*a*c^2 - 2*a*c + 7*a + 2*b^3 + 2*b^2*c - 7*b^2 - 5*b*c + 8*b + 3*c - 3)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (-a^4*b^2*c^2 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 3*a^3*b^2*c - 3*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 7*a^2*b^3*c^2 + 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^3 - 4*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 3*a*b^4*c + 4*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 + a*b^2*c^3 - 11*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 - 2*a*b*c^3 + 8*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-2*a*c - a - b + c + 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*c + a^3 - 4*a^2*b*c^2 - 2*a^2*b*c + 2*a^2*b + 2*a^2*c^2 - a^2*c - a^2 - 4*a*b^2*c + a*b^2 + 5*a*b*c - a + b^2 + b*c - 2*b - c + 1)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (2*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^3 - a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - 3*a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 6*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (-2*a + 2)/(a + b + c - 1)*$.1^2 + (-4*a^2*b*c - 2*a^2*b + a^2 - 2*a*b^2 + 2*a*b*c + 3*a*b + a*c - 2*a - b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*$.1 + (-2*a^2*b*c^2 - 2*a^2*b*c + a^2*c + a^2 - 2*a*b^2*c + 3*a*b*c + 2*a*b + a*c^2 - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) 2*c/(a + b + c - 1)*$.1^2 + (4*a*b*c + 2*a*b - 3*a + 2*b^2 + 2*b*c - 5*b - 3*c + 3)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (2*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) -2*c/(a + b + c - 1)*$.1^2 + (2*a^2 - 4*a*b*c + 2*a*b + 2*a*c - a + b + c - 1)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (2*a^2*b*c - a^2 - 2*a*b^2*c^2 + 2*a*b^2*c + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c - b^2 + b*c^2 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c) (-a - b + c + 1)/(a + b + c - 1)*$.1^2 + (-2*a^2*b - 2*a*b^2 + 2*a*b*c + 2*a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:58:36 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Denominator(Q[i,j]); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 15:58:35 on modular [Seed = 687076735] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:58:07 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Denominator(Q[3,3]); Output: Magma V2.11-10 Thu Dec 8 2005 15:58:06 on modular [Seed = 3540520608] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:57:42 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Numerator(Q[3,3]); Output: Magma V2.11-10 Thu Dec 8 2005 15:57:42 on modular [Seed = 3557755327] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a - b + c + 1)/(a + b + c - 1)*$.1^2 + (-2*a^2*b - 2*a*b^2 + 2*a*b*c + 2*a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:55:37 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:55:37 on modular [Seed = 3640788797] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [((a - 2*b*c + b + c - 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (4*a^2*b*c + a^2*b - 2*a^2 - 4*a*b^2*c^2 + 2*a*b^2*c + 2*a*b^2 + 4*a*b*c^2 + a*b*c - 6*a*b - 4*a*c + 4*a - 2*b^3*c + b^3 - 2*b^2*c^2 + 7*b^2*c - 4*b^2 + 4*b*c^2 - 9*b*c + 5*b - 2*c^2 + 4*c - 2)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*t + (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (-2*a^3*b*c + 2*a^3 - 4*a^2*b^2*c - 2*a^2*b*c^2 - 2*a^2*b*c + 5*a^2*b + 4*a^2*c - 4*a^2 - 2*a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + 4*a*b^2 - 4*a*b*c^2 + 9*a*b*c - 6*a*b + 2*a*c^2 - 4*a*c + 2*a + b^3 + b^2*c - 2*b^2 - b*c + b)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*t + (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) (2*b/(a + b + c - 1)*t^2 + (-2*a^2 + 4*a*b*c - 2*a*b - 2*a*c + a - b - c + 1)/(a^2*c + a*b*c + a*c^2 - a*c)*t + (-2*a^2*b*c + a^2 + 2*a*b^2*c^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c - 2*a - b^2*c + b^2 - b*c^2 + 3*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] [((-a^2 - 2*a*b + a*c + 3*a - b^2 - b*c + 3*b - 2)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (-2*a^3*b*c + a^3 - 4*a^2*b^2*c + 2*a^2*b*c^2 + 10*a^2*b*c + 4*a^2*b - a^2*c - 5*a^2 - 2*a*b^3*c - 2*a*b^2*c^2 + 10*a*b^2*c + 5*a*b^2 + 4*a*b*c^2 - 7*a*b*c - 12*a*b - 2*a*c^2 - 2*a*c + 7*a + 2*b^3 + 2*b^2*c - 7*b^2 - 5*b*c + 8*b + 3*c - 3)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*t + (-a^4*b^2*c^2 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 3*a^3*b^2*c - 3*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 7*a^2*b^3*c^2 + 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^3 - 4*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 3*a*b^4*c + 4*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 + a*b^2*c^3 - 11*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 - 2*a*b*c^3 + 8*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-2*a*c - a - b + c + 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (2*a^3*c + a^3 - 4*a^2*b*c^2 - 2*a^2*b*c + 2*a^2*b + 2*a^2*c^2 - a^2*c - a^2 - 4*a*b^2*c + a*b^2 + 5*a*b*c - a + b^2 + b*c - 2*b - c + 1)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*t + (2*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^3 - a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - 3*a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 6*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-2*a + 2)/(a + b + c - 1)*t^2 + (-4*a^2*b*c - 2*a^2*b + a^2 - 2*a*b^2 + 2*a*b*c + 3*a*b + a*c - 2*a - b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*t + (-2*a^2*b*c^2 - 2*a^2*b*c + a^2*c + a^2 - 2*a*b^2*c + 3*a*b*c + 2*a*b + a*c^2 - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] [(2*c/(a + b + c - 1)*t^2 + (4*a*b*c + 2*a*b - 3*a + 2*b^2 + 2*b*c - 5*b - 3*c + 3)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (2*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) (-2*c/(a + b + c - 1)*t^2 + (2*a^2 - 4*a*b*c + 2*a*b + 2*a*c - a + b + c - 1)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (2*a^2*b*c - a^2 - 2*a*b^2*c^2 + 2*a*b^2*c + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c - b^2 + b*c^2 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-2*a^2*b - 2*a*b^2 + 2*a*b*c + 2*a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:53:48 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:53:47 on modular [Seed = 3256425652] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [((a - 2*b*c + b + c - 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (2*a*b*c + a*b - a - 2*b^2*c + b^2 + 2*b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c - a*b*c)*t + (2*a^2*b^2*c - a^2*b*c - 1/2*a^2*b + 1/4*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 3*a*b^2*c - a*b^2 - a*b*c^2 + 3/2*a*b + 1/2*a*c - 1/2*a - 1/2*b^3 - b^2*c + 5/4*b^2 - 1/2*b*c^2 + 3/2*b*c - b + 1/4*c^2 - 1/2*c + 1/4)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (a^2 - 4*a*b*c + a*b + a*c - a + b*c)/(a^2*b*c + a*b^2*c - a*b*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/2*a^2*c + 1/2*a^2 - a*b^3*c - a*b^2*c^2 + a*b^2*c + 1/4*a*b^2 - 1/4*a*c^2 + 1/2*a*c - 1/4*a + 1/4*b^3 + 1/2*b^2*c - 1/2*b^2 + 1/4*b*c^2 - 1/2*b*c + 1/4*b)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*b/(a + b + c - 1)*t^2 + (-2*a + 1)/(a*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-a^2 - 2*a*b + a*c + 3*a - b^2 - b*c + 3*b - 2)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (4*a*b*c + a*b - a*c - a + b^2 - b*c - 2*b + 1)/(a^2*b*c + a*b^2*c - a*b*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b*c^2 + a^2*b*c + 1/4*a^2*b + 1/2*a^2*c - 1/2*a^2 + a*b^3*c + a*b^2*c^2 - a*b^2*c - 1/4*a*b^2 + 1/4*a*c^2 - 1/2*a*c + 1/4*a - 1/4*b^3 - 1/2*b^2*c + 1/2*b^2 - 1/4*b*c^2 + 1/2*b*c - 1/4*b)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*a*c - a - b + c + 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (2*a^2*c + a^2 - 2*a*b*c + a*b - 2*a*c - a + c)/(a^2*b*c + a*b^2*c - a*b*c)*t + (2*a^3*b*c - 1/2*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 3*a^2*b*c - a^2*b - a^2*c + 5/4*a^2 - a*b^2*c - 1/2*a*b^2 - a*b*c^2 + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*a + 2)/(a + b + c - 1)*t^2 + (-2*a*b - a + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.310 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:52:11 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); P1:=Matrix(R,3,3,[R.1,R.2,0,R.4,R.5,0,0,0,1]); I1:=Ideal(Eltseq(P1*A*Transpose(P1)-A1)); print GroebnerBasis(I1); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); print (Q1*A*Transpose(Q1)-A1); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:52:10 on modular [Seed = 3390117149] ------------------------------------- [ $.1 + (-2*b + 1)/(a + b - 1), $.2 + (-a + b)/(a + b - 1), $.4 + (a - b)/(a + b - 1), $.5 + (-2*a + 1)/(a + b - 1) ] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.270 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:51:48 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); P1:=Matrix(R,3,3,[R.1,R.2,0,R.4,R.5,0,0,0,1]); I1:=Ideal(Eltseq(P1*A*Transpose(P1)-A1)); print GroebnerBasis(I1); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,0]); print (Q1*A*Transpose(Q1)-A1); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:51:48 on modular [Seed = 3474198534] ------------------------------------- [ $.1 + (-2*b + 1)/(a + b - 1), $.2 + (-a + b)/(a + b - 1), $.4 + (a - b)/(a + b - 1), $.5 + (-2*a + 1)/(a + b - 1) ] [ 0 0 -1] [ 0 0 -1] [ 0 0 -c] [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.280 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:50:03 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); P1:=Matrix(R,3,3,[R.1,R.2,0,R.4,R.5,0,0,0,1]); I1:=Ideal(Eltseq(P1*A*Transpose(P1)-A1)); print GroebnerBasis(I1); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:50:03 on modular [Seed = 4029362086] ------------------------------------- [ $.1 + (-2*b + 1)/(a + b - 1), $.2 + (-a + b)/(a + b - 1), $.4 + (a - b)/(a + b - 1), $.5 + (-2*a + 1)/(a + b - 1) ] [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.270 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:49:12 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); P1:=Matrix(R,3,3,[R.1,R.2,R.3,0,1,0,R.7,R.8,R.9]); I1:=Ideal(Eltseq(P*A*Transpose(P)-A1)); print GroebnerBasis(I1); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:49:10 on modular [Seed = 4113443486] ------------------------------------- [ $.1 + (-a*b + b*c - b - c + 1)/(a*c + b*c - c)*$.7 + (b^2 - b*c - 2*b + 1)/(a*c + b*c - c)*$.8 + (-2*b + 1)/(a + b - 1)*$.9, $.2 + (a*b + a*c - a)/(a*c + b*c - c)*$.7 + (-a - b^2 + b*c + b)/(a*c + b*c - c)*$.8 + (-a + b)/(a + b - 1)*$.9, $.3 + b/c*$.8, $.4 + (a^2 - a*c - a - b + 1)/(a*c + b*c - c)*$.7 + (-a*b + a - b*c)/(a*c + b*c - c)*$.8 + (a - b)/(a + b - 1)*$.9, $.5 + (-a^2 + a*c)/(a*c + b*c - c)*$.7 + (a*b - a*c - 2*a - b + c + 1)/(a*c + b*c - c)*$.8 + (-2*a + 1)/(a + b - 1)*$.9, $.6 + a/c*$.7 + 1/c*$.8, $.7^2 + (b + c - 1)/(a*b - 1/2*a)*$.7*$.9 + (a*b^2 - a*b + 1/2*b^2 - b + 1/2)/(a^2*b - 1/2*a^2)*$.8^2 + (a*b - a - b*c + c)/(a^2*b - 1/2*a^2)*$.8*$.9 + (a*b*c - a*c + 1/2*c^2)/(a^2*b - 1/2*a^2)*$.9^2 + (-a*b*c + a*c - 1/2*c^2)/(a^2*b - 1/2*a^2), $.7*$.8 + (-c + 1/2)/(b - 1/2)*$.7*$.9 + (1/2*a*b - 1/2*b^2 + b - 1/2)/(a*b - 1/2*a)*$.8^2 + (1/2*a + b*c - c)/(a*b - 1/2*a)*$.8*$.9 + (1/2*a*c - 1/2*c^2)/(a*b - 1/2*a)*$.9^2 + (-1/2*a*c + 1/2*c^2)/(a*b - 1/2*a), $.7*$.9^2 + (-a*b^2*c + a*b*c - 1/4*a*c + b^2*c^2 - b*c^2 + 1/4*c^2)/(a*b^2*c + a*b*c^2 - a*b*c - 1/4*a*b - 1/4*a*c + 1/4*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)*$.7 + (-1/2*a^2*b^3 + 1/4*a^2*b^2 - a*b^4 + 3/2*a*b^3 - 1/2*a*b^2 - 1/2*b^5 + 5/4*b^4 - b^3 + 1/4*b^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^3 + (-1/2*a^2*b^2*c - 3/4*a^2*b^2 + 1/2*a^2*b + a*b^3*c - 3/2*a*b^3 - a*b^2*c + 5/2*a*b^2 + 1/2*a*b*c - a*b + 3/2*b^4*c - 3/4*b^4 - 3*b^3*c + 2*b^3 + 2*b^2*c - 7/4*b^2 - 1/2*b*c + 1/2*b)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^2*$.9 + (-1/2*a^2*b^2*c - 1/4*a^2*b*c - 1/4*a^2*b + 1/4*a^2 - 3/2*a*b^3*c - 1/2*a*b^2*c^2 + 15/4*a*b^2*c - 1/4*a*b^2 + 3/4*a*b*c^2 - 2*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a - 3/2*b^3*c^2 + 3/2*b^3*c + 9/4*b^2*c^2 - 5/2*b^2*c - b*c^2 + b*c)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8*$.9^2 + (1/2*a^2*b^2*c - 1/4*a^2*b*c + 3/2*a*b^3*c - 1/2*a*b^2*c^2 - 11/4*a*b^2*c + 1/4*a*b*c^2 + 3/2*a*b*c - 1/4*a*c + 1/2*b^3*c^2 - 1/4*b^2*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8 + (-1/2*a^2*b*c^2 - 1/4*a^2*b*c + 1/4*a^2*c + 3/2*a*b^2*c^2 - 1/4*a*b^2*c + 1/2*a*b*c^3 - 5/4*a*b*c^2 + 1/2*a*b*c + 1/4*a*c^2 - 1/4*a*c + 1/2*b^2*c^3 - 3/4*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9^3 + (1/2*a^2*b*c^2 + 1/4*a^2*b*c - 1/4*a^2*c - 3/2*a*b^2*c^2 + 1/4*a*b^2*c - 1/2*a*b*c^3 + 5/4*a*b*c^2 - 1/2*a*b*c - 1/4*a*c^2 + 1/4*a*c - 1/2*b^2*c^3 + 3/4*b^2*c^2 + 1/2*b*c^3 - 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9, $.8^4 + (2*a - 4*b*c + 2*b + 2*c - 2)/(a*b + b^2 - b)*$.8^3*$.9 + (2*a^2*b*c + a^2 + 2*a*b^2*c + 2*a*b*c^2 - 8*a*b*c + 2*a*b + 2*a*c - 2*a + 6*b^2*c^2 - 6*b^2*c + b^2 - 6*b*c^2 + 8*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2*$.9^2 + (-2*a^2*b*c - 2*a*b^2*c + 2*a*b*c^2 + 2*a*b*c - 2*b^2*c^2 + 2*b*c^2 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2 + (2*a^2*c - 4*a*b*c^2 + 2*a*b*c + 4*a*c^2 - 4*a*c - 4*b*c^3 + 6*b*c^2 - 2*b*c + 2*c^3 - 4*c^2 + 2*c)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9^3 + (-2*a^2*c + 4*a*b*c^2 - 2*a*b*c + 2*a*c + 4*b*c^3 - 2*b*c^2 - 2*c^3)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9 + (a^2*c^2 + 2*a*c^3 - 2*a*c^2 + c^4 - 2*c^3 + c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^4 + (-2*a^2*c^2 + 2*a*c^2 - 2*c^4 + 2*c^3 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^2 + (a^2*c^2 - 2*a*c^3 + c^4)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2) ] [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 1.060 seconds, Total memory usage: 4.36MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:47:06 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[c,1,1],[0,b,1],[0,0,a]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); P1:=Matrix(R,3,3,[R.1,R.2,R.3,0,1,0,R.7,R.8,R.9]); I1:=Ideal(Eltseq(P1*A*Transpose(P1)-A1)); print GroebnerBasis(I1); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:47:06 on modular [Seed = 4280559384] ------------------------------------- [ $.1 + (-2*b*c - b + 1)/(a*b + b*c + b - 1), $.2 + (-a + c)/(a*b + b*c + b - 1), $.3 + (a*b - b*c)/(a*b + b*c + b - 1), $.7 + (-a*b + b*c)/(a*b + b*c + b - 1), $.8 + (a - c)/(a*b + b*c + b - 1), $.9 + (-2*a*b - b + 1)/(a*b + b*c + b - 1) ] [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.280 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:40:45 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:40:45 on modular [Seed = 3778689153] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.270 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:36:44 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:36:44 on modular [Seed = 2504937444] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (2*a^2*b*c - a^2 - 2*a*b^2*c - 2*a*b*c^2 + 2*a*b*c + a*b + a*c - a + 2*b^2 + 4*b*c - 4*b + 2*c^2 - 4*c + 2)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*t + (a^3*b^2*c^2 - a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2 + 3*a^2*b^2*c + 3*a^2*b*c^2 - 3*a^2*b*c - a^2*b - a^2*c + a^2 + 3*a*b^3*c + 6*a*b^2*c^2 - 6*a*b^2*c - 2*a*b^2 + 3*a*b*c^3 - 6*a*b*c^2 - a*b*c + 4*a*b - 2*a*c^2 + 4*a*c - 2*a - b^3 - 3*b^2*c + 3*b^2 - 3*b*c^2 + 6*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) (-2*a/(a + b + c - 1)*t^2 + (-4*a*b*c - 2*a*c + 3*a - 2*b*c + 3*b - 2*c^2 + 5*c - 3)/(a*b*c + b^2*c + b*c^2 - b*c)*t + (-2*a^2*b^2*c^2 - 2*a^2*b*c^2 + 3*a^2*b*c + a^2*c - a^2 - 2*a*b^2*c^2 + 3*a*b^2*c - 2*a*b*c^3 + 5*a*b*c^2 - a*b*c - 2*a*b + 2*a*c^2 - 4*a*c + 2*a + b^2*c - b^2 + 2*b*c^2 - 4*b*c + 2*b + c^3 - 3*c^2 + 3*c - 1)/(a^2*b^2*c^2 + a*b^3*c^2 + a*b^2*c^3 - a*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) (2*a/(a + b + c - 1)*t^2 + (4*a*b*c - 2*a*b - a - 2*b^2 - 2*b*c + b - c + 1)/(a*b*c + b^2*c + b*c^2 - b*c)*t + (2*a^2*b*c^2 - 2*a^2*b*c - a^2*c + a^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b - a*c^2 + 3*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] [((-2*b + 2)/(a + b + c - 1)*t^2 + (-4*a*b^2*c + 6*a*b*c + 3*a*b - 3*a + 2*b^2*c + 3*b^2 + 2*b*c^2 + b*c - 6*b - 3*c + 3)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*t + (-2*a^2*b^3*c^2 + 4*a^2*b^2*c^2 + 3*a^2*b^2*c - 4*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c^2 + 3*a*b^3*c + 2*a*b^2*c^3 + a*b^2*c^2 - 8*a*b^2*c - 2*a*b^2 - 5*a*b*c^2 + 3*a*b*c + 4*a*b + 2*a*c - 2*a - b^3*c - b^3 - 2*b^2*c^2 + 3*b^2 - b*c^3 + b*c^2 + 3*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (-2*a^2*b*c + 2*a^2 + 2*a*b^2*c - 2*a*b*c^2 - 2*a*b*c + a*b + 2*a*c - a - b^2 - b*c + 2*b + c - 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*t + (-a^2*b^2*c^2 + 3*a^2*b*c - a^2 + a*b^3*c^2 - a*b^2*c^3 - a*b^2*c^2 + 3*a*b^2*c + 4*a*b*c^2 - 3*a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c^2 - b^2 + b*c^3 - b*c^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c^2 + a*b^3*c^2 + a*b^2*c^3 - a*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a^2*b - 4*a*b^2*c - 2*a*b^2 + 2*a*b*c + 3*a*b - a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*t + (-2*a^2*b*c + a^2 - 2*a*b^2*c^2 - 2*a*b^2*c + 3*a*b*c + 2*a*b + a*c - 2*a + b^2*c + b^2 + b*c^2 - 2*b - c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] [(2*c/(a + b + c - 1)*t^2 + (4*a*b*c + 2*a*b - 3*a + 2*b^2 + 2*b*c - 5*b - 3*c + 3)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (2*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) (-2*c/(a + b + c - 1)*t^2 + (2*a^2 - 4*a*b*c + 2*a*b + 2*a*c - a + b + c - 1)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (2*a^2*b*c - a^2 - 2*a*b^2*c^2 + 2*a*b^2*c + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c - b^2 + b*c^2 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-2*a^2*b - 2*a*b^2 + 2*a*b*c + 2*a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] Total time: 0.310 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:36:36 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); Output: Magma V2.11-10 Thu Dec 8 2005 15:36:35 on modular [Seed = 2655339744] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] Total time: 0.300 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:34:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); print Factorization(Numerator(D*px+py)); print (Q*B*Transpose(Q)-B); Output: Magma V2.11-10 Thu Dec 8 2005 15:34:07 on modular [Seed = 2203605009] ------------------------------------- [ <$.1 + (a*b*c - a - b - c + 1)/(a*b*c), 1>, <$.1 + (3*a*b*c - a - b - c + 1)/(a*b*c), 1> ] [0 0 0] [0 0 0] [0 0 0] Total time: 0.220 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:33:33 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); print D*px+py; print (Q*B*Transpose(Q)-B); Output: Magma V2.11-10 Thu Dec 8 2005 15:33:33 on modular [Seed = 2220842804] ------------------------------------- -t^2 + (-4*a*b*c + 2*a + 2*b + 2*c - 2)/(a*b*c)*t + (-3*a^2*b^2*c^2 + 4*a^2*b*c - a^2 + 4*a*b^2*c + 4*a*b*c^2 - 4*a*b*c - 2*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c^2) [0 0 0] [0 0 0] [0 0 0] Total time: 0.210 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:31:53 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); print (Q*B*Transpose(Q)-B); Output: Magma V2.11-10 Thu Dec 8 2005 15:31:52 on modular [Seed = 2387957147] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] Total time: 0.210 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:31:19 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,-D*px/q+py/q,-px/q,0,px/q,py/q]); print (Q*B*Transpose(Q)-B); Output: Magma V2.11-10 Thu Dec 8 2005 15:31:19 on modular [Seed = 3009964169] ------------------------------------- [0 0 0] [0 ((-16*a*b^2*c - 16*a*b*c^2 + 16*a*b*c + 8*a*b + 8*a*c - 8*a + 8*b^2 + 16*b*c - 16*b + 8*c^2 - 16*c + 8)/(a*b^2*c^2)*t^3 + (16*a^2*b^3*c^2 + 16*a^2*b^2*c^3 - 16*a^2*b^2*c^2 - 16*a^2*b^2*c - 16*a^2*b*c^2 + 16*a^2*b*c + 4*a^2*b + 4*a^2*c - 4*a^2 - 16*a*b^3*c - 32*a*b^2*c^2 + 32*a*b^2*c + 8*a*b^2 - 16*a*b*c^3 + 32*a*b*c^2 - 16*a*b + 8*a*c^2 - 16*a*c + 8*a + 4*b^3 + 12*b^2*c - 12*b^2 + 12*b*c^2 - 24*b*c + 12*b + 4*c^3 - 12*c^2 + 12*c - 4)/(a^2*b^3*c^3)*t^2 + (80*a^3*b^4*c^3 + 80*a^3*b^3*c^4 - 80*a^3*b^3*c^3 - 104*a^3*b^3*c^2 - 104*a^3*b^2*c^3 + 104*a^3*b^2*c^2 + 48*a^3*b^2*c + 48*a^3*b*c^2 - 48*a^3*b*c - 8*a^3*b - 8*a^3*c + 8*a^3 - 104*a^2*b^4*c^2 - 208*a^2*b^3*c^3 + 208*a^2*b^3*c^2 + 96*a^2*b^3*c - 104*a^2*b^2*c^4 + 208*a^2*b^2*c^3 + 88*a^2*b^2*c^2 - 192*a^2*b^2*c - 24*a^2*b^2 + 96*a^2*b*c^3 - 192*a^2*b*c^2 + 48*a^2*b*c + 48*a^2*b - 24*a^2*c^2 + 48*a^2*c - 24*a^2 + 48*a*b^4*c + 144*a*b^3*c^2 - 144*a*b^3*c - 24*a*b^3 + 144*a*b^2*c^3 - 288*a*b^2*c^2 + 72*a*b^2*c + 72*a*b^2 + 48*a*b*c^4 - 144*a*b*c^3 + 72*a*b*c^2 + 96*a*b*c - 72*a*b - 24*a*c^3 + 72*a*c^2 - 72*a*c + 24*a - 8*b^4 - 32*b^3*c + 32*b^3 - 48*b^2*c^2 + 96*b^2*c - 48*b^2 - 32*b*c^3 + 96*b*c^2 - 96*b*c + 32*b - 8*c^4 + 32*c^3 - 48*c^2 + 32*c - 8)/(a^3*b^4*c^4)*t + (48*a^4*b^5*c^4 + 48*a^4*b^4*c^5 - 48*a^4*b^4*c^4 - 80*a^4*b^4*c^3 - 80*a^4*b^3*c^4 + 80*a^4*b^3*c^3 + 52*a^4*b^3*c^2 + 52*a^4*b^2*c^3 - 52*a^4*b^2*c^2 - 16*a^4*b^2*c - 16*a^4*b*c^2 + 16*a^4*b*c + 2*a^4*b + 2*a^4*c - 2*a^4 - 80*a^3*b^5*c^3 - 160*a^3*b^4*c^4 + 160*a^3*b^4*c^3 + 104*a^3*b^4*c^2 - 80*a^3*b^3*c^5 + 160*a^3*b^3*c^4 + 128*a^3*b^3*c^3 - 208*a^3*b^3*c^2 - 48*a^3*b^3*c + 104*a^3*b^2*c^4 - 208*a^3*b^2*c^3 + 8*a^3*b^2*c^2 + 96*a^3*b^2*c + 8*a^3*b^2 - 48*a^3*b*c^3 + 96*a^3*b*c^2 - 32*a^3*b*c - 16*a^3*b + 8*a^3*c^2 - 16*a^3*c + 8*a^3 + 52*a^2*b^5*c^2 + 156*a^2*b^4*c^3 - 156*a^2*b^4*c^2 - 48*a^2*b^4*c + 156*a^2*b^3*c^4 - 312*a^2*b^3*c^3 + 12*a^2*b^3*c^2 + 144*a^2*b^3*c + 12*a^2*b^3 + 52*a^2*b^2*c^5 - 156*a^2*b^2*c^4 + 12*a^2*b^2*c^3 + 236*a^2*b^2*c^2 - 108*a^2*b^2*c - 36*a^2*b^2 - 48*a^2*b*c^4 + 144*a^2*b*c^3 - 108*a^2*b*c^2 - 24*a^2*b*c + 36*a^2*b + 12*a^2*c^3 - 36*a^2*c^2 + 36*a^2*c - 12*a^2 - 16*a*b^5*c - 64*a*b^4*c^2 + 64*a*b^4*c + 8*a*b^4 - 96*a*b^3*c^3 + 192*a*b^3*c^2 - 64*a*b^3*c - 32*a*b^3 - 64*a*b^2*c^4 + 192*a*b^2*c^3 - 144*a*b^2*c^2 - 32*a*b^2*c + 48*a*b^2 - 16*a*b*c^5 + 64*a*b*c^4 - 64*a*b*c^3 - 32*a*b*c^2 + 80*a*b*c - 32*a*b + 8*a*c^4 - 32*a*c^3 + 48*a*c^2 - 32*a*c + 8*a + 2*b^5 + 10*b^4*c - 10*b^4 + 20*b^3*c^2 - 40*b^3*c + 20*b^3 + 20*b^2*c^3 - 60*b^2*c^2 + 60*b^2*c - 20*b^2 + 10*b*c^4 - 40*b*c^3 + 60*b*c^2 - 40*b*c + 10*b + 2*c^5 - 10*c^4 + 20*c^3 - 20*c^2 + 10*c - 2)/(a^4*b^5*c^5))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t^3 + (6*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2*c^2)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + 1) ((-8*a^2*b^3*c^2 - 8*a^2*b^2*c^3 + 8*a^2*b^2*c^2 + 12*a^2*b^2*c + 12*a^2*b*c^2 - 12*a^2*b*c - 4*a^2*b - 4*a^2*c + 4*a^2 + 12*a*b^3*c + 24*a*b^2*c^2 - 24*a*b^2*c - 8*a*b^2 + 12*a*b*c^3 - 24*a*b*c^2 - 4*a*b*c + 16*a*b - 8*a*c^2 + 16*a*c - 8*a - 4*b^3 - 12*b^2*c + 12*b^2 - 12*b*c^2 + 24*b*c - 12*b - 4*c^3 + 12*c^2 - 12*c + 4)/(a^2*b^3*c^3)*t^3 + (-24*a^3*b^4*c^3 - 24*a^3*b^3*c^4 + 24*a^3*b^3*c^3 + 24*a^3*b^3*c^2 + 24*a^3*b^2*c^3 - 24*a^3*b^2*c^2 - 10*a^3*b^2*c - 10*a^3*b*c^2 + 10*a^3*b*c + 2*a^3*b + 2*a^3*c - 2*a^3 + 24*a^2*b^4*c^2 + 48*a^2*b^3*c^3 - 48*a^2*b^3*c^2 - 20*a^2*b^3*c + 24*a^2*b^2*c^4 - 48*a^2*b^2*c^3 - 16*a^2*b^2*c^2 + 40*a^2*b^2*c + 6*a^2*b^2 - 20*a^2*b*c^3 + 40*a^2*b*c^2 - 8*a^2*b*c - 12*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 10*a*b^4*c - 30*a*b^3*c^2 + 30*a*b^3*c + 6*a*b^3 - 30*a*b^2*c^3 + 60*a*b^2*c^2 - 12*a*b^2*c - 18*a*b^2 - 10*a*b*c^4 + 30*a*b*c^3 - 12*a*b*c^2 - 26*a*b*c + 18*a*b + 6*a*c^3 - 18*a*c^2 + 18*a*c - 6*a + 2*b^4 + 8*b^3*c - 8*b^3 + 12*b^2*c^2 - 24*b^2*c + 12*b^2 + 8*b*c^3 - 24*b*c^2 + 24*b*c - 8*b + 2*c^4 - 8*c^3 + 12*c^2 - 8*c + 2)/(a^3*b^4*c^4)*t^2 + (-24*a^2*b^3*c^2 - 24*a^2*b^2*c^3 + 24*a^2*b^2*c^2 + 20*a^2*b^2*c + 20*a^2*b*c^2 - 20*a^2*b*c - 4*a^2*b - 4*a^2*c + 4*a^2 + 20*a*b^3*c + 40*a*b^2*c^2 - 40*a*b^2*c - 8*a*b^2 + 20*a*b*c^3 - 40*a*b*c^2 + 4*a*b*c + 16*a*b - 8*a*c^2 + 16*a*c - 8*a - 4*b^3 - 12*b^2*c + 12*b^2 - 12*b*c^2 + 24*b*c - 12*b - 4*c^3 + 12*c^2 - 12*c + 4)/(a^2*b^3*c^3)*t + (-8*a^2*b^3*c^2 - 8*a^2*b^2*c^3 + 8*a^2*b^2*c^2 + 8*a^2*b^2*c + 8*a^2*b*c^2 - 8*a^2*b*c - 2*a^2*b - 2*a^2*c + 2*a^2 + 8*a*b^3*c + 16*a*b^2*c^2 - 16*a*b^2*c - 4*a*b^2 + 8*a*b*c^3 - 16*a*b*c^2 + 8*a*b - 4*a*c^2 + 8*a*c - 4*a - 2*b^3 - 6*b^2*c + 6*b^2 - 6*b*c^2 + 12*b*c - 6*b - 2*c^3 + 6*c^2 - 6*c + 2)/(a^2*b^3*c^3))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t^3 + (6*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2*c^2)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + 1)] [0 ((-8*a*b^2*c - 8*a*b*c^2 + 8*a*b*c + 4*a*b + 4*a*c - 4*a + 4*b^2 + 8*b*c - 8*b + 4*c^2 - 8*c + 4)/(a*b^2*c^2)*t^3 + (-24*a^2*b^3*c^2 - 24*a^2*b^2*c^3 + 24*a^2*b^2*c^2 + 16*a^2*b^2*c + 16*a^2*b*c^2 - 16*a^2*b*c - 2*a^2*b - 2*a^2*c + 2*a^2 + 16*a*b^3*c + 32*a*b^2*c^2 - 32*a*b^2*c - 4*a*b^2 + 16*a*b*c^3 - 32*a*b*c^2 + 8*a*b*c + 8*a*b - 4*a*c^2 + 8*a*c - 4*a - 2*b^3 - 6*b^2*c + 6*b^2 - 6*b*c^2 + 12*b*c - 6*b - 2*c^3 + 6*c^2 - 6*c + 2)/(a^2*b^3*c^3)*t^2 + (-24*a^2*b^3*c^2 - 24*a^2*b^2*c^3 + 24*a^2*b^2*c^2 + 28*a^2*b^2*c + 28*a^2*b*c^2 - 28*a^2*b*c - 8*a^2*b - 8*a^2*c + 8*a^2 + 28*a*b^3*c + 56*a*b^2*c^2 - 56*a*b^2*c - 16*a*b^2 + 28*a*b*c^3 - 56*a*b*c^2 - 4*a*b*c + 32*a*b - 16*a*c^2 + 32*a*c - 16*a - 8*b^3 - 24*b^2*c + 24*b^2 - 24*b*c^2 + 48*b*c - 24*b - 8*c^3 + 24*c^2 - 24*c + 8)/(a^2*b^3*c^3)*t + (-8*a^3*b^4*c^3 - 8*a^3*b^3*c^4 + 8*a^3*b^3*c^3 + 16*a^3*b^3*c^2 + 16*a^3*b^2*c^3 - 16*a^3*b^2*c^2 - 10*a^3*b^2*c - 10*a^3*b*c^2 + 10*a^3*b*c + 2*a^3*b + 2*a^3*c - 2*a^3 + 16*a^2*b^4*c^2 + 32*a^2*b^3*c^3 - 32*a^2*b^3*c^2 - 20*a^2*b^3*c + 16*a^2*b^2*c^4 - 32*a^2*b^2*c^3 - 24*a^2*b^2*c^2 + 40*a^2*b^2*c + 6*a^2*b^2 - 20*a^2*b*c^3 + 40*a^2*b*c^2 - 8*a^2*b*c - 12*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 10*a*b^4*c - 30*a*b^3*c^2 + 30*a*b^3*c + 6*a*b^3 - 30*a*b^2*c^3 + 60*a*b^2*c^2 - 12*a*b^2*c - 18*a*b^2 - 10*a*b*c^4 + 30*a*b*c^3 - 12*a*b*c^2 - 26*a*b*c + 18*a*b + 6*a*c^3 - 18*a*c^2 + 18*a*c - 6*a + 2*b^4 + 8*b^3*c - 8*b^3 + 12*b^2*c^2 - 24*b^2*c + 12*b^2 + 8*b*c^3 - 24*b*c^2 + 24*b*c - 8*b + 2*c^4 - 8*c^3 + 12*c^2 - 8*c + 2)/(a^3*b^4*c^4))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t^3 + (6*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2*c^2)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + 1) 0] Total time: 0.220 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:29:55 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; print G[5]; print G[6]; print G[7]; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,py/q]); Output: Magma V2.11-10 Thu Dec 8 2005 15:29:55 on modular [Seed = 3027198042] ------------------------------------- $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9 $.6 + $.8 $.7 0 Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:28:46 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); q:=t^2+t*D+1; print G[5]; print G[6]; print G[7]; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:28:46 on modular [Seed = 3110235626] ------------------------------------- $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9 $.6 + $.8 $.7 0 Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:24:17 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); q:=t^2+t*D+1; print G[8]; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:24:17 on modular [Seed = 2108074017] ------------------------------------- $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 0 Total time: 0.210 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:20:46 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=2*t+D; q:=t^2+t*D+1; print G[8]; print Evaluate(G[8],[0,0,0,0,0,0,0,-px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:20:46 on modular [Seed = 215472831] ------------------------------------- $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 (8*t^4 + (8*a*b*c - 4*a - 4*b - 4*c + 4)/(a*b*c)*t^3 + (-8*a^2*b^2*c^2 + 16*a^2*b*c - 4*a^2 + 16*a*b^2*c + 16*a*b*c^2 - 16*a*b*c - 8*a*b - 8*a*c + 8*a - 4*b^2 - 8*b*c + 8*b - 4*c^2 + 8*c - 4)/(a^2*b^2*c^2)*t^2 + (-8*a^3*b^3*c^3 + 20*a^3*b^2*c^2 - 12*a^3*b*c + 2*a^3 + 20*a^2*b^3*c^2 + 20*a^2*b^2*c^3 - 20*a^2*b^2*c^2 - 24*a^2*b^2*c - 24*a^2*b*c^2 + 24*a^2*b*c + 6*a^2*b + 6*a^2*c - 6*a^2 - 12*a*b^3*c - 24*a*b^2*c^2 + 24*a*b^2*c + 6*a*b^2 - 12*a*b*c^3 + 24*a*b*c^2 - 12*a*b + 6*a*c^2 - 12*a*c + 6*a + 2*b^3 + 6*b^2*c - 6*b^2 + 6*b*c^2 - 12*b*c + 6*b + 2*c^3 - 6*c^2 + 6*c - 2)/(a^3*b^3*c^3)*t)/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t^3 + (6*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2*c^2)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:20:29 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c+1; px:=2*t+D; q:=t^2+t*D+1; print G[8]; print Evaluate(G[8],[0,0,0,0,0,0,0,-px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:20:28 on modular [Seed = 837478288] ------------------------------------- $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 (8*t^4 + (20*a^2*b^2*c^2 - 10*a^2*b*c - 10*a*b^2*c - 10*a*b*c^2 - 2*a*b*c + 6*a + 6*b + 6*c - 6)/(a*b*c)*t^3 + (12*a^3*b^3*c^3 - 12*a^3*b^2*c^2 + 3*a^3*b*c - 12*a^2*b^3*c^2 - 12*a^2*b^2*c^3 - 16*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 + 22*a^2*b*c - 7*a^2 + 3*a*b^3*c + 6*a*b^2*c^2 + 22*a*b^2*c + 3*a*b*c^3 + 22*a*b*c^2 - 17*a*b*c - 14*a*b - 14*a*c + 14*a - 7*b^2 - 14*b*c + 14*b - 7*c^2 + 14*c - 7)/(a*b*c)*t^2 + (-16*a^3*b^3*c^3 + 24*a^3*b^2*c^2 - 12*a^3*b*c + 2*a^3 + 24*a^2*b^3*c^2 + 24*a^2*b^2*c^3 - 12*a^2*b^2*c^2 - 24*a^2*b^2*c - 24*a^2*b*c^2 + 18*a^2*b*c + 6*a^2*b + 6*a^2*c - 6*a^2 - 12*a*b^3*c - 24*a*b^2*c^2 + 18*a*b^2*c + 6*a*b^2 - 12*a*b*c^3 + 18*a*b*c^2 + 2*a*b*c - 12*a*b + 6*a*c^2 - 12*a*c + 8*a + 2*b^3 + 6*b^2*c - 6*b^2 + 6*b*c^2 - 12*b*c + 8*b + 2*c^3 - 6*c^2 + 8*c - 4)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 + 2*a^2*b*c - a^2 + a*b^3*c + 2*a*b^2*c^2 + 2*a*b^2*c + a*b*c^3 + 2*a*b*c^2 - 3*a*b*c - 2*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:20:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=2*a*b*c-a-b-c+1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,-px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:20:08 on modular [Seed = 987884655] ------------------------------------- (8*t^4 + (20*a^2*b^2*c^2 - 10*a^2*b*c - 10*a*b^2*c - 10*a*b*c^2 - 2*a*b*c + 6*a + 6*b + 6*c - 6)/(a*b*c)*t^3 + (12*a^3*b^3*c^3 - 12*a^3*b^2*c^2 + 3*a^3*b*c - 12*a^2*b^3*c^2 - 12*a^2*b^2*c^3 - 16*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 + 22*a^2*b*c - 7*a^2 + 3*a*b^3*c + 6*a*b^2*c^2 + 22*a*b^2*c + 3*a*b*c^3 + 22*a*b*c^2 - 17*a*b*c - 14*a*b - 14*a*c + 14*a - 7*b^2 - 14*b*c + 14*b - 7*c^2 + 14*c - 7)/(a*b*c)*t^2 + (-16*a^3*b^3*c^3 + 24*a^3*b^2*c^2 - 12*a^3*b*c + 2*a^3 + 24*a^2*b^3*c^2 + 24*a^2*b^2*c^3 - 12*a^2*b^2*c^2 - 24*a^2*b^2*c - 24*a^2*b*c^2 + 18*a^2*b*c + 6*a^2*b + 6*a^2*c - 6*a^2 - 12*a*b^3*c - 24*a*b^2*c^2 + 18*a*b^2*c + 6*a*b^2 - 12*a*b*c^3 + 18*a*b*c^2 + 2*a*b*c - 12*a*b + 6*a*c^2 - 12*a*c + 8*a + 2*b^3 + 6*b^2*c - 6*b^2 + 6*b*c^2 - 12*b*c + 8*b + 2*c^3 - 6*c^2 + 8*c - 4)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 + 2*a^2*b*c - a^2 + a*b^3*c + 2*a*b^2*c^2 + 2*a*b^2*c + a*b*c^3 + 2*a*b*c^2 - 3*a*b*c - 2*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t + 1) Total time: 0.210 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:19:32 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c+1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,-px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:19:32 on modular [Seed = 586806363] ------------------------------------- (8*t^4 + (20*a^2*b^2*c^2 - 10*a^2*b*c - 10*a*b^2*c - 10*a*b*c^2 - 2*a*b*c + 6*a + 6*b + 6*c - 6)/(a*b*c)*t^3 + (12*a^3*b^3*c^3 - 12*a^3*b^2*c^2 + 3*a^3*b*c - 12*a^2*b^3*c^2 - 12*a^2*b^2*c^3 - 16*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 + 22*a^2*b*c - 7*a^2 + 3*a*b^3*c + 6*a*b^2*c^2 + 22*a*b^2*c + 3*a*b*c^3 + 22*a*b*c^2 - 17*a*b*c - 14*a*b - 14*a*c + 14*a - 7*b^2 - 14*b*c + 14*b - 7*c^2 + 14*c - 7)/(a*b*c)*t^2 + (-16*a^3*b^3*c^3 + 24*a^3*b^2*c^2 - 12*a^3*b*c + 2*a^3 + 24*a^2*b^3*c^2 + 24*a^2*b^2*c^3 - 12*a^2*b^2*c^2 - 24*a^2*b^2*c - 24*a^2*b*c^2 + 18*a^2*b*c + 6*a^2*b + 6*a^2*c - 6*a^2 - 12*a*b^3*c - 24*a*b^2*c^2 + 18*a*b^2*c + 6*a*b^2 - 12*a*b*c^3 + 18*a*b*c^2 + 2*a*b*c - 12*a*b + 6*a*c^2 - 12*a*c + 8*a + 2*b^3 + 6*b^2*c - 6*b^2 + 6*b*c^2 - 12*b*c + 8*b + 2*c^3 - 6*c^2 + 8*c - 4)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 + 2*a^2*b*c - a^2 + a*b^3*c + 2*a*b^2*c^2 + 2*a*b^2*c + a*b*c^3 + 2*a*b*c^2 - 3*a*b*c - 2*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:19:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c+1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,-t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:19:07 on modular [Seed = 670888241] ------------------------------------- ((-4*a^2*b^2*c^2 + 2*a^2*b*c + 2*a*b^2*c + 2*a*b*c^2 - 6*a*b*c + 2*a + 2*b + 2*c - 2)/(a*b*c)*t^3 + (-4*a^3*b^3*c^3 + 4*a^3*b^2*c^2 - a^3*b*c + 4*a^2*b^3*c^2 + 4*a^2*b^2*c^3 - 8*a^2*b^2*c^2 - 2*a^2*b^2*c - 2*a^2*b*c^2 + 6*a^2*b*c - a^2 - a*b^3*c - 2*a*b^2*c^2 + 6*a*b^2*c - a*b*c^3 + 6*a*b*c^2 - 5*a*b*c - 2*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b*c)*t^2 + (4*a^2*b^2*c^2 - 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 + 6*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 8*a^2*b^2*c^2 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 6*a^2*b*c + a^2 + a*b^3*c + 2*a*b^2*c^2 - 6*a*b^2*c + a*b*c^3 - 6*a*b*c^2 + 5*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:18:04 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c-1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,R.8,t*R.8]); print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,-t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:18:03 on modular [Seed = 703790847] ------------------------------------- >> print Evaluate(G[8],[0,0,0,0,0,0,0,R.8,t*R.8]); ^ Runtime error in '*': Bad argument types Argument types given: RngUPolElt[FldFunRat], RngMPolElt ((-4*a^2*b^2*c^2 + 2*a^2*b*c + 2*a*b^2*c + 2*a*b*c^2 - 2*a*b*c + 2*a + 2*b + 2*c - 2)/(a*b*c)*t^3 + (-4*a^3*b^3*c^3 + 4*a^3*b^2*c^2 - a^3*b*c + 4*a^2*b^3*c^2 + 4*a^2*b^2*c^3 - 2*a^2*b^2*c - 2*a^2*b*c^2 + 2*a^2*b*c - a^2 - a*b^3*c - 2*a*b^2*c^2 + 2*a*b^2*c - a*b*c^3 + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c - b^2 - 2*b*c - c^2 + 1)/(a*b*c)*t^2 + (4*a^2*b^2*c^2 - 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 + 2*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c + a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + a*b*c^3 - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c + b^2 + 2*b*c + c^2 - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 - 4*a*b*c + 2*a*b + 2*a*c + 2*a + b^2 + 2*b*c + 2*b + c^2 + 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t + 1) Total time: 0.210 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:16:28 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c-1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,-t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:16:28 on modular [Seed = 3523806560] ------------------------------------- ((-4*a^2*b^2*c^2 + 2*a^2*b*c + 2*a*b^2*c + 2*a*b*c^2 - 2*a*b*c + 2*a + 2*b + 2*c - 2)/(a*b*c)*t^3 + (-4*a^3*b^3*c^3 + 4*a^3*b^2*c^2 - a^3*b*c + 4*a^2*b^3*c^2 + 4*a^2*b^2*c^3 - 2*a^2*b^2*c - 2*a^2*b*c^2 + 2*a^2*b*c - a^2 - a*b^3*c - 2*a*b^2*c^2 + 2*a*b^2*c - a*b*c^3 + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c - b^2 - 2*b*c - c^2 + 1)/(a*b*c)*t^2 + (4*a^2*b^2*c^2 - 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 + 2*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c + a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + a*b*c^3 - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c + b^2 + 2*b*c + c^2 - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 - 4*a*b*c + 2*a*b + 2*a*c + 2*a + b^2 + 2*b*c + 2*b + c^2 + 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:16:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c-1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:16:08 on modular [Seed = 3640790655] ------------------------------------- (8*t^4 + (20*a^2*b^2*c^2 - 10*a^2*b*c - 10*a*b^2*c - 10*a*b*c^2 + 2*a*b*c - 6*a - 6*b - 6*c + 6)/(a*b*c)*t^3 + (12*a^3*b^3*c^3 - 12*a^3*b^2*c^2 + 3*a^3*b*c - 12*a^2*b^3*c^2 - 12*a^2*b^2*c^3 + 16*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 22*a^2*b*c + 7*a^2 + 3*a*b^3*c + 6*a*b^2*c^2 - 22*a*b^2*c + 3*a*b*c^3 - 22*a*b*c^2 + 11*a*b*c + 14*a*b + 14*a*c + 7*b^2 + 14*b*c + 7*c^2 - 7)/(a*b*c)*t^2 + (16*a^3*b^3*c^3 - 24*a^3*b^2*c^2 + 12*a^3*b*c - 2*a^3 - 24*a^2*b^3*c^2 - 24*a^2*b^2*c^3 + 4*a^2*b^2*c^2 + 24*a^2*b^2*c + 24*a^2*b*c^2 + 2*a^2*b*c - 6*a^2*b - 6*a^2*c - 2*a^2 + 12*a*b^3*c + 24*a*b^2*c^2 + 2*a*b^2*c - 6*a*b^2 + 12*a*b*c^3 + 2*a*b*c^2 - 18*a*b*c - 4*a*b - 6*a*c^2 - 4*a*c - 2*b^3 - 6*b^2*c - 2*b^2 - 6*b*c^2 - 4*b*c - 2*c^3 - 2*c^2 + 4)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c + a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + a*b*c^3 - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c + b^2 + 2*b*c + c^2 - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 - 4*a*b*c + 2*a*b + 2*a*c + 2*a + b^2 + 2*b*c + 2*b + c^2 + 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:10:16 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Thu Dec 8 2005 15:10:16 on modular [Seed = 3741585369] ------------------------------------- [ $.1^2 - 1, $.2, $.3, $.4, $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9, $.6 + $.8, $.7, $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '207.191' ************** MAGMA ***************** Host 207.191.40.178 (207.191.40.178) Time: Thu Dec 8 14:35:51 2005 Input: 2003^Sin(2312312) Output: Magma V2.11-10 Thu Dec 8 2005 14:35:51 on modular [Seed = 2470992810] ------------------------------------- 0.00650119842810155572431836770133141839665 Total time: 0.190 seconds, Total memory usage: 3.24MB '207.191' ************** MAGMA ***************** Host 207.191.40.178 (207.191.40.178) Time: Thu Dec 8 14:35:42 2005 Input: 2003^sin(2312312) Output: Magma V2.11-10 Thu Dec 8 2005 14:35:41 on modular [Seed = 2521652917] ------------------------------------- >> 2003^sin(2312312); ^ User error: Identifier 'sin' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '207.191' ************** MAGMA ***************** Host 207.191.40.178 (207.191.40.178) Time: Thu Dec 8 14:35:23 2005 Input: 2+2 Output: Magma V2.11-10 Thu Dec 8 2005 14:35:23 on modular [Seed = 2638632008] ------------------------------------- 4 Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 12:48:27 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,2,2,[a,b,c,d]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-Transpose(A))); print GroebnerBasis(I); Q:=Matrix(K,2,2,[-1,(b+c)/d,0,1]); print (Q*A*Transpose(Q)-Transpose(A)); Output: Magma V2.11-10 Thu Dec 8 2005 12:48:27 on modular [Seed = 637409248] ------------------------------------- [ $.1 + $.4, $.2 - a/d*$.3 + (-b - c)/d*$.4, $.3^2 + (b + c)/a*$.3*$.4 + d/a*$.4^2 - d/a ] [0 0] [0 0] Total time: 0.180 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 12:46:24 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,2,2,[a,b,c,d]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-Transpose(A))); print GroebnerBasis(I); Output: Magma V2.11-10 Thu Dec 8 2005 12:46:23 on modular [Seed = 687020649] ------------------------------------- [ $.1 + $.4, $.2 - a/d*$.3 + (-b - c)/d*$.4, $.3^2 + (b + c)/a*$.3*$.4 + d/a*$.4^2 - d/a ] Total time: 0.190 seconds, Total memory usage: 3.24MB '217.136' ************** MAGMA ***************** Host 217.136.22.124 (217.136.22.124) Time: Thu Dec 8 10:53:55 2005 Input: factor(25478963) Output: Magma V2.11-10 Thu Dec 8 2005 10:53:54 on modular [Seed = 888086537] ------------------------------------- >> factor(25478963); ^ User error: Identifier 'factor' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:06:07 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:06:07 on modular [Seed = 3406869805] ------------------------------------- Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*l = 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:06:02 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:06:02 on modular [Seed = 4146891028] ------------------------------------- >> A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; ^ Runtime error in 'AssignNames': Too many names given for this object Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:05:57 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:05:57 on modular [Seed = 4079521055] ------------------------------------- >> A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; ^ Runtime error in 'AssignNames': Too many names given for this object Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*A.1 = 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:05:44 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:05:44 on modular [Seed = 4280582510] ------------------------------------- >> A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; ^ Runtime error in 'AssignNames': Too many names given for this object Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:05:37 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:05:36 on modular [Seed = 4213212514] ------------------------------------- >> A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; ^ Runtime error in 'AssignNames': Too many names given for this object Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*A.1 = 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:04:39 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:04:39 on modular [Seed = 3879508409] ------------------------------------- Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:04:23 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, c, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:04:23 on modular [Seed = 3812138413] ------------------------------------- Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:04:17 2005 Input: F := FreeAbelianGroup(3); A := quo< F | 2*a + 0*b + 0*c, b, c, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:04:17 on modular [Seed = 4013199783] ------------------------------------- >> A := quo< F | 2*a + 0*b + 0*c, b, c, d >; ^ User error: Identifier 'd' has not been declared or assigned >> print A; ^ User error: Identifier 'A' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:04:09 2005 Input: F := FreeAbelianGroup(3); A := quo< F | 2*a + 0*b + 0*c, b, c >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:04:09 on modular [Seed = 3945829787] ------------------------------------- Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*A.1 = 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:04:04 2005 Input: F := FreeAbelianGroup(3); A := quo< F | 2*a + 0*b + 0*c, b, 9*a + 6*b + 3*c >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:04:04 on modular [Seed = 2538400234] ------------------------------------- Abelian Group isomorphic to Z/6 Defined on 1 generator Relations: 6*A.1 = 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:03:57 2005 Input: F := FreeAbelianGroup(3); A := quo< F | 2*a + 0*b + 0*c, 8*a + 5*b + 2*c, 9*a + 6*b + 3*c >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:03:57 on modular [Seed = 2471030263] ------------------------------------- Abelian Group isomorphic to Z/6 Defined on 1 generator Relations: 6*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:03:38 2005 Input: F := FreeAbelianGroup(3); A := quo< F | 7*a + 4*b + c, 8*a + 5*b + 2*c, 9*a + 6*b + 3*c >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:03:38 on modular [Seed = 2672091532] ------------------------------------- Abelian Group isomorphic to Z/3 + Z Defined on 2 generators Relations: 3*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:03:16 2005 Input: F := FreeGroup(4); A := quo< F | 2*a, b, d >; Output: Magma V2.11-10 Thu Dec 8 2005 09:03:16 on modular [Seed = 2254305419] ------------------------------------- >> A := quo< F | 2*a, b, d >;; ^ Runtime error in '*': Bad argument types Argument types given: RngIntElt, GrpFPElt Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:02:05 2005 Input: F := FreeGroup(4); C:=sub< F| a,b,c,d >; f:=quo< F| C >; C Output: Magma V2.11-10 Thu Dec 8 2005 09:02:05 on modular [Seed = 2186935670] ------------------------------------- Finitely presented group C on 4 generators Generators as words in group F C.1 = a C.2 = b C.3 = c C.4 = d Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:01:55 2005 Input: F := FreeGroup(4); C:=sub< F| >; f:=quo< F| C >; C Output: Magma V2.11-10 Thu Dec 8 2005 09:01:55 on modular [Seed = 2387997044] ------------------------------------- Finitely presented group C on 1 generator Generators as words in group F C.1 = Id(F) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:01:51 2005 Input: F := FreeGroup(4); C:=sub< F| >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 09:01:51 on modular [Seed = 2320627052] ------------------------------------- Finitely presented group f on 4 generators Relations Id(f) = Id(f) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:01:35 2005 Input: F := FreeGroup(4); C:=sub< F| >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:01:34 on modular [Seed = 3060648225] ------------------------------------- Finitely presented group C on 1 generator Generators as words in group F C.1 = Id(F) Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:01:27 2005 Input: F := FreeGroup(4); D := FreeGroup(4); C:=sub< F| >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:01:26 on modular [Seed = 2993278253] ------------------------------------- >> D := FreeGroup(4); ^ User error: bad syntax Finitely presented group C on 1 generator Generators as words in group F C.1 = Id(F) Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:00:58 2005 Input: F := FreeGroup(4); D := FreeGroup(4); C:=sub< F| 2a,b,c >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:00:57 on modular [Seed = 3194339598] ------------------------------------- >> D := FreeGroup(4); ^ User error: bad syntax >> C:=sub< F| 2a,b,c >; ^ User error: bad syntax >> f:=quo< F| C >; ^ User error: Identifier 'C' has not been declared or assigned >> C; ^ User error: Identifier 'C' has not been declared or assigned Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:00:23 2005 Input: F := FreeGroup(4); D := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:00:23 on modular [Seed = 3126969828] ------------------------------------- >> D := FreeGroup(4); ^ User error: bad syntax Finitely presented group C on 3 generators Generators as words in group F C.1 = a^2 C.2 = b C.3 = c Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:00:15 2005 Input: F := FreeGroup(4); D<2*a,b,c> := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:00:15 on modular [Seed = 2793265632] ------------------------------------- >> D<2*a,b,c> := FreeGroup(4); ^ User error: bad syntax Finitely presented group C on 3 generators Generators as words in group F C.1 = a^2 C.2 = b C.3 = c Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:00:12 2005 Input: F := FreeGroup(4); D<2a,b,c> := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:00:11 on modular [Seed = 2725895648] ------------------------------------- >> D<2a,b,c> := FreeGroup(4); ^ User error: bad syntax Finitely presented group C on 3 generators Generators as words in group F C.1 = a^2 C.2 = b C.3 = c Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:53 2005 Input: F := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 08:59:53 on modular [Seed = 2926957022] ------------------------------------- Finitely presented group C on 3 generators Generators as words in group F C.1 = a^2 C.2 = b C.3 = c Finitely presented group F on 4 generators (free) Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:44 2005 Input: F := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C F Output: Magma V2.11-10 Thu Dec 8 2005 08:59:44 on modular [Seed = 2859587027] ------------------------------------- >> F; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:39 2005 Input: F := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C Output: Magma V2.11-10 Thu Dec 8 2005 08:59:39 on modular [Seed = 1452091700] ------------------------------------- Finitely presented group C on 3 generators Generators as words in group F C.1 = a^2 C.2 = b C.3 = c Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:32 2005 Input: F := FreeGroup(4); C:=sub< F| a^2,b,c >; C f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:59:32 on modular [Seed = 1384721727] ------------------------------------- >> f:=quo< F| C >; ^ User error: bad syntax >> f; ^ User error: Identifier 'f' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:25 2005 Input: F := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:59:25 on modular [Seed = 1585783114] ------------------------------------- Finitely presented group f on 4 generators Relations f.3 = Id(f) f.1^2 = Id(f) f.2 = Id(f) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:20 2005 Input: F := FreeGroup(4); C:=sub< F| a^a,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:59:20 on modular [Seed = 1518413131] ------------------------------------- Finitely presented group f on 4 generators Relations f.3 = Id(f) f.2 = Id(f) f.1 = Id(f) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:15 2005 Input: F := FreeGroup(4); C:=sub< F| aa,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:59:15 on modular [Seed = 1184708956] ------------------------------------- >> C:=sub< F| aa,b,c >; ^ User error: Identifier 'aa' has not been declared or assigned >> f:=quo< F| C >; ^ User error: Identifier 'C' has not been declared or assigned >> f; ^ User error: Identifier 'f' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:09 2005 Input: F := FreeGroup(4); C:=sub< F| a/2,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:59:08 on modular [Seed = 1117338951] ------------------------------------- >> C:=sub< F| a/2,b,c >; ^ Runtime error in '/': Bad argument types Argument types given: GrpFPElt, RngIntElt >> f:=quo< F| C >; ^ User error: Identifier 'C' has not been declared or assigned >> f; ^ User error: Identifier 'f' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:54 2005 Input: F := FreeGroup(4); C:=sub< F| 2*a,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:58:54 on modular [Seed = 1318400377] ------------------------------------- >> C:=sub< F| 2*a,b,c >; ^ Runtime error in '*': Bad argument types Argument types given: RngIntElt, GrpFPElt >> f:=quo< F| C >; ^ User error: Identifier 'C' has not been declared or assigned >> f; ^ User error: Identifier 'f' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:38 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:58:37 on modular [Seed = 1251030382] ------------------------------------- Finitely presented group f on 4 generators Relations f.3 = Id(f) f.2 = Id(f) f.1 = Id(f) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:33 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=quo< F| a >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:58:32 on modular [Seed = 1991051607] ------------------------------------- Finitely presented group f on 4 generators Relations f.1 = Id(f) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:26 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=quo< F| a >; Output: Magma V2.11-10 Thu Dec 8 2005 08:58:25 on modular [Seed = 1923681626] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:13 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=qou< F| a >; Output: Magma V2.11-10 Thu Dec 8 2005 08:58:12 on modular [Seed = 2141456290] ------------------------------------- >> f:=qou< F| a >; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:08 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=qou< F| F.2 >; Output: Magma V2.11-10 Thu Dec 8 2005 08:58:08 on modular [Seed = 2074084258] ------------------------------------- >> f:=qou< F| F.2 >; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:57:51 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=qou< F|C >; Output: Magma V2.11-10 Thu Dec 8 2005 08:57:50 on modular [Seed = 1740382096] ------------------------------------- >> f:=qou< F|C >; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:57:18 2005 Input: F := FreeGroup(4); C:=sub; f:=qou Output: Magma V2.11-10 Thu Dec 8 2005 08:57:18 on modular [Seed = 1673010164] ------------------------------------- >> f:=qou; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:56:59 2005 Input: F := FreeGroup(4); C:=sub; qou Output: Magma V2.11-10 Thu Dec 8 2005 08:56:59 on modular [Seed = 1874073581] ------------------------------------- >> qou; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:56:53 2005 Input: F := FreeGroup(4); C:=sub; qou Output: Magma V2.11-10 Thu Dec 8 2005 08:56:53 on modular [Seed = 1806701543] ------------------------------------- >> qou; ^ User error: Bad print expression Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:56:03 2005 Input: F := FreeGroup(4); C:=sub; C Output: Magma V2.11-10 Thu Dec 8 2005 08:56:03 on modular [Seed = 399273805] ------------------------------------- Finitely presented group C on 3 generators Generators as words in group F C.1 = a C.2 = b C.3 = c Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:55:58 2005 Input: F := FreeGroup(4); C:=sub; C Output: Magma V2.11-10 Thu Dec 8 2005 08:55:58 on modular [Seed = 331901780] ------------------------------------- >> C:=sub; ^ Runtime error in '+': Bad argument types Argument types given: GrpFPElt, GrpFPElt >> C; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:55:53 2005 Input: F := FreeGroup(4); C:=sub; C Output: Magma V2.11-10 Thu Dec 8 2005 08:55:53 on modular [Seed = 532965215] ------------------------------------- >> C:=sub; ^ Runtime error in '*': Bad argument types Argument types given: RngIntElt, GrpFPElt >> C; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:55:44 2005 Input: F := FreeGroup(4); C:=sub; C Output: Magma V2.11-10 Thu Dec 8 2005 08:55:44 on modular [Seed = 465593170] ------------------------------------- Finitely presented group C on 1 generator Generators as words in group F C.1 = a Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:55:25 2005 Input: F := FreeGroup(4); C:=sub; C Output: Magma V2.11-10 Thu Dec 8 2005 08:55:25 on modular [Seed = 131891043] ------------------------------------- >> C:=sub; ^ User error: bad syntax >> C; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:52:30 2005 Input: F := FreeGroup(4); C:=subGrpAb(F,a); C Output: Magma V2.11-10 Thu Dec 8 2005 08:52:30 on modular [Seed = 248870841] ------------------------------------- >> C:=subGrpAb(F,a); ^ User error: Identifier 'subGrpAb' has not been declared or assigned >> C; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:51:43 2005 Input: F := FreeGroup(4); C:=sub(F,a); C Output: Magma V2.11-10 Thu Dec 8 2005 08:51:42 on modular [Seed = 181498701] ------------------------------------- >> C:=sub(F,a); ^ User error: Identifier 'sub' has not been declared or assigned >> C; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:46:53 2005 Input: F := FreeGroup(4); rels := [ b^a = b*c ]; G := quo< GrpGPC : F | rels >; G; Output: Magma V2.11-10 Thu Dec 8 2005 08:46:52 on modular [Seed = 938233170] ------------------------------------- Consistency check (2,1,1) failed. >> G := quo< GrpGPC : F | rels >; ^ Runtime error in quo< ... >: Bad presentation for group >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:46:31 2005 Input: F := FreeGroup(4); rels := [ 2*a = 0 ]; G := quo< GrpGPC : F | rels >; G; Output: Magma V2.11-10 Thu Dec 8 2005 08:46:31 on modular [Seed = 870861101] ------------------------------------- >> rels := [ 2*a = 0 ]; ^ Runtime error in '*': Bad argument types Argument types given: RngIntElt, GrpFPElt >> G := quo< GrpGPC : F | rels >; ^ User error: Identifier 'rels' has not been declared or assigned >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:44:54 2005 Input: F := FreeGroup(4); rels := [ a = a+a ]; G := quo< GrpGPC : F | rels >; G; Output: Magma V2.11-10 Thu Dec 8 2005 08:44:54 on modular [Seed = 1071924416] ------------------------------------- >> rels := [ a = a+a ]; ^ Runtime error in '+': Bad argument types Argument types given: GrpFPElt, GrpFPElt >> G := quo< GrpGPC : F | rels >; ^ User error: Identifier 'rels' has not been declared or assigned >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:44:31 2005 Input: F := FreeGroup(4); rels := [ a = 2*a ]; G := quo< GrpGPC : F | rels >; G; Output: Magma V2.11-10 Thu Dec 8 2005 08:44:31 on modular [Seed = 1004552365] ------------------------------------- >> rels := [ a = 2*a ]; ^ Runtime error in '*': Bad argument types Argument types given: RngIntElt, GrpFPElt >> G := quo< GrpGPC : F | rels >; ^ User error: Identifier 'rels' has not been declared or assigned >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:43:27 2005 Input: F := FreeGroup(3); rels := [ b^a = b*c, b^(a^-1) = b*c^-1 ]; G := quo< GrpGPC : F | rels >; G; Output: Magma V2.11-10 Thu Dec 8 2005 08:43:20 on modular [Seed = 637427317] ------------------------------------- GrpGPC : G of infinite order on 3 PC-generators PC-Relations: b^a = b * c, b^(a^-1) = b * c^-1 Total time: 0.240 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 07:10:28 2005 Input: G :=DirichletGroup(3675); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); Output: Magma V2.11-10 Thu Dec 8 2005 07:10:26 on modular [Seed = 3256460123] ------------------------------------- Group of Dirichlet characters of modulus 3675 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 105 2 Total time: 0.230 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 07:09:47 2005 Input: G :=DirichletGroup(3675); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); Output: Magma V2.11-10 Thu Dec 8 2005 07:09:16 on modular [Seed = 3474239112] ------------------------------------- Group of Dirichlet characters of modulus 3675 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 35 2 Total time: 0.250 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 07:08:58 2005 Input: G :=DirichletGroup(3675); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); Output: Magma V2.11-10 Thu Dec 8 2005 07:08:54 on modular [Seed = 4163082629] ------------------------------------- Group of Dirichlet characters of modulus 3675 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 105 2 Total time: 0.240 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:38:54 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Coefficient(qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:38:54 on modular [Seed = 4130191596] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 105 2 [] >> Coefficient(qEigenform(D[1],12),11);Parent($1);; ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 3.91MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:37:09 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Coefficient(qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:37:09 on modular [Seed = 3996492063] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 35 2 [ Modular symbols space of level 105, weight 3, character $.2*$.3, and dimension 16 over Rational Field ] -4298657581773597647504145782596750932310358556832711002454219543201/1060354788\ 155084785331796352233842185911113329794742366431633553792856457447585916*a^\ 15 + 59792039146938966702085346471210765249524513188560315063883509432295/1\ 060354788155084785331796352233842185911113329794742366431633553792856457447\ 585916*a^14 + 2248116023755252986404452938356016622954425676202415019802671\ 354342037/30295851090145279580908467206681205311746095136992639040903815822\ 6530416413595976*a^13 - 106375893649896766138612912643006997350281429012339\ 2147140957671964814929/8482838305240678282654370817870737487288906638357938\ 931453068430342851659580687328*a^12 - 8025767737983380042130528078215951109\ 8785225632595715072720574844079253/1559345276698654096076171106226238508692\ 8137202863858329876964026365536138935087*a^11 + 454564293504204618256213652472228130198025352321178049906367490038175998773\ /42414191526203391413271854089353687436444533191789694657265342151714258297\ 90343664*a^10 + 46330602181772887430568752314793023560850992485147208453922\ 9748146003476351/3029585109014527958090846720668120531174609513699263904090\ 38158226530416413595976*a^9 - 372838139837664292581809634900339180485698524\ 348133251578366856627400443548187/84828383052406782826543708178707374872889\ 06638357938931453068430342851659580687328*a^8 - 586908927999659183743988816215739384065711763924034789343566565949655167205\ 7/6237381106794616384304684424904954034771254881145543331950785610546214455\ 5740348*a^7 + 1782427423253817741237532264633280299707060112858881754667069\ 0963198406953111017/2120709576310169570663592704467684371822226659589484732\ 863267107585712914895171832*a^6 - 88132348041601832812155114351918574435160\ 741509599262517612176007084653830584979/21207095763101695706635927044676843\ 71822226659589484732863267107585712914895171832*a^5 - 412607627058114982348201132587715925295396576903215390735685694561153455672\ 0390407/8482838305240678282654370817870737487288906638357938931453068430342\ 851659580687328*a^4 + 31511281196024029702444946733976629500285737560112208\ 52157554522243148399975157367/530177394077542392665898176116921092955556664\ 897371183215816776896428228723792958*a^3 - 117842641475610869683534570735827031431221433659772609683009641833477302642\ 068142067/42414191526203391413271854089353687436444533191789694657265342151\ 71425829790343664*a^2 + 206153836034717403743112319848430370292546919027209\ 544943233012443147466094824699047/21207095763101695706635927044676843718222\ 26659589484732863267107585712914895171832*a - 192001441789413192954514317740798175866107536955491792124958758103956350599\ 7580330285/8482838305240678282654370817870737487288906638357938931453068430\ 342851659580687328 Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^16 - 56*a^15 - 176*a^14 + 68728*a^13 - 1091948*a^12 - 22679096*a^11 + 785484976*a^10 - 2622829640*a^9 - 165895838642*a^8 + 2411431900824*a^7 - 2408011941072*a^6 - 240699299035992*a^5 + 2802555639497124*a^4 - 16324010107722792*a^3 + 61776032371713744*a^2 - 167208206400482520*a + 250585184476674681 Total time: 0.670 seconds, Total memory usage: 5.10MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:37:01 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Coefficient(qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:37:00 on modular [Seed = 3778719241] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 21 2 [] >> Coefficient(qEigenform(D[1],12),11);Parent($1);; ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.38MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:35:26 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Coefficient(qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:35:25 on modular [Seed = 3829377965] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 105, weight 3, character $.3, and dimension 12 over Rational Field ] -9627753560054457599941856824348845429104226288761/2997533989399047188118278316\ 9642133506697637827789504145539724592*a^11 - 827853708124021379076764139093823015797391865310667/69942459752644434389426\ 494062498311515627821598175509672926024048*a^10 + 40644059523131479561661764583374110235005028651212103/699424597526444343894\ 26494062498311515627821598175509672926024048*a^9 + 989817221877944422541708558763314139114963392701732223/69942459752644434389\ 426494062498311515627821598175509672926024048*a^8 - 4261878610765066657170994510860016307411024111549809945/4995889982331745313\ 530463861607022251116272971298250690923287432*a^7 - 165719595913874323498843511999184560463227059528344202087/11657076625440739\ 064904415677083051919271303599695918278821004008*a^6 - 8550237862426105241240329493675908597396021803988777427/1665296660777248437\ 843487953869007417038757657099416896974429144*a^5 - 22112515395558585770232358054851227312561970994091911441063/388569220848024\ 6354968138559027683973090434533231972759607001336*a^4 + 750904150428656166015240182381075933603548512829283147506987/77713844169604\ 92709936277118055367946180869066463945519214002672*a^3 - 1556490164649547596071612342841201796183206846608735384947771/8634871574400\ 54745548475235339485327353429896273771724357111408*a^2 - 2844096403220102441323434035539965744585803674555527965456627/2878290524800\ 18248516158411779828442451143298757923908119037136*a - 15269237696891184072333869330633977403516889841562052248686571/587406229551\ 0576500329763505710784539819250995059671594266064 Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^12 + 36*a^11 - 2526*a^10 - 62724*a^9 + 4357263*a^8 + 70031160*a^7 - 2741991732*a^6 - 10848278808*a^5 + 934352883855*a^4 + 3118035044772*a^3 + 825882174951570*a^2 + 3724663386540060*a + 4212898119152289 Total time: 0.790 seconds, Total memory usage: 5.00MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:34:16 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Coefficient(qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:34:15 on modular [Seed = 2672087196] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 15 2 [ Modular symbols space of level 105, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] 1204211855691391593183200709289952155056950680388873577693915493051674171073611\ 01921111/773542841130395463370175720228639654302353253795450016894911779720\ 33465700824042396292235962121107906560000*a^23 + 178413663326744086935855267582565035975997564952593413810130901598489068335\ 9326279354439/6446190342753295528084797668571997119186277114962083474124264\ 83100278880840200353302435299684342565888000*a^21 + 314163122012022777408157294806902082697836175760398973415297829811458435589\ 29161788688778211/154708568226079092674035144045727930860470650759090003378\ 98235594406693140164808479258447192424221581312000*a^19 + 303749024372985183227086387617877825746125847535800518380329791841520875051\ 37885941569536235047/386771420565197731685087860114319827151176626897725008\ 44745588986016732850412021198146117981060553953280000*a^17 + 724667128574550938868579220941148542781353982434056782045280667876824960971\ 064760280657235891487/42974602285021970187231984457146647461241847433080556\ 49416176554001859205601335688682901997895617105920000*a^15 + 184468297942586187153684218929043555888424249750515777012371263348874589630\ 922011126137916458194407/96692855141299432921271965028579956787794156724431\ 25211186397246504183212603005299536529495265138488320000*a^13 + 119723832376532744864758931160771929311384921478567895648026322866302922000\ 31120908049804612831370681/128923806855065910561695953371439942383725542299\ 24166948248529662005577616804007066048705993686851317760000*a^11 + 424031823655253626958763197922687939261200790731230192598618549025992165574\ 49705382056803442573887713/386771420565197731685087860114319827151176626897\ 7250084474558898601673285041202119814611798106055395328000*a^9 + 843999404082304051100875189403538073275387924102969608753468563437996319086\ 05326325926534126622691803/120677510316754362460245822188555328284298479531\ 271453493745987475871241342939223707164174667895644160000*a^7 + 272668624924044821167899473416090164973595095502511767773323075807539886151\ 8629513429037143619522487001/4834642757064971646063598251428997839389707836\ 221562605593198623252091606301502649768264747632569244160000*a^5 + 146381531928962722508261961449391819363941947189458556374555060833176093362\ 05324254941433278864341591873371/773542841130395463370175720228639654302353\ 25379545001689491177972033465700824042396292235962121107906560000*a^3 - 441582134098380488737104434011700011673746956771871179425637328829394160952\ 886464187339452627561547089377/15470856822607909267403514404572793086047065\ 07590900033789823559440669314016480847925844719242422158131200*a Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^24 + 1780*a^22 + 1308130*a^20 + 507142404*a^18 + 109322750831*a^16 + 12460050133096*a^14 + 617380119190876*a^12 + 7725436928267560*a^10 + 433854703302693743*a^8 + 769413012321755556*a^6 + 100884220801829193186*a^4 - 424438144655392806700*a^2 + 686517081253767450625 Total time: 0.720 seconds, Total memory usage: 5.38MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:33:42 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:33:41 on modular [Seed = 2454314419] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 15 2 [ Modular symbols space of level 105, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] q + (-1204211855691391593183200709289952155056950680388873577693915493051674171\ 07361101921111/773542841130395463370175720228639654302353253795450016894911\ 77972033465700824042396292235962121107906560000*a^23 - 178413663326744086935855267582565035975997564952593413810130901598489068335\ 9326279354439/6446190342753295528084797668571997119186277114962083474124264\ 83100278880840200353302435299684342565888000*a^21 - 314163122012022777408157294806902082697836175760398973415297829811458435589\ 29161788688778211/154708568226079092674035144045727930860470650759090003378\ 98235594406693140164808479258447192424221581312000*a^19 - 303749024372985183227086387617877825746125847535800518380329791841520875051\ 37885941569536235047/386771420565197731685087860114319827151176626897725008\ 44745588986016732850412021198146117981060553953280000*a^17 - 724667128574550938868579220941148542781353982434056782045280667876824960971\ 064760280657235891487/42974602285021970187231984457146647461241847433080556\ 49416176554001859205601335688682901997895617105920000*a^15 - 184468297942586187153684218929043555888424249750515777012371263348874589630\ 922011126137916458194407/96692855141299432921271965028579956787794156724431\ 25211186397246504183212603005299536529495265138488320000*a^13 - 119723832376532744864758931160771929311384921478567895648026322866302922000\ 31120908049804612831370681/128923806855065910561695953371439942383725542299\ 24166948248529662005577616804007066048705993686851317760000*a^11 - 424031823655253626958763197922687939261200790731230192598618549025992165574\ 49705382056803442573887713/386771420565197731685087860114319827151176626897\ 7250084474558898601673285041202119814611798106055395328000*a^9 - 843999404082304051100875189403538073275387924102969608753468563437996319086\ 05326325926534126622691803/120677510316754362460245822188555328284298479531\ 271453493745987475871241342939223707164174667895644160000*a^7 - 272668624924044821167899473416090164973595095502511767773323075807539886151\ 8629513429037143619522487001/4834642757064971646063598251428997839389707836\ 221562605593198623252091606301502649768264747632569244160000*a^5 - 146381531928962722508261961449391819363941947189458556374555060833176093362\ 05324254941433278864341591873371/773542841130395463370175720228639654302353\ 25379545001689491177972033465700824042396292235962121107906560000*a^3 + 198866781635917141547745587446897932027845346436277121321546088827006347496\ 9367312113184171869983705220577/1547085682260790926740351440457279308604706\ 507590900033789823559440669314016480847925844719242422158131200*a)*q^2 + (12476961992900269994914694842672726036592375736303953796713579843574932262\ 621543224704133/47443960922664255086704110840689898797210999566120934369554\ 58915618052562983874600305923805676761284935680000*a^23 + 903595974794547123846916422064344864739315690818900717059310116401468361494\ 30943/814831443454823619459664702214699014956542904734928549800611533043183\ 9059534876248060213010825216000*a^22 + 133007050168195301953005387311837515633766020831799119301997881028159849995\ 84557381087875799/284663765535985530520224665044139392783265997396725606217\ 3275349370831537790324760183554283406056770961408000*a^21 + 357865506177126080130314451026606122453653922721176124837188427611487348007\ 8020927/1810736541010719154354810449365997811014539788299841221779136740095\ 96423545219472179115844685004800*a^20 + 974897171824789797692377662628740332528541515938800040501584199494134717216\ 3693943711385988549/2846637655359855305202246650441393927832659973967256062\ 173275349370831537790324760183554283406056770961408000*a^19 + 237122859406857588886674772392788810383911185280433205170360608553915486355\ 77963699093/162966288690964723891932940442939802991308580946985709960122306\ 6086367811906975249612042602165043200*a^18 + 188215803869345293851035002929306158336603517954081240851672910216800742344\ 82775471870754798609821/142331882767992765260112332522069696391632998698362\ 80310866376746854157688951623800917771417030283854807040000*a^17 + 460936402009265941459050048574968218631190986832770111355047874993530859849\ 68626858245597/814831443454823619459664702214699014956542904734928549800611\ 5330431839059534876248060213010825216000*a^16 + 671613875219589473470896334250658076060120771084512817606627993836243816340\ 401262078657390004269049/23721980461332127543352055420344949398605499783060\ 46718477729457809026281491937300152961902838380642467840000*a^15 + 166414687021068859430755703051148105662111510593905775943082868411673194568\ 0295629438861993/1358052405758039365766107837024498358260904841224880916334\ 352555071973176589146041343368835137536000*a^14 + 755015903911301101640126665642257862869747687738633291675159959248332901172\ 46029766302320265873167909/237219804613321275433520554203449493986054997830\ 6046718477729457809026281491937300152961902838380642467840000*a^13 + 574838437834069102056863264125090372795883678164347095185930084641058975423\ 874296643258353039/40741572172741180972983235110734950747827145236746427490\ 03057665215919529767438124030106505412608000*a^12 + 108013810666016199125403395251373191360736687993098688282623461087516170068\ 58620027283698568204882031337/711659413839963826300561662610348481958164993\ 4918140155433188373427078844475811900458885708515141927403520000*a^11 + 325951975220820429899352129081277103288253284678756994495452468357504059431\ 8863033269561505201/4526841352526797885887026123414994527536349470749603054\ 44784185023991058863048680447789611712512000*a^10 + 218130343648827737565803372280913862500672801900659825070260709851180442620\ 87242361986766483003219401209/142331882767992765260112332522069696391632998\ 6983628031086637674685415768895162380091777141703028385480704000*a^9 + 861904410485912602916314470610699386422004492306488529876067923010892809355\ 68512184146137205113/814831443454823619459664702214699014956542904734928549\ 800611533043183905953487624806021301082521600*a^8 + 245536630371036020664553019781851079231832069072760081613118395881714666314\ 76674851298429413095083696427/222046618982828026926852312826941804043109202\ 33753947442849261695560308407100817162118208138892798525440000*a^7 + 660104243408122437806728808695494411609075576792469569048106937921403333179\ 24306102007660562939/127118789930549706623972652451591109977619797930566076\ 41195187722982588236403863101497992216576000*a^6 - 784270088923878604488584132605570315486605782664341985567063583588721908770\ 0640491455239991661894817790081/1423318827679927652601123325220696963916329\ 9869836280310866376746854157688951623800917771417030283854807040000*a^5 + 158972368752598138639851865415021096570100607366720440374672617783107027228\ 592084988625677414079583/81483144345482361945966470221469901495654290473492\ 85498006115330431839059534876248060213010825216000*a^4 + 126177885362772698146987322954101513463415419823229999990285659191981727936\ 1294283120226611380015581913744663/4744396092266425508670411084068989879721\ 099956612093436955458915618052562983874600305923805676761284935680000*a^3 + 107548242445181351079847960998824707446537670615451239491034955486639086890\ 72587753799864188016071573/814831443454823619459664702214699014956542904734\ 9285498006115330431839059534876248060213010825216000*a^2 - 101578065370575548158380905662701243709132216675842064292224015995999904186\ 5443782424034611565293866032975871/5693275310719710610404493300882787855665\ 31994793451212434655069874166307558064952036710856681211354192281600*a - 876320634049716527955042884895808065835717919630735226929539927608466136519\ 633711982462702897508927/32593257738192944778386588088587960598261716189397\ 1419920244613217273562381395049922408520433008640)*q^3 + (30751561629235116440071308563747291375715714291036326509348297845284326516\ 529/73807195965110835095984121577418389035918741370917441105127856253911585\ 68419271963822656712704000*a^22 + 72634138526219137808122858123253617477379\ 34416031130241562849759391724397635561/984095946201477801279788287698911853\ 812249884945565881401704750052154475789236261843020895027200*a^20 + 397522895603599006568811105692766471482533174791540340530385729497940210610\ 2408427/7380719596511083509598412157741838903591874137091744110512785625391\ 15856841927196382265671270400*a^18 + 30475970835712802812665893951199093678\ 019436221306674628593504127798434523918249469807/14761439193022167019196824\ 315483677807183748274183488221025571250782317136838543927645313425408000*a^\ 16 + 1787863494995953277267595865643103885884753221899914866306019254841276\ 46655440712404143/410039977583949083866578453207879939088437452060652450584\ 043645855064364912181775767925372928000*a^14 + 350294452309956478262589394583688280814211180328226911724501572141809287269\ 613550399271559/73807195965110835095984121577418389035918741370917441105127\ 85625391158568419271963822656712704000*a^12 + 616448361674496555865156405854511542961024017207645493093892442093067447693\ 061064754152971/30752998318796181289993383990590995431632808904548933793803\ 2734391298273684136 ** WARNING: Output too long, hence truncated. '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:33:28 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:33:27 on modular [Seed = 2504973641] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 5 2 [] >> qEigenform(D[1],12);Parent($1);; ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.220 seconds, Total memory usage: 4.50MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:32:37 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:32:36 on modular [Seed = 2287190120] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 105, weight 3, character $.1, and dimension 16 over Rational Field ] q + a*q^2 + (-355/468672*a^15 + 181/234336*a^14 - 3715/117168*a^13 + 2239/78112*a^12 - 78413/156224*a^11 + 2704/7323*a^10 - 27854/7323*a^9 + 73663/39056*a^8 - 6908345/468672*a^7 + 692717/234336*a^6 - 3457763/117168*a^5 - 52637/78112*a^4 - 4645727/156224*a^3 + 291167/117168*a^2 - 719393/58584*a + 4945/2441)*q^3 + (a^2 + 4)*q^4 + (-107/78112*a^15 - 3703/58584*a^13 - 88657/78112*a^11 - 588091/58584*a^9 - 3556185/78112*a^7 - 720082/7323*a^5 - 6160859/78112*a^3 - 408745/29292*a)*q^5 + (181/234336*a^15 + 245/78112*a^14 + 2239/78112*a^13 + 28463/234336*a^12 + 2704/7323*a^11 + 197951/117168*a^10 + 73663/39056*a^9 + 395555/39056*a^8 + 692717/234336*a^7 + 1924413/78112*a^6 - 52637/78112*a^5 + 3825007/234336*a^4 + 291167/117168*a^3 - 23593/58584*a^2 + 4945/2441*a + 1065/2441)*q^6 + (-187/78112*a^14 - 7587/78112*a^12 - 57065/39056*a^10 - 395867/39056*a^8 - 2620631/78112*a^6 - 3968375/78112*a^4 - 639803/19528*a^2 - 9730/2441)*q^7 + (a^3 + 8*a)*q^8 + (-1199/468672*a^15 + 37/4882*a^14 - 9041/78112*a^13 + 18001/58584*a^12 - 952757/468672*a^11 + 135191/29292*a^10 - 2062787/117168*a^9 + 311231/9764*a^8 - 37077241/468672*a^7 + 253210/2441*a^6 - 13759729/78112*a^5 + 8512241/58584*a^4 - 77690083/468672*a^3 + 1840435/29292*a^2 - 2881949/58584*a + 4001/4882)*q^9 + (-23/117168*a^14 - 149/19528*a^12 - 3059/29292*a^10 - 10817/19528*a^8 - 47809/117168*a^6 + 43213/9764*a^4 + 220415/29292*a^2 + 1926/2441)*q^10 + (2887/234336*a^15 + 5339/9764*a^13 + 2187631/234336*a^11 + 188875/2441*a^9 + 75768245/234336*a^7 + 1529956/2441*a^5 + 98626709/234336*a^3 + 597641/9764*a)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^16 + 46*a^14 + 823*a^12 + 7252*a^10 + 32831*a^8 + 71486*a^6 + 60809*a^4 + 15680*a^2 + 576 Total time: 0.760 seconds, Total memory usage: 5.10MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:31:14 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[10],12);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:31:11 on modular [Seed = 2186922697] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 105, weight 3, character $.1, and dimension 16 over Rational Field ] >> qEigenform(D[10],12);Parent($1);; ^ Runtime error in '[]': Sequence element 10 not defined Set of sequences over Power Structure of ModSym Total time: 0.760 seconds, Total memory usage: 4.81MB '165.91.' ************** MAGMA ***************** Host 165.91.100.60 (165.91.100.60) Time: Thu Dec 8 00:05:45 2005 Input: S:=CuspForms(Gamma0(3),14); SetPrecision(S,15); Basis(S); for i:=1 to 3 do (Coefficient(S.i,14)-6*Coefficient(S.i,7))/7; end for; Output: Magma V2.11-10 Thu Dec 8 2005 00:05:45 on modular [Seed = 3489934558] ------------------------------------- [ q + 52*q^4 - 2430*q^5 + 8748*q^6 + 44900*q^7 - 340200*q^8 + 531441*q^9 + 1370160*q^10 - 6137208*q^11 + 5904900*q^12 + 10160150*q^13 - 30275856*q^14 + O(q^15), q^2 - 54*q^4 + 128*q^5 + 729*q^6 - 3456*q^7 + 3524*q^8 + 16902*q^10 - 36608*q^11 - 39366*q^12 + 89856*q^13 + 82808*q^14 + O(q^15), q^3 + 12*q^4 + 36*q^5 - 12*q^6 - 204*q^7 - 792*q^8 + 1104*q^10 + 7524*q^11 + 700*q^12 + 1416*q^13 - 39024*q^14 + O(q^15) ] -4363608 14792 -5400 Total time: 0.300 seconds, Total memory usage: 4.96MB '165.91.' ************** MAGMA ***************** Host 165.91.100.60 (165.91.100.60) Time: Thu Dec 8 00:00:23 2005 Input: S:=CuspForms(Gamma0(3),14); SetPrecision(S,15); Basis(S); for i:=1 to 3 do Coefficient(S.i,14)-6*Coefficient(S.i,7); end for; Output: Magma V2.11-10 Thu Dec 8 2005 00:00:23 on modular [Seed = 3390977034] ------------------------------------- [ q + 52*q^4 - 2430*q^5 + 8748*q^6 + 44900*q^7 - 340200*q^8 + 531441*q^9 + 1370160*q^10 - 6137208*q^11 + 5904900*q^12 + 10160150*q^13 - 30275856*q^14 + O(q^15), q^2 - 54*q^4 + 128*q^5 + 729*q^6 - 3456*q^7 + 3524*q^8 + 16902*q^10 - 36608*q^11 - 39366*q^12 + 89856*q^13 + 82808*q^14 + O(q^15), q^3 + 12*q^4 + 36*q^5 - 12*q^6 - 204*q^7 - 792*q^8 + 1104*q^10 + 7524*q^11 + 700*q^12 + 1416*q^13 - 39024*q^14 + O(q^15) ] -30545256 103544 -37800 Total time: 0.300 seconds, Total memory usage: 4.96MB '165.91.' ************** MAGMA ***************** Host 165.91.100.60 (165.91.100.60) Time: Wed Dec 7 23:58:52 2005 Input: S:=CuspForms(Gamma0(3),14); SetPrecision(S,15); Basis(S); Output: Magma V2.11-10 Wed Dec 7 2005 23:58:52 on modular [Seed = 2388291621] ------------------------------------- [ q + 52*q^4 - 2430*q^5 + 8748*q^6 + 44900*q^7 - 340200*q^8 + 531441*q^9 + 1370160*q^10 - 6137208*q^11 + 5904900*q^12 + 10160150*q^13 - 30275856*q^14 + O(q^15), q^2 - 54*q^4 + 128*q^5 + 729*q^6 - 3456*q^7 + 3524*q^8 + 16902*q^10 - 36608*q^11 - 39366*q^12 + 89856*q^13 + 82808*q^14 + O(q^15), q^3 + 12*q^4 + 36*q^5 - 12*q^6 - 204*q^7 - 792*q^8 + 1104*q^10 + 7524*q^11 + 700*q^12 + 1416*q^13 - 39024*q^14 + O(q^15) ] Total time: 0.310 seconds, Total memory usage: 4.96MB '165.91.' ************** MAGMA ***************** Host 165.91.100.60 (165.91.100.60) Time: Wed Dec 7 23:56:23 2005 Input: S:=CuspForms(Gamma0(3),14); Basis(S); Output: Magma V2.11-10 Wed Dec 7 2005 23:56:23 on modular [Seed = 2219866964] ------------------------------------- [ q + 52*q^4 - 2430*q^5 + 8748*q^6 + 44900*q^7 + O(q^8), q^2 - 54*q^4 + 128*q^5 + 729*q^6 - 3456*q^7 + O(q^8), q^3 + 12*q^4 + 36*q^5 - 12*q^6 - 204*q^7 + O(q^8) ] Total time: 0.310 seconds, Total memory usage: 4.96MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 23:02:24 2005 Input: K := FunctionField(Rationals(), 4); (a1*a2 + a1*a3 + a2*a3) eq a1*a2*a3 - (a1 - 1)*(a2 - 1)*(a3 - 1) + (a1 + a2 + a3) - 1; Output: Magma V2.11-10 Wed Dec 7 2005 23:02:23 on modular [Seed = 1824038660] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 23:00:46 2005 Input: K := FunctionField(Rationals(), 4); (a1 - 1)*(a2 - 1)*(a3 - 1); Output: Magma V2.11-10 Wed Dec 7 2005 23:00:46 on modular [Seed = 1656131243] ------------------------------------- a1*a2*a3 - a1*a2 - a1*a3 + a1 - a2*a3 + a2 + a3 - 1 Total time: 0.180 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 23:00:04 2005 Input: K := FunctionField(Rationals(), 4); (1-a1)*(1-a2)*(1-a3); Output: Magma V2.11-10 Wed Dec 7 2005 23:00:04 on modular [Seed = 1739819908] ------------------------------------- -a1*a2*a3 + a1*a2 + a1*a3 - a1 + a2*a3 - a2 - a3 + 1 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:59:34 2005 Input: K := FunctionField(Rationals(), 4); (x-a1)*(x-a2)*(x-a3); Output: Magma V2.11-10 Wed Dec 7 2005 22:59:34 on modular [Seed = 482930845] ------------------------------------- -a1*a2*a3 + a1*a2*x + a1*a3*x - a1*x^2 + a2*a3*x - a2*x^2 - a3*x^2 + x^3 Total time: 0.180 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:59:23 2005 Input: K := FunctionField(Rationals(), 4); (x-a2)*(x-a3)*(x-a3); Output: Magma V2.11-10 Wed Dec 7 2005 22:59:22 on modular [Seed = 298188204] ------------------------------------- -a2*a3^2 + 2*a2*a3*x - a2*x^2 + a3^2*x - 2*a3*x^2 + x^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:58:34 2005 Input: K := FunctionField(Rationals(), 3); (a1+a2)*(a1+a3)*(a2+a3); Output: Magma V2.11-10 Wed Dec 7 2005 22:58:34 on modular [Seed = 381876827] ------------------------------------- a1^2*a2 + a1^2*a3 + a1*a2^2 + 2*a1*a2*a3 + a1*a3^2 + a2^2*a3 + a2*a3^2 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:58:24 2005 Input: K := FunctionField(Rationals(), 3); (a1-a2)*(a1-a3)*(a2-a3); Output: Magma V2.11-10 Wed Dec 7 2005 22:58:23 on modular [Seed = 199223155] ------------------------------------- a1^2*a2 - a1^2*a3 - a1*a2^2 + a1*a3^2 + a2^2*a3 - a2*a3^2 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:45:09 2005 Input: K := FunctionField(Rationals(), 8); //(a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y); (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y) eq a1*a2*a3*x^3 + b1*b2*b3 + (b1 + b2 + b3)*y^2 + (a1*a2*b3 + a1*a3*b2 + a2*a3*b1 - (a1*a2 + a1*a3 + a2*a3)*y)*x^2 + (a1*b2*b3 + a2*b1*b3 + a3*b1*b2 + (a1 + a2 + a3)*y^2 - (a1*b2 + a1*b3 + a2*b1 + a2*b3 + a3*b1 + a3*b2)*y)*x - (b1*b2 + b1*b3 + b2*b3 + y^2)*y ; Output: Magma V2.11-10 Wed Dec 7 2005 22:45:09 on modular [Seed = 1038733919] ------------------------------------- true Total time: 0.180 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:40:14 2005 Input: K := FunctionField(Rationals(), 8); //(a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y); (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y) eq a1*a2*a3*x^3 + (a1*a2*b3 + a1*a3*b2 + a2*a3*b1 - (a1*a2 + a1*a3 + a2*a3)*y)*x^2 + (a1*b2*b3 + a2*b1*b3 + a3*b1*b2 + (a1 + a2 + a3)*y^2 - (a1*b2 + a1*b3 + a2*b1 + a2*b3 + a3*b1 + a3*b2)*y)*x + b1*b2*b3 + (b1 + b2 + b3)*y^2 - (b1*b2 + b1*b3 + b2*b3 + y^2)*y ; Output: Magma V2.11-10 Wed Dec 7 2005 22:40:11 on modular [Seed = 738181911] ------------------------------------- true Total time: 0.200 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:35:41 2005 Input: K := FunctionField(Rationals(), 8); //(a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y); (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y) eq a1*a2*a3*x^3 + (a1*a2*b3 - a1*a2*y + a1*a3*b2 - a1*a3*y + a2*a3*b1 - a2*a3*y)*x^2 + (a1*b2*b3 - a1*b2*y - a1*b3*y + a1*y^2 + a2*b1*b3 - a2*b1*y - a2*b3*y + a2*y^2 + a3*b1*b2 - a3*b1*y - a3*b2*y + a3*y^2)*x + b1*b2*b3 - b1*b2*y - b1*b3*y + b1*y^2 - b2*b3*y + b2*y^2 + b3*y^2 - y^3 ; Output: Magma V2.11-10 Wed Dec 7 2005 22:35:41 on modular [Seed = 3624677877] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:31:08 2005 Input: K := FunctionField(Rationals(), 8); (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y); Output: Magma V2.11-10 Wed Dec 7 2005 22:31:08 on modular [Seed = 3523623182] ------------------------------------- a1*a2*a3*x^3 + a1*a2*b3*x^2 - a1*a2*x^2*y + a1*a3*b2*x^2 - a1*a3*x^2*y + a1*b2*b3*x - a1*b2*x*y - a1*b3*x*y + a1*x*y^2 + a2*a3*b1*x^2 - a2*a3*x^2*y + a2*b1*b3*x - a2*b1*x*y - a2*b3*x*y + a2*x*y^2 + a3*b1*b2*x - a3*b1*x*y - a3*b2*x*y + a3*x*y^2 + b1*b2*b3 - b1*b2*y - b1*b3*y + b1*y^2 - b2*b3*y + b2*y^2 + b3*y^2 - y^3 Total time: 0.200 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:30:51 2005 Input: K := FunctionField(Rationals(), (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y); Output: Magma V2.11-10 Wed Dec 7 2005 22:30:51 on modular [Seed = 3440977414] ------------------------------------- >> (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y);; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 22:11:42 2005 Input: G :=DirichletGroup(1575); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Wed Dec 7 2005 22:11:38 on modular [Seed = 4146270397] ------------------------------------- Group of Dirichlet characters of modulus 1575 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 Current total memory usage: 71.1MB, failed memory request: 31.6MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Integer Ring >> qEigenform(D[2],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of RngInt >> qEigenform(D[3],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[4],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr Total time: 3.430 seconds, Total memory usage: 71.10MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:52:29 2005 Input: G :=DirichletGroup(225); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:52:27 on modular [Seed = 2338281281] ------------------------------------- Group of Dirichlet characters of modulus 225 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 [ Modular symbols space of level 225, weight 3, character $.1, and dimension 2 over Rational Field, Modular symbols space of level 225, weight 3, character $.1, and dimension 2 over Rational Field, Modular symbols space of level 225, weight 3, character $.1, and dimension 4 over Rational Field, Modular symbols space of level 225, weight 3, character $.1, and dimension 4 over Rational Field ] q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 + 2 q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 + 2 q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 + 14*a^2 + 9 q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 - 238*a^2 + 17689 >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of RngSerPow >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 1.389 seconds, Total memory usage: 6.42MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:45:50 2005 Input: G :=DirichletGroup(675); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:45:50 on modular [Seed = 2203541244] ------------------------------------- Group of Dirichlet characters of modulus 675 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 5 2 [] >> qEigenform(D[1],2);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences >> qEigenform(D[2],2);Parent($1); ^ Runtime error in '[]': Sequence element 2 not defined Power Structure of PowSeqEnum >> qEigenform(D[3],2);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Power Structure of PowStr >> qEigenform(D[4],2);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 0.230 seconds, Total memory usage: 4.72MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:33:26 2005 Input: G :=DirichletGroup(1215); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:33:26 on modular [Seed = 2910418426] ------------------------------------- Group of Dirichlet characters of modulus 1215 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 5 2 [] >> qEigenform(D[1],2);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences >> qEigenform(D[2],2);Parent($1); ^ Runtime error in '[]': Sequence element 2 not defined Power Structure of PowSeqEnum >> qEigenform(D[3],2);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Power Structure of PowStr >> qEigenform(D[4],2);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 0.220 seconds, Total memory usage: 4.45MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:28:05 2005 Input: G :=DirichletGroup(1215); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Wed Dec 7 2005 21:27:44 on modular [Seed = 1502445656] ------------------------------------- Group of Dirichlet characters of modulus 1215 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 Errors: /bin/sh: line 1: 25208 Alarm clock nice -n 19 /usr/local/bin/magma '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:21:37 2005 Input: G :=DirichletGroup(405); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:21:34 on modular [Seed = 1318753729] ------------------------------------- Group of Dirichlet characters of modulus 405 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 [ Modular symbols space of level 405, weight 3, character $.1, and dimension 16 over Rational Field, Modular symbols space of level 405, weight 3, character $.1, and dimension 16 over Rational Field ] q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^16 + 48*a^14 + 918*a^12 + 9004*a^10 + 48453*a^8 + 140868*a^6 + 198472*a^4 + 96984*a^2 + 2916 q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^16 + 48*a^14 + 912*a^12 + 8704*a^10 + 43602*a^8 + 109032*a^6 + 117844*a^4 + 36000*a^2 + 81 >> qEigenform(D[3],2);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Power Structure of RngSerPow >> qEigenform(D[4],2);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 3.049 seconds, Total memory usage: 11.31MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:21:10 2005 Input: G :=DirichletGroup(405); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:21:10 on modular [Seed = 1268748540] ------------------------------------- Group of Dirichlet characters of modulus 405 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 5 2 [] >> qEigenform(D[1],2);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences >> qEigenform(D[2],2);Parent($1); ^ Runtime error in '[]': Sequence element 2 not defined Power Structure of PowSeqEnum >> qEigenform(D[3],2);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Power Structure of PowStr >> qEigenform(D[4],2);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 0.210 seconds, Total memory usage: 4.47MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:20:27 2005 Input: G :=DirichletGroup(405); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:20:25 on modular [Seed = 1167171085] ------------------------------------- Group of Dirichlet characters of modulus 405 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 405, weight 3, character $.1*$.2, and dimension 20 over Rational Field, Modular symbols space of level 405, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^20 + 1742*a^18 + 1290993*a^16 + 526424484*a^14 + 128074907970*a^12 + 18925916631936*a^10 + 1674883519400862*a^8 + 88428564569459820*a^6 + 3100955720461546389*a^4 + 68055510152463772226*a^2 + 672133169091045750841 q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^24 + 2856*a^22 + 3478968*a^20 + 2357272460*a^18 + 979084307250*a^16 + 260296493582268*a^14 + 44972144261339640*a^12 + 5000266398099461004*a^10 + 343570736862382838841*a^8 + 13461967850983877713964*a^6 + 267765601556510297904192*a^4 + 1663459504983123870604536*a^2 + 21207269458849925801875396 >> qEigenform(D[3],2);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Power Structure of RngSerPow >> qEigenform(D[4],2);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 2.370 seconds, Total memory usage: 11.23MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:19:40 2005 Input: G :=DirichletGroup(405); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); qEigenform(D[2],12);Parent($1); qEigenform(D[3],12);Parent($1); qEigenform(D[4],12);Parent($1); qEigenform(D[5],12);Parent($1); qEigenform(D[6],12);Parent($1); qEigenform(D[7],12);Parent($1); qEigenform(D[8],12);Parent($1); qEigenform(D[9],12);Parent($1); qEigenform(D[10],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:19:37 on modular [Seed = 2141955175] ------------------------------------- Group of Dirichlet characters of modulus 405 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 405, weight 3, character $.1*$.2, and dimension 20 over Rational Field, Modular symbols space of level 405, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] q + (-75172966845177826177027655884569706026546198677196542977375869681023741/2\ 211769901855602660307810712383856679743406918230929364931136698836519859741\ 843244069859801216*a^19 - 7552738132943473814982161741830663703455891635585\ 87373766935004797958588/575981745274896526121825706349962677016512218289304\ 5221174835153220103801411050114765259899*a^17 - 784518838194958988522514326745813305163450401515685373272594473386546575947\ 55/368628316975933776717968452063976113290567819705154894155189449806086643\ 290307207344976633536*a^15 - 3470483133170729781768906603215984695878622618\ 6489578710261754713708779655010971/1843141584879668883589842260319880566452\ 83909852577447077594724903043321645153603672488316768*a^13 - 727209188475665381195775305894376304647298116717075796833910337588456913990\ 85355047/737256633951867553435936904127952226581135639410309788310378899612\ 173286580614414689953267072*a^11 - 5758224767195480410489139556162556127056\ 246361617702218194111572886929630276610416783/18431415848796688835898422603\ 1988056645283909852577447077594724903043321645153603672488316768*a^9 - 121587841934540189004247901390737109962254494949443910633582498140271249466\ 470176539259/20479350943107432039887136225776450738364878872508605230843858\ 322560369071683733741387590752*a^7 - 15554821111449934404193980110163480692\ 126214952085005255697031250980512379070897856148611/23039269810995861044873\ 028253998507080660488731572180884699340612880415205644200459061039596*a^5 - 271297661984924755572096649048038905508412214458276629785797113106076873853\ 772409787447305/57598174527489652612182570634996267701651221828930452211748\ 35153220103801411050114765259899*a^3 - 487987347838286670578425364656571359026184878602182967142523113821575514659\ 68699434736182407/345589047164937915673095423809977606209907330973582713270\ 49010919320622808466300688591559394*a)*q^2 + (-1332198671522142605793090023608424979360711597386148203331041/37787935916\ 59354592294855187844728140360814217624417056076049733242742643196983152*a^1\ 8 - 1224708941679031862907868784483477992397789344194941058564217379/839731\ 909257634353843301152854384031191292048360981568016899940720609476265996256\ *a^16 - 45359631817970327997932639559715715603133811207427255262289219703/1\ 749441477620071570506877401779966731648525100752044933368541543167936408887\ 4922*a^14 - 325666701914942179008129862029362211563559553828915412704661271\ 2828503/1259597863886451530764951729281576046786938072541472352025349911080\ 914214398994384*a^12 - 6642159545681721238362211969263563281525481738512494\ 23984234362895300179/419865954628817176921650576427192015595646024180490784\ 008449970360304738132998128*a^10 - 5674425264674729308176865280361167739563\ 8330270640884749759191261347472075/9330354547307048376036679476159822568792\ 1338706775729779655548968956608473999584*a^8 - 112941055719513271540920073814330092212006970945695247540413787922444267492\ 60/787248664929032206728094830800985029241836295338420220015843694425571383\ 99937149*a^6 - 201353155687143673387451032097467612777994659259052531369475\ 4564217925002161983/1049664886572042942304126441067980038989115060451226960\ 02112492590076184533249532*a^4 - 108570403027064976849333247960661247178548\ 77736526204936164823219149559446572627/874720738810035785253438700889983365\ 8242625503760224666842707715839682044437461*a^2 - 797865655748134371693973494965638974580588654167543528147085600260159316709\ 2491769/2361745994787096620184284492402955087725508886015260660047531083276\ 71415199811447)*q^4 + (6973157488129773926528696161305835386210716301759528\ 577632858889948842919/12533362777181748408410927370175187851879305869975266\ 401276441293406945871870445049729205540224*a^19 - 211271391000069870384217673493158214036267665140495925880318890957/48950492\ 186355279388587554103340608330233987375106698544409148244426488199973719751\ 008*a^18 + 1370109783685137453617201256655764546140003373641418717568634100\ 5841179114299/6266681388590874204205463685087593925939652934987633200638220\ 646703472935935222524864602770112*a^17 - 92543339883309267031774336293946139492704150154168724172982518741051/543894\ 357626169770984306156703784537002599859723407761604546091604738757777485775\ 0112*a^16 + 226370437755144704678186406900191038757049020030506484240592166\ 03344832238224903/626668138859087420420546368508759392593965293498763320063\ 8220646703472935935222524864602770112*a^15 - 51108352131546410076003974392086035621295860787552659090226758658611945/181\ 298119208723256994768718901261512334199953241135920534848697201579585925828\ 5916704*a^14 + 101627483925058269401926569690212709123067539440102725555221\ 72114953003538664981829/313334069429543710210273184254379696296982646749381\ 6600319110323351736467967611262432301385056*a^13 - 413940370195901077823070709302803583145609514522463276941157644082954586921\ /16316830728785093129529184701113536110077995791702232848136382748142162733\ 324573250336*a^12 + 2141244014708763041177392189988502679994879145072796532\ 4122109418879338479115920781923/1253336277718174840841092737017518785187930\ 5869975266401276441293406945871870445049729205540224*a^11 - 455025687380310041278871272507918130759957245672883958759932491940234503362\ 7/3399339735163561068651913479398653356266249123271298510028413072529617236\ 10928609382*a^10 + 33294832683069843523323249202430746261955452406660623065\ 16857230436055303940466371574223/626668138859087420420546368508759392593965\ 2934987633200638220646703472935935222524864602770112*a^9 - 377841494975630937752613046909187957587507598337539735002276242598529091235\ 4609/9064905960436162849738435945063075616709997662056796026742434860078979\ 29629142958352*a^8 + 732605484908948776668800806226144287843838404833098466\ 91122495172920519586662569545780919/783335173573859275525682960635949240742\ 456616873454150079777580837934116991902815608075346264*a^7 - 299600661901575804725954660218827716100516107723510384640958514432615241161\ 5352213/4079207682196273282382296175278384027519498947925558212034095687035\ 540683331143312584*a^6 + 68144447469254693087928413080376011710217803138442\ 55868836839501060944834850856745810699683/783335173573859275525682960635949\ 240742456616873454150079777580837934116991902815608075346264*a^5 - 115941367189763439641244619029546671930037424521829300123303948012122082498\ 12853882/169966986758178053432595673969932667813312456163564925501420653626\ 480861805464304691*a^4 + 17651435453646056674409171492265326216392185256389\ 3816886764579717041136469401171542633229841/3916675867869296377628414803179\ 74620371228308436727075039888790418967058495951407804037673132*a^3 - 131048466192529284289479144792918318288052904150155564233824776330432285168\ 621815407/37770441501817345207243483104429481736291656925236650111426811916\ 995747067880956598*a^2 + 10427042922351723684558441259500154954522029725985\ 21294418925609488911106865942991223605475010/979168966967324094407103700794\ 93655092807077109181768759972197604741764623987851951009418283*a - 121780864199294714824126779885933050749620577387413855130396299806425615803\ 040066877471/15297028808236024808933610657293940103198121054720843295127858\ 82638327756249178742219)*q^5 + (-751729668451778261770276558845697060265461\ 98677196542977375869681023741/663530970556680798092343213715157003923022075\ 4692788094793410096509559579225529732209579403648*a^19 - 755273813294347381498216174183066370345589163558587373766935004797958588/17\ 279452358246895783654771190498880310495366548679135663524505459660311404233\ 150344295779697*a^17 - 7845188381949589885225143267458133051634504015156853\ 7327259447338654657594755/1105884950927801330153905356191928339871703459115\ 464682465568349418259929870921622034929900608*a^15 - 347048313317072978176890660321598469587862261864895787102617547137087796550\ 10971/552942475463900665076952678095964169935851729557732341232784174709129\ 964935460811017464950304*a^13 - 7272091884756653811957753058943763046472981\ 1671707579683391033758845691399085355047/2211769901855602660307810712383856\ 679743406918230929364931136698836519859741843244069859801216*a^11 - 575822476719548041048913955616255612705624636161770221819411157288692963027\ 6610416783/5529424754639006650769526780959641699358517295577323412327841747\ 09129964935460811017464950304*a^9 - 121587841934540189004247901390737109962\ 254494949443910633582498140271249466470176539259/61438052829322296119661408\ 677329352215094636617525815692531574967681107215051201224162772256*a^7 - 155548211114499344041939801101634806921262149520850052556970312509805123790\ 70897856148611/691178094329875831346190847619955212419814661947165426540980\ 21838641245616932601377183118788*a^5 - 271297661984924755572096649048038905508412214458276629785797113106076873853\ 772409787447305/17279452358246895783654771190498880310495366548679135663524\ 505459660311404233150344295779697*a^3 - 833576395003224586251520788466548965236092209575765680413013223014781742744\ 35000123327741801/10367671414948137470192862714299328186297219929 ** WARNING: Output too long, hence truncated. '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:17:19 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[7],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:17:19 on modular [Seed = 2024582390] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 4 over Rational Field ] q + (-1/182*a^3 - 51/182*a)*q^2 + 6*q^4 + (-1/364*a^3 - 1/12*a^2 - 51/364*a - 71/12)*q^5 + (-1/546*a^3 - 233/546*a)*q^7 + (-1/91*a^3 - 51/91*a)*q^8 + (-5/546*a^3 - 1165/546*a + 5)*q^10 + (1/6*a^2 + 71/6)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 + 142*a^2 + 8281 Total time: 0.700 seconds, Total memory usage: 4.91MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:16:53 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[3],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:16:52 on modular [Seed = 1906681284] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 4 over Rational Field ] q + q^2 - 3*q^4 + (-1/4*a + 3/4)*q^5 + (-1/2*a + 1/2)*q^7 - 7*q^8 + (-1/4*a + 3/4)*q^10 + (1/2*a - 1/2)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 - 2*a + 397 Total time: 0.690 seconds, Total memory usage: 4.91MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:16:45 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[3],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:16:45 on modular [Seed = 1689301192] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 5 2 [] >> qEigenform(D[3],12);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Set of null sequences Total time: 0.220 seconds, Total memory usage: 4.45MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:16:12 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[3],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:16:11 on modular [Seed = 1639298347] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 [ Modular symbols space of level 135, weight 3, character $.1, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field ] q + a*q^2 + (a^2 + 4)*q^4 + (1/6*a^3 + 11/6*a)*q^5 + (a^2 + 5)*q^7 + (a^3 + 8*a)*q^8 + (-a^2 - 6)*q^10 + a*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 + 17*a^2 + 36 Total time: 0.840 seconds, Total memory usage: 4.91MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:15:26 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[2],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:15:25 on modular [Seed = 449760774] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 [ Modular symbols space of level 135, weight 3, character $.1, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field ] q + a*q^2 + (a^2 + 4)*q^4 + (1/6*a^3 + 11/6*a)*q^5 + (a^2 + 5)*q^7 + (a^3 + 8*a)*q^8 + (-a^2 - 6)*q^10 + a*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 + 17*a^2 + 36 Total time: 0.820 seconds, Total memory usage: 4.91MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:14:53 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:14:52 on modular [Seed = 398705450] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 [ Modular symbols space of level 135, weight 3, character $.1, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field ] q + a*q^2 - q^4 - a*q^5 + 12*q^7 + 3*a*q^8 + 5*q^10 + 7*a*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 + 5 Total time: 0.830 seconds, Total memory usage: 4.91MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 21:06:56 2005 Input: K := FunctionField(Rationals(), 6); //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y); g0 := u^3*x^3 + v^3 + 3*v*y^2 - 3*v^2*y - y^3; //(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g1 := u^3*x^3 + v^3 + 3*v*y^2 + 3*v^2*y + y^3; //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y)*(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g2 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2*y + y^3)^2; g0*g1 eq g2; g3 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2 + y^2)^2*y^2; g2 eq g3; (u*x + v - y)*(zeta*u*x + v + y)*(zeta^2*u*x + v - y); Output: Magma V2.11-10 Wed Dec 7 2005 21:06:55 on modular [Seed = 3742042562] ------------------------------------- true true zeta^3*u^3*x^3 + zeta^3*u^2*v*x^2 - zeta^3*u^2*x^2*y + zeta^2*u^2*v*x^2 + zeta^2*u^2*x^2*y + zeta^2*u*v^2*x - zeta^2*u*x*y^2 + zeta*u^2*v*x^2 - zeta*u^2*x^2*y + zeta*u*v^2*x - 2*zeta*u*v*x*y + zeta*u*x*y^2 + u*v^2*x - u*x*y^2 + v^3 - v^2*y - v*y^2 + y^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 21:03:54 2005 Input: K := FunctionField(Rationals(), 6); //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y); g0 := u^3*x^3 + v^3 + 3*v*y^2 - 3*v^2*y - y^3; //(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g1 := u^3*x^3 + v^3 + 3*v*y^2 + 3*v^2*y + y^3; //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y)*(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g2 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2*y + y^3)^2; g0*g1 eq g2; g3 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2 + y^2)^2*y^2; g2 eq g3; (u*x + v - y)*(zeta*u*x + v + y); Output: Magma V2.11-10 Wed Dec 7 2005 21:03:53 on modular [Seed = 3658354033] ------------------------------------- true true zeta*u^2*x^2 + zeta*u*v*x - zeta*u*x*y + u*v*x + u*x*y + v^2 - y^2 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:57:38 2005 Input: K := FunctionField(Rationals(), 6); //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y); g0 := u^3*x^3 + v^3 + 3*v*y^2 - 3*v^2*y - y^3; //(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g1 := u^3*x^3 + v^3 + 3*v*y^2 + 3*v^2*y + y^3; //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y)*(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g2 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2*y + y^3)^2; g0*g1 eq g2; g3 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2 + y^2)^2*y^2; g2 eq g3; Output: Magma V2.11-10 Wed Dec 7 2005 20:57:37 on modular [Seed = 3540452320] ------------------------------------- true true Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:50:22 2005 Input: K := FunctionField(Rationals(), 6); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); g0 := a^3*x^3 + b^3 + 3*b*y^2 - 3*b^2*y - y^3; //(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); g1 := a^3*x^3 + b^3 + 3*b*y^2 + 3*b^2*y + y^3; //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)*(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); g2 := (a^3*x^3 + b^3 + 3*b*y^2)^2 - (3*b^2*y + y^3)^2; g0*g1 eq g2; g3 := (a^3*x^3 + b^3 + 3*b*y^2)^2 - (3*b^2 + y^2)^2*y^2; g2 eq g3; Output: Magma V2.11-10 Wed Dec 7 2005 20:50:22 on modular [Seed = 4163644036] ------------------------------------- true true Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:48:33 2005 Input: K := FunctionField(Rationals(), 6); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); g0 := a^3*x^3 + b^3 + 3*b*y^2 - 3*b^2*y - y^3; //(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); g1 := a^3*x^3 + b^3 + 3*b*y^2 + 3*b^2*y + y^3; //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)*(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); g2 := (a^3*x^3 + b^3 + 3*b*y^2)^2 - (3*b^2*y + y^3)^2; g0*g1 eq g2; g2; Output: Magma V2.11-10 Wed Dec 7 2005 20:48:33 on modular [Seed = 4129432869] ------------------------------------- true a^6*x^6 + 2*a^3*b^3*x^3 + 6*a^3*b*x^3*y^2 + b^6 - 3*b^4*y^2 + 3*b^2*y^4 - y^6 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:45:27 2005 Input: K := FunctionField(Rationals(), 6); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)*(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); b^3 + 3*b^2*y + 3*b*y^2 + y^3 + a^3*x^3 + (a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) + zeta^2*(a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) + zeta* (a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) eq b^3 + 3*b^2*y + 3*b*y^2 + y^3 + a^3*x^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) ; Output: Magma V2.11-10 Wed Dec 7 2005 20:45:27 on modular [Seed = 3963109369] ------------------------------------- true true Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:44:04 2005 Input: K := FunctionField(Rationals(), 6); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)*(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); zeta^3*a^3*x^3 + zeta^3*a^2*b*x^2 + zeta^3*a^2*x^2*y + zeta^2*a^2*b*x^2 + zeta^2*a^2*x^2*y + zeta^2*a*b^2*x + 2*zeta^2*a*b*x*y + zeta^2*a*x*y^2 + zeta*a^2*b*x^2 + zeta*a^2*x^2*y + zeta*a*b^2*x + 2*zeta*a*b*x*y + zeta*a*x*y^2 + a*b^2*x + 2*a*b*x*y + a*x*y^2 + b^3 + 3*b^2*y + 3*b*y^2 + y^3 eq b^3 + 3*b^2*y + 3*b*y^2 + y^3 + zeta^3*(a^3*x^3 + a^2*b*x^2 + a^2*x^2*y) + a*b^2*x + 2*a*b*x*y + a*x*y^2 + zeta^2*(a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) + zeta* (a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) ; Output: Magma V2.11-10 Wed Dec 7 2005 20:44:04 on modular [Seed = 3946784557] ------------------------------------- true true Total time: 0.180 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:41:23 2005 Input: K := FunctionField(Rationals(), 6); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)* (a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); Output: Magma V2.11-10 Wed Dec 7 2005 20:41:23 on modular [Seed = 3778358656] ------------------------------------- true zeta^3*a^3*x^3 + zeta^3*a^2*b*x^2 + zeta^3*a^2*x^2*y + zeta^2*a^2*b*x^2 + zeta^2*a^2*x^2*y + zeta^2*a*b^2*x + 2*zeta^2*a*b*x*y + zeta^2*a*x*y^2 + zeta*a^2*b*x^2 + zeta*a^2*x^2*y + zeta*a*b^2*x + 2*zeta*a*b*x*y + zeta*a*x*y^2 + a*b^2*x + 2*a*b*x*y + a*x*y^2 + b^3 + 3*b^2*y + 3*b*y^2 + y^3 Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:40:57 2005 Input: K := FunctionField(Rationals(), 6); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; (a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)* (a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); Output: Magma V2.11-10 Wed Dec 7 2005 20:40:56 on modular [Seed = 2621984918] ------------------------------------- true zeta^6*a^6*x^6 + 2*zeta^6*a^5*b*x^5 + zeta^6*a^4*b^2*x^4 - zeta^6*a^4*x^4*y^2 + 2*zeta^5*a^5*b*x^5 + 4*zeta^5*a^4*b^2*x^4 + 2*zeta^5*a^3*b^3*x^3 - 2*zeta^5*a^3*b*x^3*y^2 + 2*zeta^4*a^5*b*x^5 + 5*zeta^4*a^4*b^2*x^4 - zeta^4*a^4*x^4*y^2 + 4*zeta^4*a^3*b^3*x^3 - 4*zeta^4*a^3*b*x^3*y^2 + zeta^4*a^2*b^4*x^2 - 2*zeta^4*a^2*b^2*x^2*y^2 + zeta^4*a^2*x^2*y^4 + 4*zeta^3*a^4*b^2*x^4 + 8*zeta^3*a^3*b^3*x^3 + 4*zeta^3*a^2*b^4*x^2 - 4*zeta^3*a^2*b^2*x^2*y^2 + zeta^2*a^4*b^2*x^4 - zeta^2*a^4*x^4*y^2 + 4*zeta^2*a^3*b^3*x^3 - 4*zeta^2*a^3*b*x^3*y^2 + 5*zeta^2*a^2*b^4*x^2 - 6*zeta^2*a^2*b^2*x^2*y^2 + zeta^2*a^2*x^2*y^4 + 2*zeta^2*a*b^5*x - 4*zeta^2*a*b^3*x*y^2 + 2*zeta^2*a*b*x*y^4 + 2*zeta*a^3*b^3*x^3 - 2*zeta*a^3*b*x^3*y^2 + 4*zeta*a^2*b^4*x^2 - 4*zeta*a^2*b^2*x^2*y^2 + 2*zeta*a*b^5*x - 4*zeta*a*b^3*x*y^2 + 2*zeta*a*b*x*y^4 + a^2*b^4*x^2 - 2*a^2*b^2*x^2*y^2 + a^2*x^2*y^4 + 2*a*b^5*x - 4*a*b^3*x*y^2 + 2*a*b*x*y^4 + b^6 - 3*b^4*y^2 + 3*b^2*y^4 - y^6 Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:39:38 2005 Input: K := FunctionField(Rationals(), 6); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; Output: Magma V2.11-10 Wed Dec 7 2005 20:39:38 on modular [Seed = 2605667939] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:38:33 2005 Input: K := FunctionField(Rationals(), 6); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; Output: Magma V2.11-10 Wed Dec 7 2005 20:38:33 on modular [Seed = 2321434417] ------------------------------------- true Total time: 0.180 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:38:18 2005 Input: K := FunctionField(Rationals(), 5); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); a^3*(y^2 - mu) + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*(y^2 - mu) + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; Output: Magma V2.11-10 Wed Dec 7 2005 20:38:18 on modular [Seed = 2169850921] ------------------------------------- >> K := FunctionField(Rationals(), 5); ^ Runtime error in 'AssignNames': Argument 2 should have length at most 5 >> a^3*(y^2 - mu) + a^2*b*x^2 - a^2*x^2*y + ^ User error: Identifier 'a' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:37:49 2005 Input: K := FunctionField(Rationals(), 5); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); a^3*(y^2 - mu) + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2); Output: Magma V2.11-10 Wed Dec 7 2005 20:37:49 on modular [Seed = 3194647537] ------------------------------------- >> K := FunctionField(Rationals(), 5); ^ Runtime error in 'AssignNames': Argument 2 should have length at most 5 >> a^3*(y^2 - mu) + a^2*b*x^2 - a^2*x^2*y + ^ User error: Identifier 'a' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:33:33 2005 Input: K := FunctionField(Rationals(), 5); (a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); Output: Magma V2.11-10 Wed Dec 7 2005 20:33:33 on modular [Seed = 3026221819] ------------------------------------- zeta^3*a^3*x^3 + zeta^3*a^2*b*x^2 - zeta^3*a^2*x^2*y + zeta^2*a^2*b*x^2 - zeta^2*a^2*x^2*y + zeta^2*a*b^2*x - 2*zeta^2*a*b*x*y + zeta^2*a*x*y^2 + zeta*a^2*b*x^2 - zeta*a^2*x^2*y + zeta*a*b^2*x - 2*zeta*a*b*x*y + zeta*a*x*y^2 + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:32:32 2005 Input: K := FunctionField(Rationals(), 5); (a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); Output: Magma V2.11-10 Wed Dec 7 2005 20:32:32 on modular [Seed = 3009904033] ------------------------------------- a^3*x^3*zeta^3 + a^2*b*x^2*zeta^3 + a^2*b*x^2*zeta^2 + a^2*b*x^2*zeta - a^2*x^2*y*zeta^3 - a^2*x^2*y*zeta^2 - a^2*x^2*y*zeta + a*b^2*x*zeta^2 + a*b^2*x*zeta + a*b^2*x - 2*a*b*x*y*zeta^2 - 2*a*b*x*y*zeta - 2*a*b*x*y + a*x*y^2*zeta^2 + a*x*y^2*zeta + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 Total time: 0.180 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:31:25 2005 Input: K := FunctionField(Rationals(), 4); (a*x + b - y)*(a*x + b + y); Output: Magma V2.11-10 Wed Dec 7 2005 20:31:24 on modular [Seed = 2843567719] ------------------------------------- a^2*x^2 + 2*a*b*x + b^2 - y^2 Total time: 0.180 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:28:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:28:15 on modular [Seed = 2708837005] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:28:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:28:10 on modular [Seed = 1586130295] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:28:02 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:28:02 on modular [Seed = 1536125050] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:53 on modular [Seed = 1350859914] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:44 on modular [Seed = 1301897630] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:39 on modular [Seed = 1251894914] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:32 on modular [Seed = 1200843648] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:26 on modular [Seed = 2091405441] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:22 on modular [Seed = 2041402752] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:17 on modular [Seed = 1990351488] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:12 on modular [Seed = 1824014495] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:08 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:07 on modular [Seed = 1774011782] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:54 on modular [Seed = 1722960635] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:47 on modular [Seed = 1672957943] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:39 on modular [Seed = 416035988] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:31 on modular [Seed = 364984721] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:26 on modular [Seed = 314982035] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:11 on modular [Seed = 266019775] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 18:22:24 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); print T; print T^(-1); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:22:24 on modular [Seed = 64433808] ------------------------------------- [1/(b*c) -1/(b*c) 1/(b*c)] [0 -1/b (-b + 1)/(b*c)] [(-b - c + 1)/(a*b*c) (-a*b*c - a*b + a + b + c - 1)/(a*b^2*c) (-a*b^2*c + 2*a*b*c + a*b - a + b^2 + b*c - 2*b - c + 1)/(a*b^2*c^2)] [a*b*c/(a + b + c - 1) (a*b*c - a - b - c + 1)/(a + b + c - 1) -a*b*c/(a + b + c - 1)] [(-b^2*c + b*c)/(a + b + c - 1) (a*b^2*c - 2*a*b*c - a*b + a - b^2*c - b^2 - b*c^2 + 2*b + c - 1)/(a*b + a*c - a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1) (-a*b^2*c + a*b*c)/(a*b + a*c - a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)] [b*c^2/(a + b + c - 1) (-a*b*c^2 - a*b*c + a*c - b^2*c - b*c^2 + 2*b*c + c^2 - c)/(a*b + a*c - a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1) a*b*c^2/(a*b + a*c - a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)] [ $.1^2 - 1, $.2, $.3, $.4, $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9, $.6 + $.8, $.7, $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 18:06:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); print T; R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:06:30 on modular [Seed = 653398003] ------------------------------------- [1/(b*c) -1/(b*c) 1/(b*c)] [0 -1/b (-b + 1)/(b*c)] [(-b - c + 1)/(a*b*c) (-a*b*c - a*b + a + b + c - 1)/(a*b^2*c) (-a*b^2*c + 2*a*b*c + a*b - a + b^2 + b*c - 2*b - c + 1)/(a*b^2*c^2)] [ $.1^2 - 1, $.2, $.3, $.4, $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9, $.6 + $.8, $.7, $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 18:03:50 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); print B; R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:03:50 on modular [Seed = 603392598] ------------------------------------- [(a + b + c - 1)/(b^2*c^2) 0 0] [0 (b + c - 1)/(b*c) (a*b^2*c + a*b*c^2 - a*b*c - a*b - a*c + a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b^2*c^2)] [0 (b + c - 1)/(b*c) (b + c - 1)/(b*c)] [ $.1^2 - 1, $.2, $.3, $.4, $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9, $.6 + $.8, $.7, $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 18:02:01 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); print T*S*T^(-1)-F; B:=T*A*Transpose(T); print Transpose(B)*B^(-1)-F; R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:02:00 on modular [Seed = 3862088722] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [ $.1^2 - 1, $.2, $.3, $.4, $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9, $.6 + $.8, $.7, $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 18:01:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); print T*S*T^(-1)-F; R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:01:33 on modular [Seed = 3913139000] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> I:=Ideal(Eltseq(P*B*Transpose(P)-B)); ^ User error: Identifier 'B' has not been declared or assigned >> print GroebnerBasis(I); ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 18:01:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); print T*F*T^(-1)-S; R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:01:07 on modular [Seed = 3963143747] ------------------------------------- [-1/(a*b) (a*b^2*c + a*b*c^2 + a*b - a + b^2 + b*c - 2*b - c + 1)/(a*b^3*c + a*b^2*c^2 - a*b^2*c) (b^2 + b*c - 2*b - 2*c + 1)/(b^2*c + b*c^2 - b*c)] [(-a*b^3 - 2*a*b^2*c + 2*a*b^2 - a*b*c^2 + 2*a*b*c - a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b) (a^2*b^4*c + 2*a^2*b^3*c^2 - 2*a^2*b^3*c - a^2*b^3 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 - 2*a^2*b^2*c + a^2*b^2 - a^2*b*c^2 + 3*a^2*b*c + a^2*b - a^2 - a*b^4 - 4*a*b^3*c + a*b^3 - 4*a*b^2*c^2 + 7*a*b^2*c + 3*a*b^2 - a*b*c^3 + 5*a*b*c^2 - a*b*c - 5*a*b - 2*a*c + 2*a - b^4 - b^3*c + 4*b^3 + b^2*c^2 + 4*b^2*c - 6*b^2 + b*c^3 - 5*b*c + 4*b - c^2 + 2*c - 1)/(a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c + a*b^4*c + 2*a*b^3*c^2 - 2*a*b^3*c + a*b^2*c^3 - 2*a*b^2*c^2 + a*b^2*c) (-a^2*b^3*c - 2*a^2*b^2*c^2 + 2*a^2*b^2*c + a^2*b^2 - a^2*b*c^3 + 2*a^2*b*c^2 + a^2*b*c - a^2*b - a^2*c + a*b^3 + 3*a*b^2*c - a*b^2 + 2*a*b*c^2 - 3*a*b*c - a*b - a*c^2 + a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + a*b*c^3 - 2*a*b*c^2 + a*b*c)] [(-a^2*b^4*c - 2*a^2*b^3*c^2 + 2*a^2*b^3*c + a^2*b^3 - a^2*b^2*c^3 + 2*a^2*b^2*c^2 + a^2*b^2*c - 2*a^2*b^2 + a^2*b*c^2 - 4*a^2*b*c + a^2 + a*b^4 + 3*a*b^3*c - 3*a*b^3 + 3*a*b^2*c^2 - 8*a*b^2*c + a*b^2 + a*b*c^3 - 5*a*b*c^2 + 3*a*b*c + 3*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) (a^3*b^5*c^2 + 2*a^3*b^4*c^3 - 2*a^3*b^4*c^2 - 2*a^3*b^4*c + a^3*b^3*c^4 - 2*a^3*b^3*c^3 - 5*a^3*b^3*c^2 + 3*a^3*b^3*c + a^3*b^3 - 3*a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 4*a^3*b^2*c - a^3*b^2 + a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3 - 2*a^2*b^5*c - 8*a^2*b^4*c^2 + 4*a^2*b^4*c + 2*a^2*b^4 - 9*a^2*b^3*c^3 + 16*a^2*b^3*c^2 + 10*a^2*b^3*c - 3*a^2*b^3 - 3*a^2*b^2*c^4 + 11*a^2*b^2*c^3 + 4*a^2*b^2*c^2 - 21*a^2*b^2*c - 3*a^2*b^2 + 2*a^2*b*c^3 - 12*a^2*b*c^2 + 6*a^2*b*c + 7*a^2*b + 3*a^2*c - 3*a^2 - a*b^5*c + a*b^5 - a*b^4*c^2 + 10*a*b^4*c - a*b^4 + a*b^3*c^3 + 14*a*b^3*c^2 - 18*a*b^3*c - 6*a*b^3 + a*b^2*c^4 + 6*a*b^2*c^3 - 22*a*b^2*c^2 + 4*a*b^2*c + 14*a*b^2 + a*b*c^4 - 7*a*b*c^3 + 6*a*b*c^2 + 11*a*b*c - 11*a*b + 3*a*c^2 - 6*a*c + 3*a + b^5 + 3*b^4*c - 5*b^4 + 3*b^3*c^2 - 12*b^3*c + 10*b^3 + b^2*c^3 - 9*b^2*c^2 + 18*b^2*c - 10*b^2 - 2*b*c^3 + 9*b*c^2 - 12*b*c + 5*b + c^3 - 3*c^2 + 3*c - 1)/(a^3*b^4*c^2 + a^3*b^3*c^3 - a^3*b^3*c^2 + a^2*b^5*c^2 + 2*a^2*b^4*c^3 - 2*a^2*b^4*c^2 + a^2*b^3*c^4 - 2*a^2*b^3*c^3 + a^2*b^3*c^2) (-a^2*b^4*c - 2*a^2*b^3*c^2 + 2*a^2*b^3*c + a^2*b^3 - a^2*b^2*c^3 + 2*a^2*b^2*c^2 + 2*a^2*b^2*c - a^2*b^2 + a^2*b*c^2 - 3*a^2*b*c - a^2*b + a^2 + a*b^4 + 4*a*b^3*c - a*b^3 + 4*a*b^2*c^2 - 6*a*b^2*c - 3*a*b^2 + a*b*c^3 - 4*a*b*c^2 + 5*a*b + 2*a*c - 2*a + b^4 + 2*b^3*c - 4*b^3 + b^2*c^2 - 6*b^2*c + 6*b^2 - 2*b*c^2 + 6*b*c - 4*b + c^2 - 2*c + 1)/(a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c + a*b^4*c + 2*a*b^3*c^2 - 2*a*b^3*c + a*b^2*c^3 - 2*a*b^2*c^2 + a*b^2*c)] >> I:=Ideal(Eltseq(P*B*Transpose(P)-B)); ^ User error: Identifier 'B' has not been declared or assigned >> print GroebnerBasis(I); ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:58:43 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); print F; print T; Output: Magma V2.11-10 Wed Dec 7 2005 17:58:42 on modular [Seed = 2910422931] ------------------------------------- [1 0 0] [0 0 1] [0 -1 (2*a*b*c - a - b - c + 1)/(a*b*c)] [1/(b*c) -1/(b*c) 1/(b*c)] [0 -1/b (-b + 1)/(b*c)] [(-b - c + 1)/(a*b*c) (-a*b*c - a*b + a + b + c - 1)/(a*b^2*c) (-a*b^2*c + 2*a*b*c + a*b - a + b^2 + b*c - 2*b - c + 1)/(a*b^2*c^2)] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:57:05 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); print PrimaryRationalForm(S); Output: Magma V2.11-10 Wed Dec 7 2005 17:57:05 on modular [Seed = 3127808359] ------------------------------------- [1 0 0] [0 0 1] [0 -1 (2*a*b*c - a - b - c + 1)/(a*b*c)] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:54:45 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); print RationalForm(S); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.(7-i)*Matrix(K,4,4,L[i]); end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:54:45 on modular [Seed = 3177813652] ------------------------------------- [1 0 0 0] [0 0 1 0] [0 0 0 1] [0 1 0 0] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:51:33 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.(7-i)*Matrix(K,4,4,L[i]); end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print G; Output: Magma V2.11-10 Wed Dec 7 2005 17:51:33 on modular [Seed = 2236719596] ------------------------------------- [ $.1^2 + $.1*$.3 - $.1*$.5 - $.1*$.6 - $.3*$.5 - $.4*$.5, $.1*$.2 + $.1*$.5 + $.2*$.3 + $.3*$.5, $.1*$.3*$.5^2 + 2*$.1*$.5^3 + 3*$.1*$.5^2*$.6 + 2*$.1*$.5*$.6^2 + $.1*$.6^3 - $.1*$.6 + 2*$.2*$.3*$.5^2 + 3*$.2*$.4*$.5^2 + 2*$.2*$.4*$.5*$.6 + $.2*$.4*$.6^2 + $.2*$.5^3 + 3*$.2*$.5^2*$.6 + 2*$.2*$.5*$.6^2 + $.2*$.5 + $.2*$.6^3 + $.3*$.5^3 + 2*$.4*$.5^3 + 2*$.4*$.5^2*$.6 + $.4*$.5*$.6^2 - $.5^4 - 2*$.5^3*$.6 - 3*$.5^2*$.6^2 + $.5^2 - 2*$.5*$.6^3 + $.5*$.6 - $.6^4 + $.6^2, $.1*$.4 - $.1*$.5 - $.2*$.3 - $.2*$.5 - $.2*$.6 - $.3*$.5 - $.4*$.5 - $.4*$.6, $.1*$.5^4 + 2*$.1*$.5^3*$.6 + 3*$.1*$.5^2*$.6^2 + 2*$.1*$.5*$.6^3 + $.1*$.6^4 - $.1*$.6^2 + $.2*$.3*$.5^3 + 2*$.2*$.4*$.5^3 + 3*$.2*$.4*$.5^2*$.6 + 2*$.2*$.4*$.5*$.6^2 + $.2*$.4*$.6^3 + $.2*$.5^4 + 2*$.2*$.5^3*$.6 + 3*$.2*$.5^2*$.6^2 + 2*$.2*$.5*$.6^3 + $.2*$.5*$.6 + $.2*$.6^4 + $.3*$.5^4 + 2*$.4*$.5^4 + 3*$.4*$.5^3*$.6 + 2*$.4*$.5^2*$.6^2 + $.4*$.5*$.6^3 - $.5^4*$.6 - 2*$.5^3*$.6^2 - 3*$.5^2*$.6^3 + $.5^2*$.6 - 2*$.5*$.6^4 + $.5*$.6^2 - $.6^5 + $.6^3, $.2^2 + $.2*$.4 + $.2*$.5 + $.4*$.5, $.3^2 + $.3*$.4 + $.3*$.5 + $.4^2 + $.4*$.5 + $.4*$.6 + $.5^2 + $.5*$.6 + $.6^2 - 1, $.3*$.4*$.5 + $.4^2*$.5 + $.4^2*$.6 + $.4*$.5^2 + $.4*$.5*$.6 + $.4*$.6^2 + $.5^2*$.6 + $.5*$.6^2 + $.6^3 - $.6, $.3*$.6 - $.4*$.5, $.4^2*$.5^2 + $.4^2*$.5*$.6 + $.4^2*$.6^2 + $.4*$.5^2*$.6 + $.4*$.5*$.6^2 + $.4*$.6^3 + $.5^2*$.6^2 + $.5*$.6^3 + $.6^4 - $.6^2 ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:50:55 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.i*Matrix(K,4,4,L[i]); end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print G; Output: Magma V2.11-10 Wed Dec 7 2005 17:50:55 on modular [Seed = 2287774939] ------------------------------------- [ $.1^2 + $.1*$.2 - $.1*$.5 + $.2^2 - $.2*$.5 - $.2*$.6 + $.3^2 + $.3*$.4 + $.3*$.6 + $.4^2 - $.4*$.5 - 1, $.1*$.2*$.5 + $.2^2*$.5 + $.2^2*$.6 - $.2*$.5^2 - $.2*$.5*$.6 - $.2*$.6^2 + $.3^2*$.6 + $.3*$.4*$.6 - $.3*$.5^2 + $.3*$.5*$.6 + $.3*$.6^2 + $.4^2*$.6 - $.4*$.5^2 - $.4*$.5*$.6 + $.4*$.6^2 - $.5^3 + $.6^3 - $.6, $.1*$.3 + $.1*$.5 - $.3*$.5 - $.3*$.6 - $.5^2 - $.5*$.6, $.1*$.4 + $.2*$.5 + $.3*$.5 + $.5^2, $.1*$.6 - $.2*$.5 - $.3*$.5 - $.4*$.5 - $.4*$.6 - $.5^2 - $.5*$.6 - $.6^2, $.2^2*$.5^2 + $.2^2*$.5*$.6 + $.2^2*$.6^2 - $.2*$.5^2*$.6 - $.2*$.5*$.6^2 - $.2*$.6^3 + $.3*$.4*$.6^2 - 2*$.3*$.5^3 - 3*$.3*$.5^2*$.6 - 3*$.3*$.5*$.6^2 + $.3*$.5 - $.4^2*$.5^2 - 2*$.4^2*$.5*$.6 - $.4^2*$.6^2 - $.4*$.5^3 - 3*$.4*$.5^2*$.6 - 4*$.4*$.5*$.6^2 + $.4*$.5 - 2*$.4*$.6^3 + $.4*$.6 - 2*$.5^4 - 4*$.5^3*$.6 - 4*$.5^2*$.6^2 + $.5^2 - 2*$.5*$.6^3 + $.5*$.6, $.2*$.3 + $.2*$.5 + $.3*$.5 + $.5^2, $.2*$.4 + $.2*$.6 + $.4*$.5 + $.5*$.6, $.3^3 - 3*$.3*$.4*$.6 + 3*$.3*$.5*$.6 - $.3 - $.4^3 + 3*$.4^2*$.5 - 3*$.4*$.5^2 + $.4 + $.5^3 - $.5 - $.6^3 + $.6, $.3^2*$.4 + $.3^2*$.6 + $.3*$.4^2 + 2*$.3*$.4*$.6 + $.3*$.6^2 + $.4^3 - $.4^2*$.5 + $.4^2*$.6 + $.4*$.5^2 + $.4*$.6^2 - $.4 + $.5^2*$.6 + $.5*$.6^2 + $.6^3 - $.6, $.3^2*$.5 + $.3*$.4*$.6 + $.3*$.5^2 - $.3*$.5*$.6 - $.4^2*$.5 - $.4^2*$.6 + $.4*$.5^2 - $.4*$.6^2, $.3^2*$.6^2 + $.3*$.5^3 + 2*$.3*$.5^2*$.6 + 4*$.3*$.5*$.6^2 - $.3*$.5 + $.3*$.6^3 + $.4^2*$.5^2 + 2*$.4^2*$.5*$.6 + 2*$.4^2*$.6^2 + $.4*$.5^2*$.6 + 3*$.4*$.5*$.6^2 - $.4*$.5 + 3*$.4*$.6^3 - $.4*$.6 + $.5^4 + 2*$.5^3*$.6 + 3*$.5^2*$.6^2 - $.5^2 + 2*$.5*$.6^3 - $.5*$.6 + $.6^4 - $.6^2, $.3*$.4^2*$.6 + $.3*$.4*$.6^2 - $.4^3*$.5 + $.4^2*$.5^2 + $.4^2*$.6^2 - $.4*$.5^3 + $.4*$.5 + $.4*$.6^3 - $.5^3*$.6 - $.5^2*$.6^2 - $.5*$.6^3 + $.5*$.6, $.3*$.4*$.5 + $.3*$.5*$.6 + $.4^2*$.5 + $.4^2*$.6 + $.4*$.5*$.6 + $.4*$.6^2, $.3*$.4*$.6^3 - $.3*$.5^4 - 2*$.3*$.5^3*$.6 - 3*$.3*$.5^2*$.6^2 + $.3*$.5^2 - 2*$.3*$.5*$.6^3 - $.4^2*$.5^3 - 2*$.4^2*$.5^2*$.6 - 3*$.4^2*$.5*$.6^2 - $.4^2*$.6^3 - $.4*$.5^3*$.6 - 2*$.4*$.5^2*$.6^2 + $.4*$.5^2 - 3*$.4*$.5*$.6^3 + $.4*$.5*$.6 - $.4*$.6^4 - $.5^5 - 2*$.5^4*$.6 - 3*$.5^3*$.6^2 + $.5^3 - 2*$.5^2*$.6^3 + $.5^2*$.6 - $.5*$.6^4 + $.5*$.6^2, $.3*$.5^5 + 2*$.3*$.5^4*$.6 + 3*$.3*$.5^3*$.6^2 - $.3*$.5^3 + 2*$.3*$.5^2*$.6^3 + $.3*$.5*$.6^4 + $.4^2*$.5^4 + 2*$.4^2*$.5^3*$.6 + 3*$.4^2*$.5^2*$.6^2 + 2*$.4^2*$.5*$.6^3 + $.4^2*$.6^4 + $.4*$.5^4*$.6 + 2*$.4*$.5^3*$.6^2 - $.4*$.5^3 + 3*$.4*$.5^2*$.6^3 - $.4*$.5^2*$.6 + 2*$.4*$.5*$.6^4 + $.4*$.6^5 + $.5^6 + 2*$.5^5*$.6 + 3*$.5^4*$.6^2 - $.5^4 + 2*$.5^3*$.6^3 - $.5^3*$.6 + $.5^2*$.6^4 - $.5^2*$.6^2, $.4^3*$.5^2 + $.4^3*$.5*$.6 + $.4^3*$.6^2 - $.4^2*$.5^3 + $.4^2*$.6^3 + $.4*$.5^4 - $.4*$.5^2 - $.4*$.5*$.6^3 + $.5^4*$.6 + $.5^3*$.6^2 + $.5^2*$.6^3 - $.5^2*$.6 ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:49:34 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.i*Matrix(K,4,4,L[1]); end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print Factorization(G[1]); Output: Magma V2.11-10 Wed Dec 7 2005 17:49:33 on modular [Seed = 2470950514] ------------------------------------- [ <$.1 + $.2 + $.3 + $.4 + $.5 + $.6 - 1, 1>, <$.1 + $.2 + $.3 + $.4 + $.5 + $.6 + 1, 1> ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:47:31 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.i*Matrix(K,4,4,L[1]); end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:47:30 on modular [Seed = 2520953608] ------------------------------------- [ $.1^2 + 2*$.1*$.2 + 2*$.1*$.3 + 2*$.1*$.4 + 2*$.1*$.5 + 2*$.1*$.6 + $.2^2 + 2*$.2*$.3 + 2*$.2*$.4 + 2*$.2*$.5 + 2*$.2*$.6 + $.3^2 + 2*$.3*$.4 + 2*$.3*$.5 + 2*$.3*$.6 + $.4^2 + 2*$.4*$.5 + 2*$.4*$.6 + $.5^2 + 2*$.5*$.6 + $.6^2 - 1 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:47:04 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.i*Matrix(K,4,4,L[1]); end for; Output: Magma V2.11-10 Wed Dec 7 2005 17:47:04 on modular [Seed = 2555166188] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:46:24 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); print Matrix(K,4,4,L[1]); Output: Magma V2.11-10 Wed Dec 7 2005 17:46:24 on modular [Seed = 1672955455] ------------------------------------- [1 0 0 0] [0 0 1 0] [0 1 0 0] [0 0 0 1] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:45:40 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); print #L; Output: Magma V2.11-10 Wed Dec 7 2005 17:45:40 on modular [Seed = 1722958702] ------------------------------------- 6 Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:43:53 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); print KernelMatrix(M); Output: Magma V2.11-10 Wed Dec 7 2005 17:43:53 on modular [Seed = 1774014428] ------------------------------------- [ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1] [ 0 1 0 0 0 0 0 1 -1 1 0 0 0 0 -1 1] [ 0 0 1 0 0 0 0 1 -1 0 1 0 0 -1 0 1] [ 0 0 0 1 0 0 -1 1 0 -1 0 1 1 -1 -1 1] [ 0 0 0 0 1 0 -1 1 -1 0 1 -1 0 0 0 0] [ 0 0 0 0 0 1 -1 0 0 -1 1 0 0 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:42:44 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); print KroneckerProduct(Transpose(S),I4); Output: Magma V2.11-10 Wed Dec 7 2005 17:42:44 on modular [Seed = 1824016916] ------------------------------------- [ 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0] [ 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0] [ 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0] [ 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1] [-1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0] [ 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0] [ 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0] [ 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0] [-1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0] [ 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0] [ 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0] [ 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0] [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:41:56 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); print KroneckerProduct(I4,S); Output: Magma V2.11-10 Wed Dec 7 2005 17:41:56 on modular [Seed = 1990348894] ------------------------------------- [ 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0] [ 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 1 -1 -1 1 0 0 0 0 0 0 0 0] [ 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 1 -1 -1 1 0 0 0 0] [ 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:36:45 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); B1:=Matrix(K,4,4,[[0,-1,1,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B2:=Matrix(K,4,4,[[0,0,0,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B3:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,1],[0,0,0,0],[0,0,0,0]]); B4:=Matrix(K,4,4,[[0,0,0,0],[0,1,-1,0],[0,-1,1,0],[0,0,0,0]]); B5:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,0],[0,0,0,1],[0,0,0,0]]); B6:=Matrix(K,4,4,[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]); B:=[B1,B2,B3,B4,B5,B6]; R:=PolynomialRing(K,6); P:=0; for i in [1..6] do P:= P + R.i*B[i]; end for; print P; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:36:45 on modular [Seed = 1367699888] ------------------------------------- [ $.6 -$.1 + $.3 + $.5 $.1 $.2] [ 0 $.4 -$.4 + $.6 $.3] [ 0 -$.4 + $.6 $.4 $.5] [ 0 0 0 $.6] [ $.1, $.2, $.3, $.4^2 - $.4*$.6, $.5, $.6^2 - 1 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:30:58 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); B1:=Matrix(K,4,4,[[0,-1,1,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B2:=Matrix(K,4,4,[[0,0,0,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B3:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,1],[0,0,0,0],[0,0,0,0]]); B4:=Matrix(K,4,4,[[0,0,0,0],[0,1,-1,0],[0,-1,1,0],[0,0,0,0]]); B5:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,0],[0,0,0,1],[0,0,0,0]]); B6:=Matrix(K,4,4,[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]); B:=[B1,B2,B3,B4,B5,B6]; R:=PolynomialRing(K,6); P:=0; for i in [1..6] do P:= P + R.i*B[i]; end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:30:58 on modular [Seed = 148646326] ------------------------------------- [ $.1, $.2, $.3, $.4^2 - $.4*$.6, $.5, $.6^2 - 1 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:30:25 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); B1:=Matrix(K,4,4,[[0,-1,1,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B2:=Matrix(K,4,4,[[0,0,0,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B3:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,1],[0,0,0,0],[0,0,0,0]]); B4:=Matrix(K,4,4,[[0,0,0,0],[0,1,-1,0],[0,-1,1,0],[0,0,0,0]]); B5:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,0],[0,0,0,1],[0,0,0,0]]); B6:=Matrix(K,4,4,[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]); B:=[B1,B2,B3,B4,B5,B6]; R:=PolynomialRing(K,6); P:=0; for i in [1..6] do P:= P + R.i*B[i]; end for; Output: Magma V2.11-10 Wed Dec 7 2005 17:30:25 on modular [Seed = 331821710] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:29:25 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); B1:=Matrix(K,4,4,[[0,-1,1,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B2:=Matrix(K,4,4,[[0,0,0,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B3:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,1],[0,0,0,0],[0,0,0,0]]); B4:=Matrix(K,4,4,[[0,0,0,0],[0,1,-1,0],[0,-1,1,0],[0,0,0,0]]); B5:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,0],[0,0,0,1],[0,0,0,0]]); B6:=Matrix(K,4,4,[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]); B:=[B1,B2,B3,B4,B5,B6]; for i in [1..6] do print B[i]*A-A*B[i]; end for; Output: Magma V2.11-10 Wed Dec 7 2005 17:29:25 on modular [Seed = 3963145652] ------------------------------------- [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] Total time: 0.180 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:23:13 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); R:=PolynomialRing(K,16); P:=Matrix(R,4,4,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9,R.10,R.11,R.12,R.13,R.14,R.15,R.16]); I:=Ideal(Eltseq(P*A-A*P)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:23:13 on modular [Seed = 2910424673] ------------------------------------- [ $.1 - $.16, $.2 + $.3 - $.8 - $.12, $.5, $.6 - $.11, $.7 + $.11 - $.16, $.9, $.10 + $.11 - $.16, $.13, $.14, $.15 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:00:38 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,0,1,1],[0,1,1,1],[0,0,1,1],[0,0,0,1]]); R:=PolynomialRing(K,16); P:=Matrix(R,4,4,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9,R.10,R.11,R.12,R.13,R.14,R.15,R.16]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:00:29 on modular [Seed = 1502429497] ------------------------------------- [ $.1 - $.6 + $.8 + $.14 + $.15, $.2 + $.6 - $.13 - 2*$.14 - 2*$.15 - $.16, $.3 - $.8 - $.15, $.4 + $.8 + $.13 + $.14 + 2*$.15, $.5 + $.6 - $.8 - 2*$.13 - 2*$.14 - 3*$.15 - $.16, $.6^2 - 3*$.6*$.14 - 2*$.6*$.15 - $.6*$.16 - $.8*$.13 - $.8*$.16 + 3*$.14^2 + 4*$.14*$.15 + 2*$.14*$.16 + 2*$.15^2 + $.15*$.16 + $.16^2 - 1, $.6*$.8 + $.6*$.14 + $.6*$.15 - $.8*$.14 - $.8*$.15 - $.14^2 - 2*$.14*$.15 - $.15^2, $.6*$.13 - $.6*$.14 - $.8*$.13 - $.8*$.16 + 2*$.14^2 + 3*$.14*$.15 + $.14*$.16 + 2*$.15^2 + $.15*$.16 + $.16^2 - 1, $.6*$.14^2 + 2*$.6*$.14*$.16 + $.6*$.16^2 - $.6 - $.8*$.13*$.15 + $.8*$.13*$.16 - $.8*$.14^2 - $.8*$.14*$.15 - $.8*$.14*$.16 - $.8*$.15^2 - $.8*$.15*$.16 + $.8 - $.13*$.15^2 - $.13*$.16^2 + $.13 + 2*$.14^2*$.15 - 2*$.14^2*$.16 + 3*$.14*$.15^2 - $.14*$.16^2 + $.15^3 - $.16^3 + $.16, $.6*$.14*$.15 - $.6*$.14*$.16 + 1/2*$.6*$.15^2 + 1/2*$.6*$.15*$.16 - 1/2*$.6*$.16^2 + 1/2*$.6 + 1/2*$.8*$.13*$.15 - 1/2*$.8*$.13*$.16 + 1/2*$.8*$.14^2 + $.8*$.14*$.16 + 1/2*$.8*$.15*$.16 - 1/2*$.8 + 1/2*$.13*$.15^2 + 1/2*$.13*$.16^2 - 1/2*$.13 - $.14^3 - 7/2*$.14^2*$.15 + 1/2*$.14^2*$.16 - 4*$.14*$.15^2 - $.14*$.15*$.16 + 1/2*$.14 - 3/2*$.15^3 - 1/2*$.15^2*$.16 - 1/2*$.15*$.16^2 + 1/2*$.15 + 1/2*$.16^3 - 1/2*$.16, $.6*$.14*$.16^2 + $.6*$.14 - 1/2*$.6*$.15^3 + 1/2*$.6*$.15^2*$.16 - 1/2*$.6*$.15*$.16^2 + 3/2*$.6*$.15 + $.6*$.16^3 - $.6*$.16 + 1/2*$.8*$.13*$.15^2 + $.8*$.13*$.16^2 - 1/2*$.8*$.14^3 - 3/2*$.8*$.14^2*$.15 - 1/2*$.8*$.14^2*$.16 - 2*$.8*$.14*$.15^2 - 1/2*$.8*$.14*$.15*$.16 - 3/2*$.8*$.14*$.16^2 + 1/2*$.8*$.14 - 1/2*$.8*$.15^3 - 1/2*$.8*$.15^2*$.16 + $.8*$.16 + 1/2*$.13*$.15^3 - 1/2*$.13*$.15 - $.13*$.16^3 + $.13*$.16 - $.14^3*$.15 + 3/2*$.14^3*$.16 - 5/2*$.14^2*$.15^2 + 5/2*$.14^2*$.15*$.16 - 1/2*$.14^2*$.16^2 - $.14^2 - 2*$.14*$.15^3 + 2*$.14*$.15^2*$.16 - 1/2*$.14*$.15*$.16^2 - 3/2*$.14*$.15 - 1/2*$.14*$.16^3 - 1/2*$.14*$.16 - 1/2*$.15^4 + $.15^3*$.16 + 1/2*$.15^2*$.16^2 - 1/2*$.15^2 - 1/2*$.15*$.16^3 - $.16^4 + $.16^2, $.6*$.15^4 - 2*$.6*$.15^3*$.16 + 3*$.6*$.15^2*$.16^2 - 2*$.6*$.15^2 - 2*$.6*$.15*$.16^3 + 6*$.6*$.15*$.16 + $.6*$.16^4 - 2*$.6*$.16^2 + $.6 - $.8*$.13*$.15^3 + $.8*$.13*$.15 + $.8*$.13*$.16^3 - $.8*$.13*$.16 + 2*$.8*$.14^3*$.15 - 2*$.8*$.14^3*$.16 + 5*$.8*$.14^2*$.15^2 - 3*$.8*$.14^2*$.15*$.16 - $.8*$.14^2*$.16^2 + $.8*$.14^2 + 6*$.8*$.14*$.15^3 - 4*$.8*$.14*$.15^2*$.16 + 2*$.8*$.14*$.15*$.16^2 - 2*$.8*$.14*$.15 - 2*$.8*$.14*$.16^3 + 4*$.8*$.14*$.16 + 2*$.8*$.15^4 - $.8*$.15^3*$.16 - $.8*$.15^2*$.16^2 - $.8*$.15^2 + $.8*$.15*$.16^3 + $.8*$.16^2 - $.8 - $.13*$.15^4 + 2*$.13*$.15^2 - $.13*$.16^4 + 2*$.13*$.16^2 - $.13 + 3*$.14^3*$.15^2 - 4*$.14^3*$.15*$.16 + 3*$.14^3*$.16^2 - 5*$.14^3 + 7*$.14^2*$.15^3 - 7*$.14^2*$.15^2*$.16 + 4*$.14^2*$.15*$.16^2 - 11*$.14^2*$.15 + 2*$.14^2*$.16^3 - 4*$.14^2*$.16 + 6*$.14*$.15^4 - 5*$.14*$.15^3*$.16 + 3*$.14*$.15^2*$.16^2 - 12*$.14*$.15^2 + $.14*$.15*$.16^3 - 8*$.14*$.15*$.16 + $.14*$.16^4 - 5*$.14*$.16^2 + 4*$.14 + 2*$.15^5 - 2*$.15^4*$.16 - 6*$.15^3 + $.15^2*$.16^3 - 3*$.15^2*$.16 - 4*$.15*$.16^2 + 4*$.15 - $.16^5 + 2*$.16^3 - $.16, $.7 + $.8 + $.13 + $.14 + $.15, $.8^2 + $.8*$.13 + $.8*$.14 + 2*$.8*$.15 - $.15*$.16, $.9 - $.13 - $.14 - $.15, $.10 - $.13 - $.14 - $.15, $.11 - $.15 - $.16, $.12 + $.13 + $.14 + $.15, $.13^2 + $.13*$.15 + $.13*$.16 + $.14^2 + $.14*$.15 + $.14*$.16 + $.15^2 + $.15*$.16 + $.16^2 - 1, $.13*$.14 + $.13*$.15 + $.14*$.15 + $.15^2 + $.15*$.16, $.13*$.15*$.16 - $.14^3 - 2*$.14^2*$.15 - $.14^2*$.16 - 2*$.14*$.15^2 - $.14*$.15*$.16 - $.14*$.16^2 + $.14 - $.15^3 + $.15, $.14^4 + 3*$.14^3*$.15 + $.14^3*$.16 + 4*$.14^2*$.15^2 + 2*$.14^2*$.15*$.16 + $.14^2*$.16^2 - $.14^2 + 3*$.14*$.15^3 + 2*$.14*$.15^2*$.16 + $.14*$.15*$.16^2 - 2*$.14*$.15 + $.15^4 + $.15^3*$.16 + $.15^2*$.16^2 - $.15^2 ] Total time: 9.570 seconds, Total memory usage: 22.40MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 16:58:26 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); R:=PolynomialRing(K,16); P:=Matrix(R,4,4,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9,R.10,R.11,R.12,R.13,R.14,R.15,R.16]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 16:58:22 on modular [Seed = 586562497] ------------------------------------- [ $.1 - $.13 - $.14 - $.15 - $.16, $.2 + $.13 + $.15, $.3 + $.13 + $.14, $.4 - $.13, $.5 + $.12 - $.13, $.6 - $.11 - $.12 - $.14, $.7 + $.11 + $.13 - $.15 - $.16, $.8 + $.12 + $.14 + $.15, $.9 - $.12 - $.13 - $.14 - $.15, $.10 + $.11 + $.12 + $.13 - $.16, $.11^2 + $.11*$.13 - $.11*$.15 - $.11*$.16 - $.13*$.15 - $.14*$.15, $.11*$.12 + $.11*$.15 + $.12*$.13 + $.13*$.15, $.11*$.13*$.15^2 + 2*$.11*$.15^3 + 3*$.11*$.15^2*$.16 + 2*$.11*$.15*$.16^2 + $.11*$.16^3 - $.11*$.16 + 2*$.12*$.13*$.15^2 + 3*$.12*$.14*$.15^2 + 2*$.12*$.14*$.15*$.16 + $.12*$.14*$.16^2 + $.12*$.15^3 + 3*$.12*$.15^2*$.16 + 2*$.12*$.15*$.16^2 + $.12*$.15 + $.12*$.16^3 + $.13*$.15^3 + 2*$.14*$.15^3 + 2*$.14*$.15^2*$.16 + $.14*$.15*$.16^2 - $.15^4 - 2*$.15^3*$.16 - 3*$.15^2*$.16^2 + $.15^2 - 2*$.15*$.16^3 + $.15*$.16 - $.16^4 + $.16^2, $.11*$.14 - $.11*$.15 - $.12*$.13 - $.12*$.15 - $.12*$.16 - $.13*$.15 - $.14*$.15 - $.14*$.16, $.11*$.15^4 + 2*$.11*$.15^3*$.16 + 3*$.11*$.15^2*$.16^2 + 2*$.11*$.15*$.16^3 + $.11*$.16^4 - $.11*$.16^2 + $.12*$.13*$.15^3 + 2*$.12*$.14*$.15^3 + 3*$.12*$.14*$.15^2*$.16 + 2*$.12*$.14*$.15*$.16^2 + $.12*$.14*$.16^3 + $.12*$.15^4 + 2*$.12*$.15^3*$.16 + 3*$.12*$.15^2*$.16^2 + 2*$.12*$.15*$.16^3 + $.12*$.15*$.16 + $.12*$.16^4 + $.13*$.15^4 + 2*$.14*$.15^4 + 3*$.14*$.15^3*$.16 + 2*$.14*$.15^2*$.16^2 + $.14*$.15*$.16^3 - $.15^4*$.16 - 2*$.15^3*$.16^2 - 3*$.15^2*$.16^3 + $.15^2*$.16 - 2*$.15*$.16^4 + $.15*$.16^2 - $.16^5 + $.16^3, $.12^2 + $.12*$.14 + $.12*$.15 + $.14*$.15, $.13^2 + $.13*$.14 + $.13*$.15 + $.14^2 + $.14*$.15 + $.14*$.16 + $.15^2 + $.15*$.16 + $.16^2 - 1, $.13*$.14*$.15 + $.14^2*$.15 + $.14^2*$.16 + $.14*$.15^2 + $.14*$.15*$.16 + $.14*$.16^2 + $.15^2*$.16 + $.15*$.16^2 + $.16^3 - $.16, $.13*$.16 - $.14*$.15, $.14^2*$.15^2 + $.14^2*$.15*$.16 + $.14^2*$.16^2 + $.14*$.15^2*$.16 + $.14*$.15*$.16^2 + $.14*$.16^3 + $.15^2*$.16^2 + $.15*$.16^3 + $.16^4 - $.16^2 ] Total time: 3.500 seconds, Total memory usage: 14.17MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 16:58:14 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); R:=PolynomialRing(K,16); P:=Matrix(R,4,4,R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9,R.10,R.11,R.12,R.13,R.14,R.15,R.16]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 16:58:12 on modular [Seed = 670252750] ------------------------------------- >> 0,R.11,R.12,R.13,R.14,R.15,R.16]); ^ User error: bad syntax >> I:=Ideal(Eltseq(P*A*Transpose(P)-A)); ^ User error: Identifier 'P' has not been declared or assigned >> print GroebnerBasis(I); ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '68.238.' ************** MAGMA ***************** Host 68.238.133.139 (68.238.133.139) Time: Wed Dec 7 15:10:38 2005 Input: [[1,0],[0,1]]*2 Output: Magma V2.11-10 Wed Dec 7 2005 15:10:38 on modular [Seed = 1218202344] ------------------------------------- >> [[1,0],[0,1]]*2 ^ Runtime error in '*': Bad argument types Argument types given: SeqEnum[SeqEnum[RngIntElt]], RngIntElt Total time: 0.190 seconds, Total memory usage: 3.24MB '68.238.' ************** MAGMA ***************** Host 68.238.133.139 (68.238.133.139) Time: Wed Dec 7 15:10:14 2005 Input: ((1,0),(0,1)) Output: Magma V2.11-10 Wed Dec 7 2005 15:10:13 on modular [Seed = 1335578100] ------------------------------------- >> ((1,0),(0,1)) ^ Runtime error in elt< ... >: No permutation group context in which to create cycle Total time: 0.190 seconds, Total memory usage: 3.24MB '68.238.' ************** MAGMA ***************** Host 68.238.133.139 (68.238.133.139) Time: Wed Dec 7 15:09:13 2005 Input: {{1,0},{0,1}}.{{2,0},{0,2}} Output: Magma V2.11-10 Wed Dec 7 2005 15:09:12 on modular [Seed = 47579252] ------------------------------------- >> {{1,0},{0,1}}.{{2,0},{0,2}}; ^ Runtime error in '.': Bad argument types Argument types given: SetEnum[SetEnum], SetEnum[SetEnum] Total time: 0.180 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:54:21 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; allconjs := procedure(x) print "x = " * Sprint(x) * ", x^q = " * Sprint(x^q) * ", x^{q^2} = " * Sprint(x^(q^2)); end procedure; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; allconjs(1); allconjs(z^2); allconjs(z^4); print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:54:20 on modular [Seed = 1301885886] ------------------------------------- xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 x = 1, x^q = 1, x^{q^2} = 1 x = z^2, x^q = 1461501624495162299985421915440573293973338433026*z^2, x^{q^2} = 1627965160026674480212199743920457792*z^2 x = z^4, x^q = 1627965160026674480212199743920457792*z^4, x^{q^2} = 1461501624495162299985421915440573293973338433026*z^4 trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0, norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:54:01 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; allconjs := procedure(x) print "x = " * Sprint(x); print "x^q = " * Sprint(x^q); print "x^{q^2} = " * Sprint(x^(q^2)); end procedure; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; allconjs(1); allconjs(z^2); allconjs(z^4); print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:54:00 on modular [Seed = 249196745] ------------------------------------- xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 x = 1 x^q = 1 x^{q^2} = 1 x = z^2 x^q = 1461501624495162299985421915440573293973338433026*z^2 x^{q^2} = 1627965160026674480212199743920457792*z^2 x = z^4 x^q = 1627965160026674480212199743920457792*z^4 x^{q^2} = 1461501624495162299985421915440573293973338433026*z^4 trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0, norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:53:44 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; allconjs := procedure(x) print "x = " * Sprint(x); print "x^q = " * Sprint(x^q); print "x^{q^2} = " * Sprint(x^(q^2)); end function; trace := procedure(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; allconjs(1); allconjs(z^2); allconjs(z^4); print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:53:42 on modular [Seed = 3239427871] ------------------------------------- >> end function; ^ User error: bad syntax >> end function; ^ User error: bad syntax xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 >> allconjs(1); ^ User error: Identifier 'allconjs' has not been declared or assigned >> allconjs(z^2); ^ User error: Identifier 'allconjs' has not been declared or assigned >> allconjs(z^4); ^ User error: Identifier 'allconjs' has not been declared or assigned >> print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); ^ User error: Identifier 'trace' has not been declared or assigned >> print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm ^ User error: Identifier 'trace' has not been declared or assigned >> print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm ^ User error: Identifier 'trace' has not been declared or assigned Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:53:35 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; allconjs := procedure(x) print "x = " * Sprint(x); print "x^q = " * Sprint(x^q); print "x^{q^2} = " * Sprint(x^(q^2); end function; trace := procedure(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; allconjs(1); allconjs(z^2); allconjs(z^4); print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:53:35 on modular [Seed = 3043085074] ------------------------------------- >> print "x^{q^2} = " * Sprint(x^(q^2); ^ User error: bad syntax >> end function; ^ User error: bad syntax >> end function; ^ User error: bad syntax xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 >> allconjs(1); ^ User error: Identifier 'allconjs' has not been declared or assigned >> allconjs(z^2); ^ User error: Identifier 'allconjs' has not been declared or assigned >> allconjs(z^4); ^ User error: Identifier 'allconjs' has not been declared or assigned >> print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); ^ User error: Identifier 'trace' has not been declared or assigned >> print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm ^ User error: Identifier 'trace' has not been declared or assigned >> print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm ^ User error: Identifier 'trace' has not been declared or assigned Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:38:51 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:38:50 on modular [Seed = 1485578502] ------------------------------------- xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0, norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:38:13 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:38:12 on modular [Seed = 1401367139] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0, norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:34:37 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) "=========="; "*** x =", x; "*** x^q =", x^q; "*** x^(q^2) =", x^(q^2); "*** x^(-p) =", x^(-p); "*** x^(p^2) =", x^(p^2); "=========="; return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 - p + p^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:34:37 on modular [Seed = 738152818] ------------------------------------- ========== *** x = 1 *** x^q = 1 *** x^(q^2) = 1 *** x^(-p) = 1 *** x^(p^2) = 1 ========== trace(1) = 3, norm(1) = 1 ========== *** x = z^2 *** x^q = 1461501624495162299985421915440573293973338433026*z^2 *** x^(q^2) = 1627965160026674480212199743920457792*z^2 *** x^(-p) = (1461501624483766543865235194079087895765895228475*i + 1461501624483766543865235194079087895765895228475)*z^4 *** x^(p^2) = 1461501624495162299985421915440573293973338433026*z^2 ========== trace(z^2) = 0, norm(z^2) = 1461501624495162299985421915440573293973338433027*i\ *z^2 ========== *** x = z^4 *** x^q = 1627965160026674480212199743920457792*z^4 *** x^(q^2) = 1461501624495162299985421915440573293973338433026*z^4 *** x^(-p) = (1461501624483766543865235194079087895765895228483*i + 13023721280213395841697597951363662336)*z^2 *** x^(p^2) = 1627965160026674480212199743920457792*z^4 ========== trace(z^4) = 0, norm(z^4) = 1627965160026674480212199743920457793*z^4 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:34:15 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) "=========="; "*** x =", x; "*** x^q =", x^q; "*** x^(q^2) =", x^(q^2); "*** x^(-p) =", x^(-p); "*** x^(p^2) =", x^(p^2); return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 - p + p^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:34:14 on modular [Seed = 788157452] ------------------------------------- ========== *** x = 1 *** x^q = 1 *** x^(q^2) = 1 *** x^(-p) = 1 *** x^(p^2) = 1 trace(1) = 3, norm(1) = 1 ========== *** x = z^2 *** x^q = 1461501624495162299985421915440573293973338433026*z^2 *** x^(q^2) = 1627965160026674480212199743920457792*z^2 *** x^(-p) = (1461501624483766543865235194079087895765895228475*i + 1461501624483766543865235194079087895765895228475)*z^4 *** x^(p^2) = 1461501624495162299985421915440573293973338433026*z^2 trace(z^2) = 0, norm(z^2) = 1461501624495162299985421915440573293973338433027*i\ *z^2 ========== *** x = z^4 *** x^q = 1627965160026674480212199743920457792*z^4 *** x^(q^2) = 1461501624495162299985421915440573293973338433026*z^4 *** x^(-p) = (1461501624483766543865235194079087895765895228483*i + 13023721280213395841697597951363662336)*z^2 *** x^(p^2) = 1627965160026674480212199743920457792*z^4 trace(z^4) = 0, norm(z^4) = 1627965160026674480212199743920457793*z^4 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:32:25 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 - p + p^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:32:25 on modular [Seed = 552885245] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1461501624495162299985421915440573293973338433027*i\ *z^2 trace(z^4) = 0, norm(z^4) = 1627965160026674480212199743920457793*z^4 Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:32:06 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^(-p) + x^(p^2); end function; norm := function(x) return x^(1 - p + p^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:32:06 on modular [Seed = 988692888] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = (1461501624483766543865235194079087895765895228475*i + 1461501624483766543865235194079087895765895228475)*z^4 + 1461501624495162299985421915440573293973338433027*z^2, norm(z^2) = 1461501624495162299985421915440573293973338433027*i*z^2 trace(z^4) = 1627965160026674480212199743920457793*z^4 + (1461501624483766543865235194079087895765895228483*i + 13023721280213395841697597951363662336)*z^2, norm(z^4) = 1627965160026674480212199743920457793*z^4 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:31:30 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 - p + p^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:31:30 on modular [Seed = 1038698171] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1461501624495162299985421915440573293973338433027*i\ *z^2 trace(z^4) = 0, norm(z^4) = 1627965160026674480212199743920457793*z^4 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:30:52 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 - q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:30:51 on modular [Seed = 820267855] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1461501624495162299985421915440573293973338433026*z\ ^2 trace(z^4) = 0, norm(z^4) = 1627965160026674480212199743920457792*z^4 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:26:29 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; norm := function(x) return x * x^(p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:26:29 on modular [Seed = 182353531] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0, norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:26:17 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; norm := function(x) return x * x^(p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) =" * Sprint(trace(1)) * ", norm(1) =" * Sprint(norm(1)); print "trace(z^2) =" * Sprint(trace(z^2)) * ", norm(z^2) =" * Sprint(norm(z^2)); print "trace(z^4) =" * Sprint(trace(z^4)) * ", norm(z^4) =" * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:26:17 on modular [Seed = 232356160] ------------------------------------- trace(1) =3, norm(1) =1 trace(z^2) =0, norm(z^2) =1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) =0, norm(z^4) =239777610269004652875425159283878870062987786775*i Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:25:28 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; norm := function(x) return x * x^(p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1), ", norm(1) =", norm(1); "trace(z^2) =", trace(z^2), ", norm(z^2) =", norm(z^2); "trace(z^4) =", trace(z^4), ", norm(z^4) =", norm(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 14:25:27 on modular [Seed = 131298107] ------------------------------------- trace(1) = 3 , norm(1) = 1 trace(z^2) = 0 , norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0 , norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:16:46 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(-p^2) + x^(p^4); end function; norm := function(x) return x * x^(-p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1), ", norm(1) =", norm(1); "trace(z^2) =", trace(z^2), ", norm(z^2) =", norm(z^2); "trace(z^4) =", trace(z^4), ", norm(z^4) =", norm(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 14:16:46 on modular [Seed = 331834649] ------------------------------------- trace(1) = 3 , norm(1) = 1 trace(z^2) = (13023721280213395841697597951363662336*i + 1461501624483766543865235194079087895765895228483)*z^4 + 1627965160026674480212199743920457793*z^2 , norm(z^2) = 1461501624495162299985421915440573293973338433026*z^2 trace(z^4) = 1461501624495162299985421915440573293973338433027*z^4 + (1461501624483766543865235194079087895765895228475*i + 13023721280213395841697597951363662344)*z^2 , norm(z^4) = 1627965160026674480212199743920457792*z^4 Total time: 0.270 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:16:22 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; norm := function(x) return x * x^(-p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1), ", norm(1) =", norm(1); "trace(z^2) =", trace(z^2), ", norm(z^2) =", norm(z^2); "trace(z^4) =", trace(z^4), ", norm(z^4) =", norm(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 14:16:22 on modular [Seed = 381837316] ------------------------------------- trace(1) = 3 , norm(1) = 1 trace(z^2) = 0 , norm(z^2) = 1461501624495162299985421915440573293973338433026*z^2 trace(z^4) = 0 , norm(z^4) = 1627965160026674480212199743920457792*z^4 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:14:19 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; norm := function(x) return x * x^(p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1), ", norm(1) =", norm(1); "trace(z^2) =", trace(z^2), ", norm(z^2) =", norm(z^2); "trace(z^4) =", trace(z^4), ", norm(z^4) =", norm(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 14:14:18 on modular [Seed = 3896287094] ------------------------------------- trace(1) = 3 , norm(1) = 1 trace(z^2) = 0 , norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0 , norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:12:39 2005 Input: F := FunctionField(Rationals(), 4); (x - h0)*(x - h1)*(x - h2) eq x^3 - (h0 + h1 + h2)*x^2 + (h0*h1 + h0*h2 + h1*h2)*x - (h0*h1*h2); Output: Magma V2.11-10 Wed Dec 7 2005 14:12:39 on modular [Seed = 3979975391] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:12:00 2005 Input: F := FunctionField(Rationals(), 4); (x - h0)*(x - h1)*(x - h2); Output: Magma V2.11-10 Wed Dec 7 2005 14:11:59 on modular [Seed = 3878917901] ------------------------------------- x^3 - x^2*h0 - x^2*h1 - x^2*h2 + x*h0*h1 + x*h0*h2 + x*h1*h2 - h0*h1*h2 Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:11:50 2005 Input: F := FunctionField(Rationals(), 4); (x - h0)*(x - h1)*(x - h2); Output: Magma V2.11-10 Wed Dec 7 2005 14:11:49 on modular [Seed = 4163667992] ------------------------------------- -h0*h1*h2 + h0*h1*x + h0*h2*x - h0*x^2 + h1*h2*x - h1*x^2 - h2*x^2 + x^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:55:46 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "3*xi =", 3*xi; Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); "trace(z^6) =", trace(z^6); "trace(z^8) =", trace(z^8); Output: Magma V2.11-10 Wed Dec 7 2005 13:55:46 on modular [Seed = 2287795119] ------------------------------------- 3*xi = 91343851531049391571590536870049093357328680676*i + 91343851531049391571590536870049093357328680676 trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 trace(z^6) = 91343851531049391571590536870049093357328680676*i + 91343851531049391571590536870049093357328680676 trace(z^8) = 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:55:27 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); "trace(z^6) =", trace(z^6); "trace(z^8) =", trace(z^8); Output: Magma V2.11-10 Wed Dec 7 2005 13:55:27 on modular [Seed = 2371485319] ------------------------------------- xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 trace(z^6) = 91343851531049391571590536870049093357328680676*i + 91343851531049391571590536870049093357328680676 trace(z^8) = 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:53:46 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); "trace(z^6) =", trace(z^6); "trace(z^8) =", trace(z^8); Output: Magma V2.11-10 Wed Dec 7 2005 13:53:46 on modular [Seed = 2672554800] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 trace(z^6) = 91343851531049391571590536870049093357328680676*i + 91343851531049391571590536870049093357328680676 trace(z^8) = 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:53:20 2005 Input: p := 115792089237314936872688561244471742058375878355761205198700409522629664518163; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); "trace(z^6) =", trace(z^6); "trace(z^8) =", trace(z^8); Output: Magma V2.11-10 Wed Dec 7 2005 13:53:19 on modular [Seed = 1790838496] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 trace(z^6) = 723700557733218355454303507777948387864849239723507532491877559516\ 4354032385*i + 723700557733218355454303507777948387864849239723507532491877\ 5595164354032385 trace(z^8) = 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:29:59 2005 Input: p := 115792089237314936872688561244471742058375878355761205198700409522629664518163; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(i) =", trace(i); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 13:29:58 on modular [Seed = 1167142006] ------------------------------------- trace(1) = 3 trace(i) = 3*i trace(z^2) = 0 trace(z^4) = 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:28:52 2005 Input: p := 115792089237314936872688561244471742058375878355761205198700409522629664518163; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 13:28:51 on modular [Seed = 1569273771] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:28:29 2005 Input: p := 26959946667149205758383469736921695435015736735261155141423417423923; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 13:28:29 on modular [Seed = 1451375758] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:28:13 2005 Input: p := 6277101719531269400517043710060892862318604713139674509723; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 13:28:12 on modular [Seed = 738148738] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:27:53 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 13:27:53 on modular [Seed = 754468496] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:27:25 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; "trace(1) =", trace(1, p); "trace(z^2) =", trace(z^2, p); "trace(z^4) =", trace(z^4, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:27:24 on modular [Seed = 569723832] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:26:10 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; //a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; "trace(1) =", trace(1, p); "trace(z^2) =", trace(z^2, p); "trace(z^4) =", trace(z^4, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:26:10 on modular [Seed = 1005539692] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:25:05 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := 1; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:25:05 on modular [Seed = 1021859368] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = 1 u1 = 1 u2 = 1 3 3 3 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:24:27 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := 1 + z^2 + z^4; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:24:27 on modular [Seed = 837114623] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^4 + z^2 + 1 u1 = 1627965160026674480212199743920457792*z^4 + 1461501624495162299985421915440573293973338433026*z^2 + 1 u2 = 1461501624495162299985421915440573293973338433026*z^4 + 1627965160026674480212199743920457792*z^2 + 1 3 3 3 Total time: 0.430 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:23:32 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := (5 + 3*i) + (11 - 2*i)*z^2 + (47 + i)*z^4; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:23:31 on modular [Seed = 182346172] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = (i + 47)*z^4 + (1461501624496790265145448589920785493717258890817*i + 11)*z^2 + 3*i + 5 u1 = (1627965160026674480212199743920457792*i + 76514362521253700569973387964261516224)*z^4 + (3255930320053348960424399487840915586*i + 1461501624478882648385155170638451296534133855096)*z^2 + 3*i + 5 u2 = (1461501624495162299985421915440573293973338433026*i + 1461501624420275902624194889350812105752997374548)*z^4 + (1461501624493534334825395240960361094229417975235*i + 17907616760293419282334197183125035712)*z^2 + 3*i + 5 9*i + 15 9*i + 15 9*i + 15 Total time: 0.460 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:22:04 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := (5 + 3*i) + (11 - 2*i)*z^2 + (47 + i)*z^4; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -(a+1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:22:04 on modular [Seed = 47606107] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = (i + 47)*z^4 + (1461501624496790265145448589920785493717258890817*i + 11)*z^2 + 3*i + 5 u1 = (1627965160026674480212199743920457792*i + 76514362521253700569973387964261516224)*z^4 + (3255930320053348960424399487840915586*i + 1461501624478882648385155170638451296534133855096)*z^2 + 3*i + 5 : false u2 = (1461501624495162299985421915440573293973338433026*i + 1461501624420275902624194889350812105752997374548)*z^4 + (1461501624493534334825395240960361094229417975235*i + 17907616760293419282334197183125035712)*z^2 + 3*i + 5 : false 1461501624495162299985421915440573293973338433027 9*i + 15 9*i + 15 9*i + 15 9*i + 15 Total time: 0.460 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:20:47 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2 + z^4; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -(a+1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:20:46 on modular [Seed = 131294188] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^4 + z^2 u1 = 1627965160026674480212199743920457792*z^4 + 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1461501624495162299985421915440573293973338433026*z^4 + 1627965160026674480212199743920457792*z^2 : false 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.420 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:19:47 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -(a+1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:19:47 on modular [Seed = 416054478] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : true u2 = 1627965160026674480212199743920457792*z^2 : true 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:19:37 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq (-a-1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:19:36 on modular [Seed = 499743152] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : true u2 = 1627965160026674480212199743920457792*z^2 : true 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:19:30 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq (p-a-1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:19:29 on modular [Seed = 381842098] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : true u2 = 1627965160026674480212199743920457792*z^2 : true 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:19:16 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq (p-a+1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:19:16 on modular [Seed = 3896282202] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : true 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:18:51 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq (p-a)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:18:30 on modular [Seed = 4045772176] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : true 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:17:58 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq (p-a)*u0; "u2 =", u2, ":", u2 eq a*u0; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:17:58 on modular [Seed = 2470967250] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : true 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:16:27 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -a*u0; "u2 =", u2, ":", u2 eq a*u0; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:16:24 on modular [Seed = 636567756] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : true 0 0 0 0 Total time: 0.270 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:15:49 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -a*u0; "u2 =", u2, ":", u1 eq a*u0; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:15:48 on modular [Seed = 3778380839] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : false 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:15:32 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -a*u0; "u2 =", u2, ":", u1 eq a*u0; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:15:32 on modular [Seed = 4197359411] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : false 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:15:03 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq a*u0; "u2 =", u2, ":", u1 eq -a*u0; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:15:03 on modular [Seed = 4230519126] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : false 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:13:50 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1) mod u; 1627965160026674480212199743920457792 eq 18*u^3 + 18*u^2 + 9*u + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:13:50 on modular [Seed = 3778380978] ------------------------------------- true 0 true 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:13:37 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1) mod u; 1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 9*u - 1 eq 0; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:13:36 on modular [Seed = 3946811062] ------------------------------------- true 0 true 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:13:25 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1) mod u; 1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1 eq 6*u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:13:25 on modular [Seed = 3590501278] ------------------------------------- true 0 true 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:12:56 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1) mod u; (1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1) div u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:12:55 on modular [Seed = 2775695503] ------------------------------------- true 0 6 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:12:13 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 12*u^2 - 9*u - 1) mod u; (1627965160026674480212199743920457792 - 18*u^3 - 12*u^2 - 9*u - 1) div u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:12:09 on modular [Seed = 1468215712] ------------------------------------- true 0 2693242448394 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.270 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:11:26 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 12*u^2 - 3*u - 1) mod u; (1627965160026674480212199743920457792 - 18*u^3 - 12*u^2 - 3*u - 1) div u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:11:25 on modular [Seed = 1906644915] ------------------------------------- true 0 2693242448400 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:11:05 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 12*u^2 - 3*u - 1) mod u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:11:04 on modular [Seed = 1656095957] ------------------------------------- true 0 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:10:45 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (2*1627965160026674480212199743920457792 - 36*u^3 - 24*u^2 - 6*u - 1) mod u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:10:45 on modular [Seed = 1874528709] ------------------------------------- true 1 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:09:45 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (2*1627965160026674480212199743920457792 - (36*u^3 + 24*u^2 + 6*u + 1) - 1) mod u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:09:44 on modular [Seed = 603407684] ------------------------------------- true 0 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:08:45 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 2*1627965160026674480212199743920457792 - (36*u^3 + 24*u^2 + 6*u + 1); Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:08:45 on modular [Seed = 348156470] ------------------------------------- true 2417851628615822402623201 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:08:40 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 2*1627965160026674480212199743920457792 - (36*u^3 + 18*u^2 + 6*u + 1); Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:08:39 on modular [Seed = 432897841] ------------------------------------- true 3626777442921040361486407 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:08:16 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 2*1627965160026674480212199743920457792 - 36*u^3 + 24*u^2 + 6*u + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:08:16 on modular [Seed = 533428480] ------------------------------------- true 12089258143062952558425639 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:08:10 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 2*1627965160026674480212199743920457792 - 36*u^3 + 18*u^2 + 6*u + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:08:10 on modular [Seed = 80765441] ------------------------------------- true 10880332328757734599562433 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:07:46 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 36*u^3 + 18*u^2 + 6*u + 1; 2*1627965160026674480212199743920457792; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:07:46 on modular [Seed = 215510001] ------------------------------------- true 3255930320049722182981478447479429177 3255930320053348960424399487840915584 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:07:36 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 36*u^3 + 18*u^2 + 6*u + 1; 1627965160026674480212199743920457792; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:07:36 on modular [Seed = 4129460502] ------------------------------------- true 3255930320049722182981478447479429177 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:07:31 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 36*u^3 + 18*u^2 + 6*u + 1;] 1627965160026674480212199743920457792; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:07:30 on modular [Seed = 4045772315] ------------------------------------- true 3255930320049722182981478447479429177 >> 36*u^3 + 18*u^2 + 6*u + 1;] ^ User error: bad syntax 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:07:16 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 36*u^3 + 18*u^2 + 6*u + 1 - 1627965160026674480212199743920457792; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:07:16 on modular [Seed = 4230519584] ------------------------------------- true 1627965160023047702769278703558971385 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:06:14 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:06:14 on modular [Seed = 3862069624] ------------------------------------- true 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:05:51 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := Sqrt((t - 1) div 6); "u =", u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:05:51 on modular [Seed = 3778381462] ------------------------------------- u = 448873741399.0000000000000000 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:04:18 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); F := FunctionField(Fp12, 3); (a + b*z^2 + c*z^4)^(p^2); Output: Magma V2.11-10 Wed Dec 7 2005 13:04:17 on modular [Seed = 3963128773] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 >> (a + b*z^2 + c*z^4)^(p^2); ^ Runtime error in '^': Argument 1 (213598699840675693484172798829876117292923333\ 2694664962512687420744793421079272493992902162490761) is too large Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:04:00 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); F := FunctionField(Fp12, 3); a + b*z^2 + c*z^4; Output: Magma V2.11-10 Wed Dec 7 2005 13:03:59 on modular [Seed = 3607343312] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 a + z^2*b + z^4*c Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:03:42 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); F := FunctionField(Fp12, 3); a + b*z^2 + c*z^4; Output: Magma V2.11-10 Wed Dec 7 2005 13:03:42 on modular [Seed = 3523654180] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 a + z^2*b + z^4*c Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:03:16 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); F := FunctionField(Fp2, 3); a + b*z^2 + c*z^4; Output: Magma V2.11-10 Wed Dec 7 2005 13:03:16 on modular [Seed = 3708402993] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 >> a + b*z^2 + c*z^4; ^ Runtime error in '*': Bad argument types Argument types given: FldFunRatMElt, FldFinElt Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:03:02 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); F := FunctionField(Fp2); a + b*z^2 + c*z^4; Output: Magma V2.11-10 Wed Dec 7 2005 13:03:02 on modular [Seed = 3624712198] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 >> F := FunctionField(Fp2); ^ Runtime error in 'AssignNames': Argument 2 should have length at most 1 >> a + b*z^2 + c*z^4; ^ User error: Identifier 'a' has not been declared or assigned Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:01:05 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:01:04 on modular [Seed = 3256263338] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:59:55 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z; u1 := u0^(p^2); u2 := u0^(p^4); u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:59:55 on modular [Seed = 3441012190] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975233*z 1461501624496790265145448589920785493717258890817*z Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:57:43 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^2)^(p^j); end for; u0 := z; u1 := u0^(p^2); u2 := u0^(p^4); u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:57:43 on modular [Seed = 3357321559] ------------------------------------- 0 : z^2 1 : 1627965160026674480212199743920457792*i*z^2 2 : 1461501624495162299985421915440573293973338433026*z^2 3 : i*z^2 4 : 1627965160026674480212199743920457792*z^2 5 : 1461501624495162299985421915440573293973338433026*i*z^2 6 : z^2 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975233*z 1461501624496790265145448589920785493717258890817*z Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:55:35 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; (z^2)^(p^3); u0 := z; u1 := u0^(p^2); u2 := u0^(p^4); u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:55:34 on modular [Seed = 2993083308] ------------------------------------- i*z^2 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975233*z 1461501624496790265145448589920785493717258890817*z Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:55:25 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; (z^2)^(p^6); u0 := z; u1 := u0^(p^2); u2 := u0^(p^4); u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:55:25 on modular [Seed = 3177831608] ------------------------------------- z^2 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975233*z 1461501624496790265145448589920785493717258890817*z Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:53:47 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:53:47 on modular [Seed = 2809382148] ------------------------------------- false 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975233*z 1461501624496790265145448589920785493717258890817*z Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:53:34 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := i*z^2; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:53:34 on modular [Seed = 2725691444] ------------------------------------- false 0 0 0 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:53:23 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := i*z^3; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:53:22 on modular [Seed = 2910441271] ------------------------------------- false i*z^3 i*z^3 1461501624496790265145448589920785493717258890818*i*z^3 i*z^3 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:52:33 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z^3; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:52:33 on modular [Seed = 2520970495] ------------------------------------- false z^3 z^3 1461501624496790265145448589920785493717258890818*z^3 z^3 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:51:18 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:51:18 on modular [Seed = 2420437646] ------------------------------------- true 0 0 0 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:51:00 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:51:00 on modular [Seed = 2605709719] ------------------------------------- true 0 0 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:50:47 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:50:46 on modular [Seed = 2253579640] ------------------------------------- 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 0 0 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:49:33 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:49:32 on modular [Seed = 2169889317] ------------------------------------- 0 0 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:48:19 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u := z^2; trace(z^2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:48:19 on modular [Seed = 2354638823] ------------------------------------- 0 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:47:02 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u := z^2; c0 := u; c1 := u^(p^4); c2 := u^(p^2); c0 + c1 + c2; Output: Magma V2.11-10 Wed Dec 7 2005 12:47:02 on modular [Seed = 1973491307] ------------------------------------- 0 Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:45:48 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u := z^2; u; -u^(p^2); u^(p^4); Output: Magma V2.11-10 Wed Dec 7 2005 12:45:48 on modular [Seed = 1889807540] ------------------------------------- z^2 1627965160026674480212199743920457793*z^2 1627965160026674480212199743920457792*z^2 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:44:39 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u := z^2; u; u^(p^2); u^(p^4); Output: Magma V2.11-10 Wed Dec 7 2005 12:44:38 on modular [Seed = 2075076597] ------------------------------------- z^2 1461501624495162299985421915440573293973338433026*z^2 1627965160026674480212199743920457792*z^2 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:43:56 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; z^6 eq xi; Output: Magma V2.11-10 Wed Dec 7 2005 12:43:56 on modular [Seed = 1722956338] ------------------------------------- true Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:43:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; z^6 eq xi; Output: Magma V2.11-10 Wed Dec 7 2005 12:43:46 on modular [Seed = 1639267643] ------------------------------------- true Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:43:35 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; z^6; xi; Output: Magma V2.11-10 Wed Dec 7 2005 12:43:34 on modular [Seed = 1824010274] ------------------------------------- 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:42:53 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; z^6; 1/xi; Output: Magma V2.11-10 Wed Dec 7 2005 12:42:53 on modular [Seed = 1468205890] ------------------------------------- 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 8*i + 1461501624496790265145448589920785493717258890811 Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:40:33 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; z := 1/(-8 + 8*i); z; Output: Magma V2.11-10 Wed Dec 7 2005 12:40:33 on modular [Seed = 1367674056] ------------------------------------- 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 Total time: 0.180 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:40:19 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; z := 1/(-8 + 8*i); z; Output: Magma V2.11-10 Wed Dec 7 2005 12:40:18 on modular [Seed = 1552417238] ------------------------------------- >> z := 1/(-8 + 8*i); ^ User error: Identifier 'i' has not been declared or assigned >> z; ^ User error: Identifier 'z' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:40:01 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ext; z := 1/(-8 + 8*i); z; Output: Magma V2.11-10 Wed Dec 7 2005 12:40:01 on modular [Seed = 1200822846] ------------------------------------- >> Fp2 := ext; ^ User error: bad syntax >> z := 1/(-8 + 8*i); ^ User error: Identifier 'i' has not been declared or assigned >> z; ^ User error: Identifier 'z' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:38:14 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ext; z := 1/(-8 + 8*i); z; Output: Magma V2.11-10 Wed Dec 7 2005 12:38:10 on modular [Seed = 920808252] ------------------------------------- >> Fp2 := ext; ^ User error: Identifier 'i' has not been declared or assigned >> z := 1/(-8 + 8*i); ^ User error: Identifier 'i' has not been declared or assigned >> z; ^ User error: Identifier 'z' has not been declared or assigned Total time: 0.230 seconds, Total memory usage: 3.24MB '88.106.' ************** MAGMA ***************** Host 88.106.224.134 (88.106.224.134) Time: Wed Dec 7 11:14:03 2005 Input: a*x^2+b*x+c=0, x Output: Magma V2.11-10 Wed Dec 7 2005 11:14:03 on modular [Seed = 2537807573] ------------------------------------- >> a*x^2+b*x+c=0, x; ^ User error: Identifier 'a' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:54:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print G; Output: Magma V2.11-10 Wed Dec 7 2005 10:54:28 on modular [Seed = 2405157170] ------------------------------------- [ $.1 + (-a^2 + a*c - b - c + 1)/(a*c + b*c - c)*$.7 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.8 - $.9, $.2 + (a^2 + a*c - a)/(a*c + b*c - c)*$.7 + (-a*b + a*c)/(a*c + b*c - c)*$.8, $.3 + (-a^2 + a*b)/(a*c + b*c - c)*$.7 + (2*a*b - a)/(a*c + b*c - c)*$.8, $.4 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.7 + (-b^2 - b*c + b)/(a*c + b*c - c)*$.8, $.5 + (-a*b + a*c)/(a*c + b*c - c)*$.7 + (-a + b^2 - b*c - 2*b + c + 1)/(a*c + b*c - c)*$.8 - $.9, $.6 + (2*a*b - a)/(a*c + b*c - c)*$.7 + (a*b - b^2 + 2*b - 1)/(a*c + b*c - c)*$.8, $.7^2 + (b + c - 1)/(a*b - 1/2*a)*$.7*$.9 + (a*b^2 - a*b + 1/2*b^2 - b + 1/2)/(a^2*b - 1/2*a^2)*$.8^2 + (a*b - a - b*c + c)/(a^2*b - 1/2*a^2)*$.8*$.9 + (a*b*c - a*c + 1/2*c^2)/(a^2*b - 1/2*a^2)*$.9^2 + (-a*b*c + a*c - 1/2*c^2)/(a^2*b - 1/2*a^2), $.7*$.8 + (-c + 1/2)/(b - 1/2)*$.7*$.9 + (1/2*a*b - 1/2*b^2 + b - 1/2)/(a*b - 1/2*a)*$.8^2 + (1/2*a + b*c - c)/(a*b - 1/2*a)*$.8*$.9 + (1/2*a*c - 1/2*c^2)/(a*b - 1/2*a)*$.9^2 + (-1/2*a*c + 1/2*c^2)/(a*b - 1/2*a), $.7*$.9^2 + (-a*b^2*c + a*b*c - 1/4*a*c + b^2*c^2 - b*c^2 + 1/4*c^2)/(a*b^2*c + a*b*c^2 - a*b*c - 1/4*a*b - 1/4*a*c + 1/4*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)*$.7 + (-1/2*a^2*b^3 + 1/4*a^2*b^2 - a*b^4 + 3/2*a*b^3 - 1/2*a*b^2 - 1/2*b^5 + 5/4*b^4 - b^3 + 1/4*b^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^3 + (-1/2*a^2*b^2*c - 3/4*a^2*b^2 + 1/2*a^2*b + a*b^3*c - 3/2*a*b^3 - a*b^2*c + 5/2*a*b^2 + 1/2*a*b*c - a*b + 3/2*b^4*c - 3/4*b^4 - 3*b^3*c + 2*b^3 + 2*b^2*c - 7/4*b^2 - 1/2*b*c + 1/2*b)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^2*$.9 + (-1/2*a^2*b^2*c - 1/4*a^2*b*c - 1/4*a^2*b + 1/4*a^2 - 3/2*a*b^3*c - 1/2*a*b^2*c^2 + 15/4*a*b^2*c - 1/4*a*b^2 + 3/4*a*b*c^2 - 2*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a - 3/2*b^3*c^2 + 3/2*b^3*c + 9/4*b^2*c^2 - 5/2*b^2*c - b*c^2 + b*c)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8*$.9^2 + (1/2*a^2*b^2*c - 1/4*a^2*b*c + 3/2*a*b^3*c - 1/2*a*b^2*c^2 - 11/4*a*b^2*c + 1/4*a*b*c^2 + 3/2*a*b*c - 1/4*a*c + 1/2*b^3*c^2 - 1/4*b^2*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8 + (-1/2*a^2*b*c^2 - 1/4*a^2*b*c + 1/4*a^2*c + 3/2*a*b^2*c^2 - 1/4*a*b^2*c + 1/2*a*b*c^3 - 5/4*a*b*c^2 + 1/2*a*b*c + 1/4*a*c^2 - 1/4*a*c + 1/2*b^2*c^3 - 3/4*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9^3 + (1/2*a^2*b*c^2 + 1/4*a^2*b*c - 1/4*a^2*c - 3/2*a*b^2*c^2 + 1/4*a*b^2*c - 1/2*a*b*c^3 + 5/4*a*b*c^2 - 1/2*a*b*c - 1/4*a*c^2 + 1/4*a*c - 1/2*b^2*c^3 + 3/4*b^2*c^2 + 1/2*b*c^3 - 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9, $.8^4 + (2*a - 4*b*c + 2*b + 2*c - 2)/(a*b + b^2 - b)*$.8^3*$.9 + (2*a^2*b*c + a^2 + 2*a*b^2*c + 2*a*b*c^2 - 8*a*b*c + 2*a*b + 2*a*c - 2*a + 6*b^2*c^2 - 6*b^2*c + b^2 - 6*b*c^2 + 8*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2*$.9^2 + (-2*a^2*b*c - 2*a*b^2*c + 2*a*b*c^2 + 2*a*b*c - 2*b^2*c^2 + 2*b*c^2 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2 + (2*a^2*c - 4*a*b*c^2 + 2*a*b*c + 4*a*c^2 - 4*a*c - 4*b*c^3 + 6*b*c^2 - 2*b*c + 2*c^3 - 4*c^2 + 2*c)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9^3 + (-2*a^2*c + 4*a*b*c^2 - 2*a*b*c + 2*a*c + 4*b*c^3 - 2*b*c^2 - 2*c^3)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9 + (a^2*c^2 + 2*a*c^3 - 2*a*c^2 + c^4 - 2*c^3 + c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^4 + (-2*a^2*c^2 + 2*a*c^2 - 2*c^4 + 2*c^3 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^2 + (a^2*c^2 - 2*a*c^3 + c^4)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2) ] Total time: 1.439 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:53:33 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.8,R.7,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print G; Output: Magma V2.11-10 Wed Dec 7 2005 10:53:21 on modular [Seed = 2287783664] ------------------------------------- [ $.1 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.7 + (-a^2 + a*c - b - c + 1)/(a*c + b*c - c)*$.8 - $.9, $.2 + (-a*b + a*c)/(a*c + b*c - c)*$.7 + (a^2 + a*c - a)/(a*c + b*c - c)*$.8, $.3 + (2*a*b - a)/(a*c + b*c - c)*$.7 + (-a^2 + a*b)/(a*c + b*c - c)*$.8, $.4 + (-b^2 - b*c + b)/(a*c + b*c - c)*$.7 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.8, $.5 + (-a + b^2 - b*c - 2*b + c + 1)/(a*c + b*c - c)*$.7 + (-a*b + a*c)/(a*c + b*c - c)*$.8 - $.9, $.6 + (a*b - b^2 + 2*b - 1)/(a*c + b*c - c)*$.7 + (2*a*b - a)/(a*c + b*c - c)*$.8, $.7^2 + (a - c)/(a*b + 1/2*b - 1/2)*$.7*$.9 + (a^2*b - 1/2*a^2)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.8^2 + (a*b + a*c - a)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.8*$.9 + (a*b*c - a*c + 1/2*c^2)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.9^2 + (-a*b*c + a*c - 1/2*c^2)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2), $.7*$.8 + (b*c + 1/2*b - 1/2)/(a*b + 1/2*b - 1/2)*$.7*$.9 + (-1/2*a^2*b + 1/2*a*b^2 - a*b + 1/2*a)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.8^2 + (-a*b*c + 1/2*b^2 - b + 1/2)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.8*$.9 + (1/2*b^2*c - 1/2*b*c^2 - b*c + 1/2*c)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.9^2 + (-1/2*b^2*c + 1/2*b*c^2 + b*c - 1/2*c)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2), $.7*$.9^2 + (-a^2*b^4*c + a^2*b^3*c^2 + 2*a^2*b^3*c - a^2*b^2*c - a*b^4*c + a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^2 - 3*a*b^2*c + a*b*c - 1/4*b^4*c + 1/4*b^3*c^2 + b^3*c - 1/2*b^2*c^2 - 3/2*b^2*c + 1/4*b*c^2 + b*c - 1/4*c)/(a^2*b^4*c - 2*a^2*b^3*c - 1/4*a^2*b^3 + a^2*b^2*c + 1/2*a^2*b^2 - 1/4*a^2*b + a*b^4*c^2 + a*b^4*c - 1/4*a*b^4 - 2*a*b^3*c^2 - 7/2*a*b^3*c + 1/2*a*b^3 + a*b^2*c^2 + 4*a*b^2*c - 3/2*a*b*c - 1/2*a*b + 1/4*a - 1/4*b^4*c - 1/4*b^4 - 1/4*b^3*c^2 + 1/2*b^3*c + b^3 + 1/2*b^2*c^2 - 3/2*b^2 - 1/4*b*c^2 - 1/2*b*c + b + 1/4*c - 1/4)*$.7 + (-1/2*a^5*b^3 - a^4*b^4 + 3/4*a^4*b^3 + 1/4*a^4*b^2 - 1/2*a^3*b^5 + 1/2*a^3*b^4 + 1/2*a^3*b^3 - 1/2*a^3*b^2 - 1/4*a^2*b^5 + 3/4*a^2*b^4 - 3/4*a^2*b^3 + 1/4*a^2*b^2)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.8^3 + (-3/2*a^4*b^3*c - 3/4*a^4*b^3 + 3/4*a^4*b^2 - a^3*b^4*c - 3/2*a^3*b^4 - a^3*b^3*c + 5/2*a^3*b^3 + 2*a^3*b^2*c - 1/2*a^3*b^2 - 1/2*a^3*b + 1/2*a^2*b^5*c - 3/4*a^2*b^5 - 3*a^2*b^4*c + 5/4*a^2*b^4 + 4*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 9/4*a^2*b^2 - 1/2*a^2*b*c + a^2*b - 1/2*a*b^5 - 1/2*a*b^4*c + 2*a*b^4 + 3/2*a*b^3*c - 3*a*b^3 - 3/2*a*b^2*c + 2*a*b^2 + 1/2*a*b*c - 1/2*a*b)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.8^2*$.9 + (-3/2*a^3*b^4*c - 3/2*a^3*b^3*c^2 + 3/2*a^3*b^3*c - 1/2*a^2*b^5*c - 1/2*a^2*b^4*c^2 - 3/4*a^2*b^4*c - 1/4*a^2*b^4 - 5/4*a^2*b^3*c^2 + 11/4*a^2*b^3*c + 3/4*a^2*b^3 + 7/4*a^2*b^2*c^2 - 5/4*a^2*b^2*c - 3/4*a^2*b^2 - 1/4*a^2*b*c + 1/4*a^2*b + 1/4*a*b^5*c - 1/4*a*b^5 + 1/4*a*b^4*c^2 - 9/4*a*b^4*c + 3/4*a*b^4 - 7/4*a*b^3*c^2 + 5*a*b^3*c - 1/2*a*b^3 + 11/4*a*b^2*c^2 - 4*a*b^2*c - 1/2*a*b^2 - 5/4*a*b*c^2 + 3/4*a*b*c + 3/4*a*b + 1/4*a*c - 1/4*a - 1/4*b^5 - 1/2*b^4*c + 5/4*b^4 - 1/4*b^3*c^2 + 2*b^3*c - 5/2*b^3 + 3/4*b^2*c^2 - 3*b^2*c + 5/2*b^2 - 3/4*b*c^2 + 2*b*c - 5/4*b + 1/4*c^2 - 1/2*c + 1/4)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.8*$.9^2 + (3/2*a^3*b^4*c + 1/2*a^3*b^3*c^2 - 3*a^3*b^3*c + 3/2*a^3*b^2*c + 1/2*a^2*b^5*c - 1/2*a^2*b^4*c^2 - 1/4*a^2*b^4*c + 9/4*a^2*b^3*c^2 - 9/4*a^2*b^3*c - 7/4*a^2*b^2*c^2 + 13/4*a^2*b^2*c - 5/4*a^2*b*c + 1/4*a*b^5*c - 1/4*a*b^4*c^2 - 3/4*a*b^4*c + 7/4*a*b^3*c^2 + 1/2*a*b^3*c - 11/4*a*b^2*c^2 + 1/2*a*b^2*c + 5/4*a*b*c^2 - 3/4*a*b*c + 1/4*a*c + 1/4*b^3*c^2 - 3/4*b^2*c^2 + 3/4*b*c^2 - 1/4*c^2)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.8 + (-3/2*a^2*b^4*c^2 - 1/4*a^2*b^4*c - 1/2*a^2*b^3*c^3 + 9/4*a^2*b^3*c^2 + 3/4*a^2*b^3*c - 3/4*a^2*b^2*c^2 - 3/4*a^2*b^2*c + 1/4*a^2*b*c + 1/2*a*b^5*c^2 - 1/4*a*b^5*c - 1/2*a*b^4*c^3 - 13/4*a*b^4*c^2 + 3/4*a*b^4*c + 1/2*a*b^3*c^3 + 25/4*a*b^3*c^2 - 1/2*a*b^3*c - 19/4*a*b^2*c^2 - 1/2*a*b^2*c + 5/4*a*b*c^2 + 3/4*a*b*c - 1/4*a*c - 1/4*b^5*c - 1/4*b^4*c^2 + 5/4*b^4*c + b^3*c^2 - 5/2*b^3*c - 3/2*b^2*c^2 + 5/2*b^2*c + b*c^2 - 5/4*b*c - 1/4*c^2 + 1/4*c)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.9^3 + (3/2*a^2*b^4*c^2 + 1/4*a^2*b^4*c + 1/2*a^2*b^3*c^3 - 9/4*a^2*b^3*c^2 - 3/4*a^2*b^3*c + 3/4*a^2*b^2*c^2 + 3/4*a^2*b^2*c - 1/4*a^2*b*c - 1/2*a*b^5*c^2 + 1/4*a*b^5*c + 1/2*a*b^4*c^3 + 13/4*a*b^4*c^2 - 3/4*a*b^4*c - 1/2*a*b^3*c^3 - 25/4*a*b^3*c^2 + 1/2*a*b^3*c + 19/4*a*b^2*c^2 + 1/2*a*b^2*c - 5/4*a*b*c^2 - 3/4*a*b*c + 1/4*a*c + 1/4*b^5*c + 1/4*b^4*c^2 - 5/4*b^4*c - b^3*c^2 + 5/2*b^3*c + 3/2*b^2*c^2 - 5/2*b^2*c - b*c^2 + 5/4*b*c + 1/4*c^2 - 1/4*c)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.9, $.8^4 + (4*a*b*c + 2*a*b - 2*a + 2*b^2 + 2*b*c - 4*b - 2*c + 2)/(a^2*b + a*b^2 - a*b)*$.8^3*$.9 + (2*a^2*b^3*c + 6*a^2*b^2*c^2 + 2*a^2*b^2*c + a^2*b^2 - 4*a^2*b*c - 2*a^2*b + a^2 + 2*a*b^4*c + 2*a*b^3*c^2 + 2*a*b^3 + 2*a*b^2*c^2 - 4*a*b^2*c - 6*a*b^2 - 4*a*b*c^2 + 6*a*b + 2*a*c - 2*a + b^4 + 2*b^3*c - 4*b^3 + b^2*c^2 - 6*b^2*c + 6*b^2 - 2*b*c^2 + 6*b*c - 4*b + c^2 - 2*c + 1)/(a^4*b^2 + 2*a^3*b^3 - 2*a^3*b^2 + a^2*b^4 - 2*a^2*b^3 + a^2*b^2)*$.8^2*$.9^2 + (-2*a^2*b^3*c - 2*a^2*b^2*c^2 + 4*a^2*b^2*c - 2*a^2*b*c - 2*a*b^4*c + 2*a*b^3*c^2 + 6*a*b^3*c - 6*a*b^2*c^2 - 6*a*b^2*c + 4*a*b*c^2 + 2*a*b*c - b^2*c^2 + 2*b*c^2 - c^2)/(a^4*b^2 + 2*a^3*b^3 - 2*a^3*b^2 + a^2*b^4 - 2*a^2*b^3 + a^2*b^2)*$.8^2 + (4*a*b^2*c^2 + 2*a*b^2*c + 4*a*b*c^3 - 2*a*b*c^2 - 4*a*b*c - 2*a*c^2 + 2*a*c + 2*b^3*c + 4*b^2*c^2 - 6*b^2*c + 2*b*c^3 - 8*b*c^2 + 6*b*c - 2*c^3 + 4*c^2 - 2*c)/(a^4*b + 2*a^3*b^2 - 2*a^3*b + a^2*b^3 - 2*a^2*b^2 + a^2*b)*$.8*$.9^3 + (-4*a*b^3*c^2 - 2*a*b^3*c - 4*a*b^2*c^3 + 6*a*b^2*c^2 + 6*a*b^2*c - 2*a*b*c^2 - 6*a*b*c + 2*a*c - 2*b^4*c + 8*b^3*c - 2*b^2*c^3 - 12*b^2*c + 2*b*c^3 + 8*b*c - 2*c)/(a^4*b^2 + 2*a^3*b^3 - 2*a^3*b^2 + a^2*b^4 - 2*a^2*b^3 + a^2*b^2)*$.8*$.9 + (b^2*c^2 + 2*b*c^3 - 2*b*c^2 + c^4 - 2*c^3 + c^2)/(a^4 + 2*a^3*b - 2*a^3 + a^2*b^2 - 2*a^2*b + a^2)*$.9^4 + (-2*b^4*c^2 + 6*b^3*c^2 - 2*b^2*c^4 - 2*b^2*c^3 - 7*b^2*c^2 + 2*b*c^3 + 4*b*c^2 - c^2)/(a^4*b^2 + 2*a^3*b^3 - 2*a^3*b^2 + a^2*b^4 - 2*a^2*b^3 + a^2*b^2)*$.9^2 + (b^4*c^2 - 2*b^3*c^3 - 4*b^3*c^2 + b^2*c^4 + 4*b^2*c^3 + 6*b^2*c^2 - 2*b*c^3 - 4*b*c^2 + c^2)/(a^4*b^2 + 2*a^3*b^3 - 2*a^3*b^2 + a^2*b^4 - 2*a^2*b^3 + a^2*b^2) ] Total time: 12.500 seconds, Total memory usage: 10.35MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:53:14 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.8,R.7,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:53:01 on modular [Seed = 1956655648] ------------------------------------- Total time: 12.470 seconds, Total memory usage: 10.35MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:51:18 2005 Input: K:=RationalField(); A:=Matrix(K,3,3,[[2,1,1],[1,2,1],[1,1,2]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:51:17 on modular [Seed = 2091400064] ------------------------------------- [ $.1^2 + 1/3*$.1*$.5 + 1/3*$.1*$.6 + 1/3*$.1*$.8 + 1/3*$.1*$.9 + 1/3*$.2*$.4 + 2/3*$.2*$.5 + 1/3*$.2*$.6 + 1/3*$.2*$.7 + 2/3*$.2*$.8 + 1/3*$.2*$.9 + 1/3*$.3*$.4 + 1/3*$.3*$.5 + 2/3*$.3*$.6 + 1/3*$.3*$.7 + 1/3*$.3*$.8 + 2/3*$.3*$.9 - $.4*$.5 - $.4*$.6 + 1/3*$.4*$.8 + 1/3*$.4*$.9 - $.5^2 - $.5*$.6 + 1/3*$.5*$.7 + 2/3*$.5*$.8 + 1/3*$.5*$.9 - $.6^2 + 1/3*$.6*$.7 + 1/3*$.6*$.8 + 2/3*$.6*$.9 - $.7*$.8 - $.7*$.9 - $.8^2 - $.8*$.9 - $.9^2, $.1*$.2 - 1/3*$.1*$.5 - 1/3*$.1*$.8 - 1/3*$.2*$.4 - 1/3*$.2*$.7 + $.4*$.5 - 1/3*$.4*$.8 - 1/3*$.5*$.7 + $.7*$.8 + 1/3, $.1*$.3 - 1/3*$.1*$.6 - 1/3*$.1*$.9 - 1/3*$.3*$.4 - 1/3*$.3*$.7 + $.4*$.6 - 1/3*$.4*$.9 - 1/3*$.6*$.7 + $.7*$.9 + 1/3, $.1*$.4 + 1/2*$.1*$.5 + 1/2*$.1*$.6 + 1/2*$.2*$.4 + $.2*$.5 + 1/2*$.2*$.6 + 1/2*$.3*$.4 + 1/2*$.3*$.5 + $.3*$.6 - 1/2, $.1*$.5^2 + 1/3*$.1*$.5*$.9 + 1/3*$.1*$.6*$.8 + $.1*$.6*$.9 - 2/3*$.1*$.8*$.9 - $.1*$.9^2 - 11/24*$.1 - $.2*$.4*$.5 - 1/6*$.2*$.4*$.9 - 1/6*$.2*$.5*$.9 - 1/6*$.2*$.6*$.7 - 1/6*$.2*$.6*$.8 + 1/3*$.2*$.6*$.9 + 1/3*$.2*$.7*$.9 + 1/3*$.2*$.8*$.9 - 1/3*$.2*$.9^2 - 1/24*$.2 - 1/6*$.3*$.4*$.8 - 1/2*$.3*$.4*$.9 - 1/6*$.3*$.5*$.7 + 1/3*$.3*$.5*$.8 - 1/6*$.3*$.5*$.9 - 1/2*$.3*$.6*$.7 - 1/6*$.3*$.6*$.8 + 1/3*$.3*$.7*$.8 + $.3*$.7*$.9 - 1/3*$.3*$.8^2 + 1/3*$.3*$.8*$.9 + 1/3*$.3 - 1/6*$.4*$.5*$.9 - 1/6*$.4*$.6*$.8 - 1/2*$.4*$.6*$.9 + 1/3*$.4*$.8*$.9 + 1/2*$.4*$.9^2 + 3/8*$.4 + 1/6*$.5^2*$.9 + 1/3*$.5*$.6*$.7 - 1/6*$.5*$.6*$.8 - 1/6*$.5*$.6*$.9 - 1/6*$.5*$.7*$.9 - 1/6*$.5*$.8*$.9 + 1/6*$.5*$.9^2 + 1/8*$.5 + 1/2*$.6^2*$.7 + 1/6*$.6^2*$.8 - 1/6*$.6*$.7*$.8 - 1/2*$.6*$.7*$.9 + 1/6*$.6*$.8^2 - 1/6*$.6*$.8*$.9 - 7/24*$.7 - 5/24*$.8 - 1/3*$.9, $.1*$.5*$.6 - 1/2*$.1*$.5*$.9 - 1/2*$.1*$.6*$.8 + $.1*$.8*$.9 + 3/8*$.1 - $.3*$.4*$.5 + 1/2*$.3*$.4*$.8 + 1/2*$.3*$.5*$.7 - $.3*$.7*$.8 - 3/8*$.3 + 1/2*$.4*$.5*$.9 - 1/2*$.4*$.8*$.9 - 1/8*$.4 - 1/2*$.5*$.6*$.7 + 1/2*$.6*$.7*$.8 + 1/8*$.6 - 1/8*$.7 + 1/8*$.9, $.1*$.5*$.8 + 1/3*$.1*$.5*$.9 + 1/3*$.1*$.6*$.8 + $.1*$.6*$.9 - 2/3*$.1 - $.2*$.5*$.7 - 1/3*$.2*$.5*$.9 - 1/3*$.2*$.6*$.7 + 1/3*$.2*$.6*$.9 - 1/3*$.2 - 1/3*$.3*$.5*$.7 + 1/3*$.3*$.5*$.8 - $.3*$.6*$.7 - 1/3*$.3*$.6*$.8 - 1/3*$.3 + 2/3*$.7 + 1/3*$.8 + 1/3*$.9, $.1*$.5*$.9^2 - 3/2*$.1*$.5 - $.1*$.6*$.8*$.9 - 1/2*$.1*$.6 + 3/4*$.1*$.8 + 1/4*$.1*$.9 - $.3*$.5*$.7*$.9 - 1/2*$.3*$.5 + $.3*$.6*$.7*$.8 + 1/2*$.3*$.6 + 1/4*$.3*$.8 - 1/4*$.3*$.9 + 3/4*$.5*$.7 + 1/4*$.5*$.9 + 1/4*$.6*$.7 - 1/4*$.6*$.9 - 3/4*$.7*$.8 - 1/4*$.7*$.9 - 1/4*$.8*$.9 + 1/4*$.9^2 - 1/4, $.1*$.6^2 - $.1*$.6*$.9 + $.1*$.9^2 - 9/8*$.1 - $.3*$.4*$.6 + 1/2*$.3*$.4*$.9 + 1/2*$.3*$.6*$.7 - $.3*$.7*$.9 - 3/8*$.3 + 1/2*$.4*$.6*$.9 - 1/2*$.4*$.9^2 + 3/8*$.4 - 1/2*$.6^2*$.7 + 1/2*$.6*$.7*$.9 + 1/8*$.6 + 3/8*$.7 + 1/8*$.9, $.1*$.7 + 1/2*$.1*$.8 + 1/2*$.1*$.9 + 1/2*$.2*$.7 + $.2*$.8 + 1/2*$.2*$.9 + 1/2*$.3*$.7 + 1/2*$.3*$.8 + $.3*$.9 - 1/2, $.1*$.8^2 + 2/3*$.1*$.8*$.9 + $.1*$.9^2 - 4/3*$.1 - $.2*$.7*$.8 - 1/3*$.2*$.7*$.9 - 1/3*$.2*$.8*$.9 + 1/3*$.2*$.9^2 - 2/3*$.2 - 1/3*$.3*$.7*$.8 - $.3*$.7*$.9 + 1/3*$.3*$.8^2 - 1/3*$.3*$.8*$.9 - 2/3*$.3 + 2/3*$.7 + 1/3*$.8 + 1/3*$.9, $.2^2 - 2/3*$.2*$.5 - 2/3*$.2*$.8 + $.5^2 - 2/3*$.5*$.8 + $.8^2 - 1, $.2*$.3 - 1/3*$.2*$.6 - 1/3*$.2*$.9 - 1/3*$.3*$.5 - 1/3*$.3*$.8 + $.5*$.6 - 1/3*$.5*$.9 - 1/3*$.6*$.8 + $.8*$.9 + 1/3, $.2*$.4*$.6 - 1/2*$.2*$.4*$.9 - 1/2*$.2*$.6*$.7 + $.2*$.7*$.9 + 3/8*$.2 - $.3*$.4*$.5 + 1/2*$.3*$.4*$.8 + 1/2*$.3*$.5*$.7 - $.3*$.7*$.8 - 3/8*$.3 + 1/2*$.4*$.5*$.9 - 1/2*$.4*$.6*$.8 - 1/2*$.5*$.7*$.9 - 1/8*$.5 + 1/2*$.6*$.7*$.8 + 1/8*$.6 - 1/8*$.8 + 1/8*$.9, $.2*$.4*$.8 + 1/3*$.2*$.4*$.9 - $.2*$.5*$.7 - 1/3*$.2*$.5*$.9 - 1/3*$.2*$.6*$.7 + 1/3*$.2*$.6*$.8 + 1/3*$.3*$.4*$.8 + $.3*$.4*$.9 - 1/3*$.3*$.5*$.7 + 1/3*$.3*$.5*$.9 - $.3*$.6*$.7 - 1/3*$.3*$.6*$.8 - 2/3*$.4 - 1/3*$.5 - 1/3*$.6 + 2/3*$.7 + 1/3*$.8 + 1/3*$.9, $.2*$.4*$.9^2 - 3/2*$.2*$.4 - $.2*$.6*$.7*$.9 - 1/2*$.2*$.6 + 3/4*$.2*$.7 + 1/4*$.2*$.9 - $.3*$.4*$.8*$.9 - 1/2*$.3*$.4 + $.3*$.6*$.7*$.8 + 1/2*$.3*$.6 + 1/4*$.3*$.7 - 1/4*$.3*$.9 + 3/4*$.4*$.8 + 1/4*$.4*$.9 + 1/4*$.6*$.8 - 1/4*$.6*$.9 - 3/4*$.7*$.8 - 1/4*$.7*$.9 - 1/4*$.8*$.9 + 1/4*$.9^2 - 1/4, $.2*$.5*$.6 - 1/2*$.2*$.5*$.9 - 1/2*$.2*$.6*$.8 + $.2*$.8*$.9 + 3/8*$.2 - $.3*$.5^2 + $.3*$.5*$.8 - $.3*$.8^2 + 9/8*$.3 + 1/2*$.5^2*$.9 - 1/2*$.5*$.6*$.8 - 1/2*$.5*$.8*$.9 - 1/8*$.5 + 1/2*$.6*$.8^2 - 3/8*$.6 - 1/8*$.8 - 3/8*$.9, $.2*$.5*$.9^2 - 3/2*$.2*$.5 - $.2*$.6*$.8*$.9 - 1/2*$.2*$.6 + 3/4*$.2*$.8 + 1/4*$.2*$.9 - $.3*$.5*$.8*$.9 - 1/2*$.3*$.5 + $.3*$.6*$.8^2 - 3/2*$.3*$.6 + 1/4*$.3*$.8 + 3/4*$.3*$.9 + 3/4*$.5*$.8 + 1/4*$.5*$.9 + 1/4*$.6*$.8 + 3/4*$.6*$.9 - 3/4*$.8^2 - 1/2*$.8*$.9 - 3/4*$.9^2 + 1/2, $.2*$.6^2 - $.2*$.6*$.9 + $.2*$.9^2 - 9/8*$.2 - $.3*$.5*$.6 + 1/2*$.3*$.5*$.9 + 1/2*$.3*$.6*$.8 - $.3*$.8*$.9 - 3/8*$.3 + 1/2*$.5*$.6*$.9 - 1/2*$.5*$.9^2 + 3/8*$.5 - 1/2*$.6^2*$.8 + 1/2*$.6*$.8*$.9 + 1/8*$.6 + 3/8*$.8 + 1/8*$.9, $.3^2 - 2/3*$.3*$.6 - 2/3*$.3*$.9 + $.6^2 - 2/3*$.6*$.9 + $.9^2 - 1, $.4^2 + $.4*$.5 + $.4*$.6 + $.5^2 + $.5*$.6 + $.6^2 - 1, $.4*$.5*$.8 + 1/3*$.4*$.5*$.9 + 1/3*$.4*$.6*$.8 + $.4*$.6*$.9 - 2/3*$.4 - $.5^2*$.7 - 1/3*$.5^2*$.9 - 2/3*$.5*$.6*$.7 + 1/3*$.5*$.6*$.8 + 1/3*$.5*$.6*$.9 - 1/3*$.5 - $.6^2*$.7 - 1/3*$.6^2*$.8 - 1/3*$.6 + 4/3*$.7 + 2/3*$.8 + 2/3*$.9, $.4*$.5*$.9^2 - 3/2*$.4*$.5 - $.4*$.6*$.8*$.9 - 1/2*$.4*$.6 + 3/4*$.4*$.8 + 1/4*$.4*$.9 - $.5*$.6*$.7*$.9 - 1/2*$.5*$.6 + 3/4*$.5*$.7 + 1/4*$.5*$.9 + $.6^2*$.7*$.8 + 1/2*$.6^2 + 1/4*$.6*$.7 + 1/4*$.6*$.8 - 1/2*$.6*$.9 - 3/2*$.7*$.8 - 1/2*$.7*$.9 - 1/2*$.8*$.9 + 1/2*$.9^2 - 3/4, $.4*$.7 + 1/2*$.4*$.8 + 1/2*$.4*$.9 + 1/2*$.5*$.7 + $.5*$.8 + 1/2*$.5*$.9 + 1/2*$.6*$.7 + 1/2*$.6*$.8 + $.6*$.9 - 1/2, $.4*$.8^2 + 2/3*$.4*$.8*$.9 + $.4*$.9^2 - 4/3*$.4 - $.5*$.7*$.8 - 1/3*$.5*$.7*$.9 - 1/3*$.5*$.8*$.9 + 1/3*$.5*$.9^2 - 2/3*$.5 - 1/3*$.6*$.7*$.8 - $.6*$.7*$.9 + 1/3*$.6*$.8^2 - 1/3*$.6*$.8*$.9 - 2/3*$.6 + 2/3*$.7 + 1/3*$.8 + 1/3*$.9, $.5^2*$.9^2 - 3/2*$.5^2 - 2*$.5*$.6*$.8*$.9 - $.5*$.6 + 3/2*$.5*$.8 + 1/2*$.5*$.9 + $.6^2*$.8^2 - 3/2*$.6^2 + 1/2*$.6*$.8 + 3/2*$.6*$.9 - 3/2*$.8^2 - $.8*$.9 - 3/2*$.9^2 + 3/2, $.7^2 + $.7*$.8 + $.7*$.9 + $.8^2 + $.8*$.9 + $.9^2 - 1 ] Total time: 0.210 seconds, Total memory usage: 3.86MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:50:11 2005 Input: K:=RationalField(); A:=Matrix(K,3,3,[[0,1,1],[-1,0,1],[-1,-1,0]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:50:11 on modular [Seed = 1739792605] ------------------------------------- [ $.1 - $.3 - $.4 + $.6 + $.7 - $.9, $.2 + $.3 - $.5 - $.6 + $.8 + $.9, $.4*$.8 + $.4*$.9 - $.5*$.7 + $.5*$.9 - $.6*$.7 - $.6*$.8 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:33:41 2005 Input: K:=RationalField(); A:=Matrix(K,3,3,[[1,1,1],[0,1,1],[0,0,1]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.8,R.7,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:33:40 on modular [Seed = 1183974811] ------------------------------------- [ $.1 - $.7 - $.8 - $.9, $.2 + $.8, $.3 + $.7, $.4 - $.7 - $.8, $.5 - $.7 - $.9, $.6 + $.7 + $.8, $.7^2 + $.7*$.8 + $.7*$.9 - $.8*$.9, $.8^2 + 2*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:31:07 2005 Input: K:=RationalField(); A:=Matrix(K,3,3,[[1,1,1],[0,1,1],[0,0,1]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:31:06 on modular [Seed = 1083442035] ------------------------------------- [ $.1 - $.7 - $.8 - $.9, $.2 + $.7, $.3 + $.8, $.4 - $.7 - $.8, $.5 - $.8 - $.9, $.6 + $.7 + $.8, $.7^2 + 2*$.7*$.9 + $.9^2 - 1, $.7*$.8 - $.7*$.9 + $.8^2 + $.8*$.9, $.7*$.9^2 - 1/2*$.8^3 - 1/2*$.8^2*$.9 - 1/2*$.8*$.9^2 + 1/2*$.8 + 1/2*$.9^3 - 1/2*$.9, $.8^4 + 2*$.8^2*$.9^2 - $.8^2 + 2*$.8*$.9 + $.9^4 - $.9^2 ] Total time: 0.220 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:25:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print G; Output: Magma V2.11-10 Wed Dec 7 2005 10:25:28 on modular [Seed = 920882135] ------------------------------------- [ $.1 + (-a^2 + a*c - b - c + 1)/(a*c + b*c - c)*$.7 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.8 - $.9, $.2 + (a^2 + a*c - a)/(a*c + b*c - c)*$.7 + (-a*b + a*c)/(a*c + b*c - c)*$.8, $.3 + (-a^2 + a*b)/(a*c + b*c - c)*$.7 + (2*a*b - a)/(a*c + b*c - c)*$.8, $.4 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.7 + (-b^2 - b*c + b)/(a*c + b*c - c)*$.8, $.5 + (-a*b + a*c)/(a*c + b*c - c)*$.7 + (-a + b^2 - b*c - 2*b + c + 1)/(a*c + b*c - c)*$.8 - $.9, $.6 + (2*a*b - a)/(a*c + b*c - c)*$.7 + (a*b - b^2 + 2*b - 1)/(a*c + b*c - c)*$.8, $.7^2 + (b + c - 1)/(a*b - 1/2*a)*$.7*$.9 + (a*b^2 - a*b + 1/2*b^2 - b + 1/2)/(a^2*b - 1/2*a^2)*$.8^2 + (a*b - a - b*c + c)/(a^2*b - 1/2*a^2)*$.8*$.9 + (a*b*c - a*c + 1/2*c^2)/(a^2*b - 1/2*a^2)*$.9^2 + (-a*b*c + a*c - 1/2*c^2)/(a^2*b - 1/2*a^2), $.7*$.8 + (-c + 1/2)/(b - 1/2)*$.7*$.9 + (1/2*a*b - 1/2*b^2 + b - 1/2)/(a*b - 1/2*a)*$.8^2 + (1/2*a + b*c - c)/(a*b - 1/2*a)*$.8*$.9 + (1/2*a*c - 1/2*c^2)/(a*b - 1/2*a)*$.9^2 + (-1/2*a*c + 1/2*c^2)/(a*b - 1/2*a), $.7*$.9^2 + (-a*b^2*c + a*b*c - 1/4*a*c + b^2*c^2 - b*c^2 + 1/4*c^2)/(a*b^2*c + a*b*c^2 - a*b*c - 1/4*a*b - 1/4*a*c + 1/4*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)*$.7 + (-1/2*a^2*b^3 + 1/4*a^2*b^2 - a*b^4 + 3/2*a*b^3 - 1/2*a*b^2 - 1/2*b^5 + 5/4*b^4 - b^3 + 1/4*b^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^3 + (-1/2*a^2*b^2*c - 3/4*a^2*b^2 + 1/2*a^2*b + a*b^3*c - 3/2*a*b^3 - a*b^2*c + 5/2*a*b^2 + 1/2*a*b*c - a*b + 3/2*b^4*c - 3/4*b^4 - 3*b^3*c + 2*b^3 + 2*b^2*c - 7/4*b^2 - 1/2*b*c + 1/2*b)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^2*$.9 + (-1/2*a^2*b^2*c - 1/4*a^2*b*c - 1/4*a^2*b + 1/4*a^2 - 3/2*a*b^3*c - 1/2*a*b^2*c^2 + 15/4*a*b^2*c - 1/4*a*b^2 + 3/4*a*b*c^2 - 2*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a - 3/2*b^3*c^2 + 3/2*b^3*c + 9/4*b^2*c^2 - 5/2*b^2*c - b*c^2 + b*c)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8*$.9^2 + (1/2*a^2*b^2*c - 1/4*a^2*b*c + 3/2*a*b^3*c - 1/2*a*b^2*c^2 - 11/4*a*b^2*c + 1/4*a*b*c^2 + 3/2*a*b*c - 1/4*a*c + 1/2*b^3*c^2 - 1/4*b^2*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8 + (-1/2*a^2*b*c^2 - 1/4*a^2*b*c + 1/4*a^2*c + 3/2*a*b^2*c^2 - 1/4*a*b^2*c + 1/2*a*b*c^3 - 5/4*a*b*c^2 + 1/2*a*b*c + 1/4*a*c^2 - 1/4*a*c + 1/2*b^2*c^3 - 3/4*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9^3 + (1/2*a^2*b*c^2 + 1/4*a^2*b*c - 1/4*a^2*c - 3/2*a*b^2*c^2 + 1/4*a*b^2*c - 1/2*a*b*c^3 + 5/4*a*b*c^2 - 1/2*a*b*c - 1/4*a*c^2 + 1/4*a*c - 1/2*b^2*c^3 + 3/4*b^2*c^2 + 1/2*b*c^3 - 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9, $.8^4 + (2*a - 4*b*c + 2*b + 2*c - 2)/(a*b + b^2 - b)*$.8^3*$.9 + (2*a^2*b*c + a^2 + 2*a*b^2*c + 2*a*b*c^2 - 8*a*b*c + 2*a*b + 2*a*c - 2*a + 6*b^2*c^2 - 6*b^2*c + b^2 - 6*b*c^2 + 8*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2*$.9^2 + (-2*a^2*b*c - 2*a*b^2*c + 2*a*b*c^2 + 2*a*b*c - 2*b^2*c^2 + 2*b*c^2 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2 + (2*a^2*c - 4*a*b*c^2 + 2*a*b*c + 4*a*c^2 - 4*a*c - 4*b*c^3 + 6*b*c^2 - 2*b*c + 2*c^3 - 4*c^2 + 2*c)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9^3 + (-2*a^2*c + 4*a*b*c^2 - 2*a*b*c + 2*a*c + 4*b*c^3 - 2*b*c^2 - 2*c^3)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9 + (a^2*c^2 + 2*a*c^3 - 2*a*c^2 + c^4 - 2*c^3 + c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^4 + (-2*a^2*c^2 + 2*a*c^2 - 2*c^4 + 2*c^3 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^2 + (a^2*c^2 - 2*a*c^3 + c^4)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2) ] Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:21:12 2005 Input: K:=FunctionField(RationalField(),2); A:=Matrix(K,2,2,[a,1,0,b]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); print Transpose(A)*(A^(-1)); Output: Magma V2.11-10 Wed Dec 7 2005 10:21:11 on modular [Seed = 1055621328] ------------------------------------- [ $.1 - 1/b*$.3 - $.4, $.2 + a/b*$.3, $.3^2 + 1/a*$.3*$.4 + b/a*$.4^2 - b/a ] [ 1 -1/b] [ 1/a (a*b - 1)/(a*b)] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:19:43 2005 Input: K:=FunctionField(RationalField(),2); A:=Matrix(K,2,2,[a,1,0,b]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:19:42 on modular [Seed = 1005487548] ------------------------------------- [ $.1 - 1/b*$.3 - $.4, $.2 + a/b*$.3, $.3^2 + 1/a*$.3*$.4 + b/a*$.4^2 - b/a ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:16:52 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[0,1,-1,0]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:16:52 on modular [Seed = 771265216] ------------------------------------- [ -$.1*$.4 + $.2*$.3 + 1 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:16:06 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[1,0,0,1]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:16:06 on modular [Seed = 381921705] ------------------------------------- [ $.1^2 - $.4^2, $.1*$.2 + $.3*$.4, $.1*$.3 + $.2*$.4, $.1*$.4^2 - $.1 - $.2*$.3*$.4, $.2^2 + $.4^2 - 1, $.3^2 + $.4^2 - 1 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:14:12 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[1,1,0,1]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); B:=Matrix(K,2,2,[1,-1,1,0]); print B^3; Output: Magma V2.11-10 Wed Dec 7 2005 10:14:12 on modular [Seed = 331787358] ------------------------------------- [ $.1 - $.3 - $.4, $.2 + $.3, $.3^2 + $.3*$.4 + $.4^2 - 1 ] [-1 0] [ 0 -1] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:14:05 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[1,1,0,1]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); B:=Matrix(K,2,2,[1,-1,1,0]); print B^2; Output: Magma V2.11-10 Wed Dec 7 2005 10:14:04 on modular [Seed = 482976553] ------------------------------------- [ $.1 - $.3 - $.4, $.2 + $.3, $.3^2 + $.3*$.4 + $.4^2 - 1 ] [ 0 -1] [ 1 -1] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:12:37 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[1,1,0,1]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:12:37 on modular [Seed = 97566185] ------------------------------------- [ $.1 - $.3 - $.4, $.2 + $.3, $.3^2 + $.3*$.4 + $.4^2 - 1 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:12:27 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[1 1 0 1]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:12:27 on modular [Seed = 249280220] ------------------------------------- >> A:=Matrix(K,2,2,[1 1 0 1]); ^ User error: bad syntax >> I:=Ideal(Eltseq(P*A*Transpose(P)-A)); ^ User error: Identifier 'A' has not been declared or assigned >> print GroebnerBasis(I); ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 09:00:35 2005 Input: ps Output: Magma V2.11-10 Wed Dec 7 2005 09:00:34 on modular [Seed = 1184099918] ------------------------------------- >> ps; ^ User error: Identifier 'ps' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 09:00:24 2005 Input: who Output: Magma V2.11-10 Wed Dec 7 2005 09:00:24 on modular [Seed = 1301865663] ------------------------------------- >> who; ^ User error: Identifier 'who' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 07:23:32 2005 Input: G :=DirichletGroup(3675); G; X :=Elements(G); X; Y Output: Magma V2.11-10 Wed Dec 7 2005 07:23:29 on modular [Seed = 2893626342] ------------------------------------- Group of Dirichlet characters of modulus 3675 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] >> Y; ^ User error: Identifier 'Y' has not been declared or assigned Total time: 0.220 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 07:01:29 2005 Input: G :=DirichletGroup(1715); G; X :=Elements(G); X; Output: Magma V2.11-10 Wed Dec 7 2005 07:01:26 on modular [Seed = 4146232293] ------------------------------------- Group of Dirichlet characters of modulus 1715 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] Total time: 0.270 seconds, Total memory usage: 3.34MB '212.120' ************** MAGMA ***************** Host 212.120.88.54 (212.120.88.54) Time: Wed Dec 7 06:27:11 2005 Input: Q:=GaloisField(35098201); P:=PolynomialRing(Q,2); I:=ideal; Groebner(I); I; Output: Magma V2.11-10 Wed Dec 7 2005 06:27:10 on modular [Seed = 2588876432] ------------------------------------- Ideal of Polynomial ring of rank 2 over GF(35098201) Lexicographical Order Variables: x, y Dimension 0 Groebner basis: [ x + 33784728*y^33 + 15744019*y^32 + 14466235*y^31 + 14937582*y^30 + 9988153*y^29 + 24849537*y^28 + 13827463*y^27 + 10851940*y^26 + 25333828*y^25 + 29238403*y^24 + 35087366*y^23 + 3185785*y^22 + 12125255*y^21 + 11305600*y^20 + 713800*y^19 + 11882241*y^18 + 23388419*y^17 + 12677392*y^16 + 20159861*y^15 + 31143912*y^14 + 33062327*y^13 + 11580434*y^12 + 10629964*y^11 + 14094725*y^10 + 30606411*y^9 + 20913610*y^8 + 23355486*y^7 + 32139384*y^6 + 35026862*y^5 + 11038274*y^4 + 26690476*y^3 + 752845*y^2 + 9514380*y + 16409093, y^34 + 34*y^33 + 561*y^32 + 5984*y^31 + 46376*y^30 + 278256*y^29 + 1344904*y^28 + 5379616*y^27 + 18156204*y^26 + 17353055*y^25 + 25833537*y^24 + 5312152*y^23 + 21881025*y^22 + 15430534*y^21 + 23145801*y^20 + 30861068*y^19 + 27872968*y^18 + 17124952*y^17 + 27873223*y^16 + 30858484*y^15 + 23140973*y^14 + 15452362*y^13 + 21951303*y^12 + 5362540*y^11 + 25762851*y^10 + 17184126*y^9 + 18014271*y^8 + 5332492*y^7 + 1358334*y^6 + 297330*y^5 + 54824*y^4 + 7259*y^3 + 714*y^2 + 34*y + 1 ] Total time: 0.200 seconds, Total memory usage: 3.43MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 05:47:41 2005 Input: G :=DirichletGroup(2205); G; X :=Elements(G); X; YqEigenform(D[2],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 05:47:40 on modular [Seed = 4214097202] ------------------------------------- Group of Dirichlet characters of modulus 2205 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] >> YqEigenform(D[2],12);Parent($1);; ^ User error: Identifier 'D' has not been declared or assigned Set of sequences over Group of Dirichlet characters of modulus 2205 over Rational Field Total time: 0.200 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 04:17:20 2005 Input: G :=DirichletGroup(735,CyclotomicField(3)); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Wed Dec 7 2005 04:17:15 on modular [Seed = 1840809715] ------------------------------------- Group of Dirichlet characters of modulus 735 over Cyclotomic Field of order 3 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 15 2 Current total memory usage: 94.3MB, failed memory request: 6.9MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1);; ^ User error: Identifier 'D' has not been declared or assigned Integer Ring Total time: 4.540 seconds, Total memory usage: 94.31MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 04:16:58 2005 Input: G :=DirichletGroup(735,CyclotomicField(2)); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Wed Dec 7 2005 04:16:53 on modular [Seed = 1519366618] ------------------------------------- Group of Dirichlet characters of modulus 735 over Cyclotomic Field of order 2 and degree 1 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 15 2 Current total memory usage: 94.4MB, failed memory request: 6.9MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1);; ^ User error: Identifier 'D' has not been declared or assigned Integer Ring Total time: 4.750 seconds, Total memory usage: 94.44MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 04:16:32 2005 Input: G :=DirichletGroup(735,CyclotomicField(3)); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Wed Dec 7 2005 04:16:28 on modular [Seed = 1602925622] ------------------------------------- Group of Dirichlet characters of modulus 735 over Cyclotomic Field of order 3 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 15 2 Current total memory usage: 94.3MB, failed memory request: 6.9MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1);; ^ User error: Identifier 'D' has not been declared or assigned Integer Ring Total time: 4.540 seconds, Total memory usage: 94.31MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 02:50:58 2005 Input: G :=DirichletGroup(945); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Wed Dec 7 2005 02:50:37 on modular [Seed = 2321505729] ------------------------------------- Group of Dirichlet characters of modulus 945 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 Errors: /bin/sh: line 1: 17804 Alarm clock nice -n 19 /usr/local/bin/magma '71.107.' ************** MAGMA ***************** Host 71.107.113.52 (71.107.113.52) Time: Wed Dec 7 02:39:50 2005 Input: 2*5 Output: Magma V2.11-10 Wed Dec 7 2005 02:39:50 on modular [Seed = 2141892031] ------------------------------------- 10 Total time: 0.190 seconds, Total memory usage: 3.24MB '71.107.' ************** MAGMA ***************** Host 71.107.113.52 (71.107.113.52) Time: Wed Dec 7 02:39:45 2005 Input: 2*5 Output: Magma V2.11-10 Wed Dec 7 2005 02:39:45 on modular [Seed = 1757264559] ------------------------------------- 10 Total time: 0.200 seconds, Total memory usage: 3.24MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:51:38 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Norm(Coefficient(qEigenform(D[5],12),11));Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 22:51:37 on modular [Seed = 670185583] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 84 2 [ Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 24 over Rational Field ] 18446744073709551616 Rational Field Total time: 0.770 seconds, Total memory usage: 5.66MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:50:53 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[5],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 22:50:53 on modular [Seed = 552943420] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 84 2 [ Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 24 over Rational Field ] q + (-3154406192093409505440711462312941399852135706366107785045295182763186079\ 28428971344315339315045192045991571277525409859827663237/189995137843409027\ 617487587962791777330586787708604501827307026332673400350064776414910901612\ 630255870623517588050261987626733147108813190983697047190466795042504704*a^\ 23 + 2046713682813363551390719098081181580173919339183486331065409475571468\ 3675074421826010030637789674368851790787507432663/2808359505892003487982033\ 118596455881223287644722412087617624775237881768519330067813745105864690267\ 99198059093922373442235037760563158855880380306685952*a^22 + 806085296019685616144143947305046259006184635473299530724479575153186911398\ 1703355075673148713595769683513371437405201356793596757/1979116019202177371\ 015495707945747680526945705297963560701114857632014586979841420988655225131\ 565165318994974875523562371111803615716804072746844241567362448359424*a^21 - 594405873337565451457161013922161608814448343984476963661369465357742143789\ 108061944860489619859248937016028710929714103/29253744853041702999812844985\ 379748762742579632525125912683591408727935088743021539726511519423856958249\ 79782228358056689948310005866238082087294861312*a^20 - 127078750369517880433428667565193737420182227514108638612875763685033102954\ 84450452511051459387657914085644406883824703311612957974143/131941067946811\ 824734366380529716512035129713686530904046740990508800972465322761399243681\ 6754376776879329983250349041580741202410477869381831229494378241632239616*a\ ^19 + 115897702670860128055252011542100564997697371756860793042945494076141\ 7293168266372740607310102015816304808280296702173753557/1950249656869446866\ 654189665691983250849505308835008394178906093915195672582868102648434101294\ 923797216653188152238704459965540003910825388058196574208*a^18 - 801499971193552088123081136029973976441096960863138512484575007977890572381\ 15071165551896087726591097773005912938438963324363775295845/183251483259460\ 867686619972957939600048791269009070700064918042373334683979614946387838446\ 77144121901101805322921514466399183366812192630303211520755253356003328*a^1\ 7 - 93330274076902537849306929164657193536041977241136632494472590444585539\ 94399609467017138279145038758128068431535279159280137/270868007898534287035\ 304120234997673729097959560417832524848068599332732303176125367838069624294\ 97183564627613225537561943965833387650352611919396864*a^16 - 160919378681959651143345172158986210922269327431250461763250808566962073817\ 32467735601535570410881248587594453528434391417688187643051238151/593734805\ 760653211304648712383724304158083711589389068210334457289604376093952426296\ 596567539469549595698492462657068711333541084715041221824053272470208734507\ 8272*a^15 + 930823436402097914986428110859561869946661959994923991858824647\ 9834075435528806342021349063684299269281970919942420846948297780909/8776123\ 455912510899943853495613924628822773889757537773805077422618380526622906461\ 917953455827157087474939346685074170069844930017598714246261884583936*a^14 - 474400382573859373647744547665029784867702395709311220243662874342868577144\ 5285136571120498923793361328837341169135487875556688018469551624565/1236947\ 512001360856884684817466092300329341065811227225438196786020009116862400888\ 117909515707228228324371859297202226481944877259823002545466777650979601530\ 22464*a^13 + 17988148604197499466251189088797157487150628007331167213079988\ 6307322737443992720470915872459364480678344919118947877240625593563463/1828\ 359053315106437488302811586234297671411227032820369542724463045495943046438\ 84623290696996399105989061236389272378543121769375366639880130455928832*a^1\ 2 - 12077424639649578807401618997354726212754990341571861908738338954117449\ 72720643252948983280401574685353079380400382511684554800633233480797161501/\ 274877224889191301529929959436909400073186903513606050097377063560002025969\ 422419581757670157161828516527079843822716995987750502182889454548172811328\ 80034004992*a^11 + 70284478037285411782526662558559283016529395011013468253\ 869939853103932274198592143503509473048061476405720062849997393995361863252\ 831/40630201184780143055295618035249651059364693934062674878727210289899909\ 845476418805175710443644245775346941419838306342915948750081475528917879095\ 296*a^10 - 2275304208864881464093247236199218006147517499081666579033912627\ 397424583384308713264189478116631015764793505153774432667065177337510840579\ 90943591/103078959333446738073723734788841025027445088817602268786516398835\ 000759738533407343159126308935685693697654941433518873495406438318583545455\ 56480424830012751872*a^9 + 403624905434348741317428949808993986230233383536\ 112752648511952539299938277142816228039602617489234439902591785861675654751\ 84356950263245/152363254442925536457358567632186191472617602252735030795227\ 038587124661920536570519408914163665921657551030324393648785934807812805533\ 23344204660736*a^8 - 536863438013163848909686699813041801759222656207768838\ 267880930401537559768267360633888692139739679973718735231373760309391297820\ 07148112131983017571925/742168507200816514130810890479655380197604639486736\ 335262918071612005470117440532870745709424336936994623115578321335889166926\ 355893801527280066590587760918134784*a^7 + 242618769363091951561890266658236970786051566545089358542446419991784632475\ 9615479435587616221016583062314751340274420889162350986348673460167/1097015\ 431989063862492981686951740578602846736219692221725634677827297565827863307\ 739744181978394635934367418335634271258730616252199839280782735572992*a^6 - 815956106990515446621243672771862838230476458486154599474863160605389301281\ 596951288384298500076971815038264402537613999569586869147708263122822732015\ 3/2147478319446807043202577808100854688071772683700047266385758309062515827\ 886112652982481798102826785285367811279864976531154300798303823863657600088\ 50625265664*a^5 + 975751497144935641984257969495268233146482100189621161871\ 354807426121472387178178979731347495347495642696083323138270981563010814969\ 727776757/95227034026828460285849104770116369670386001407959394247016899116\ 952913700335356574630571352291201035969393952746030491209254883003458270901\ 2791296*a^4 - 4214282065319867674104287233039127493846095089120457888384077\ 344530577620215203451054009473787984621124170460749704387807920726244007592\ 9249811447833429/2236956582757090670002685216771723633408096545520882569151\ 831571940120654048034013523418539690444568005591470083192683886619063331566\ 483191310000092194013184*a^3 + 55633661622414685684625703326627082510787615\ 166410840928619784281858668663122204801269538594403523438448295127656780573\ 08196809025593323578383/330649423704265487103642724896237394688840282666525\ 674468808677489419839237275543661911706084344448041560395669257050316698801\ 6770953412184072192*a^2 - 9101952422759966972741079240584783508003137941846\ 924367312699275266145545290771848082338986500695633276654813330189767231611\ 35230292180453657443745361/606813309124644821506804800556565655763915078537\ 565801093704310964659465616328671203184282685124937067489005556421626488340\ 67473187024826153157554584256*a - 35192976472321009415300881048173611070273\ 254363617242903149999763674532848021383532918588950585547163810834389696410\ 705627955328530894755797/29898132207055255995338064678840910254705610050141\ 572127170923528774218681032583158086634303054872688943178138496188722213071\ 621554483255426)*q^2 + (627357858570778026301225016805693743264686948371932\ 321960948998522285788281795776772741615744531099314194630611362594028543/24\ 789857418426363455665403343370682139538163921238851899415376828487322017681\ 546396936563673651931777276544672985684441035157174854510970806654303638346\ 79296*a^22 - 15000640249084634727201454165311500627583747217900570190425267\ 315204987672052824682556923615126109025694218272440859805934647/25822768144\ 194128599651461816011127228685587417957137395224350863007627101751610830142\ 253826720762267996400701026754626078288723806782261256931566289944576*a^20 + 261243656123494266218453149588110177061989661234056624486624459104178885424\ 04887623205088104918756819355333376244859688263085101/172151787627960857331\ 009745440074181524570582786380915968162339086717514011677405534281692178138\ 41511997600467351169750718859149204521507504621044193296384*a^18 + 149350881659913989744502200475655919197936739705278835234382217831881625963\ 515709098691901658277964490975352277315632696968509719/23909970503883452407\ 084686866676969656190358720330682773355880428710765834955195213094679469185\ 8909888855562046544024315539710405618354270897514502684672*a^16 + 739444599610863832847240003865105196326492376110116060569125492543687683824\ 509741170623047203201900 ** WARNING: Output too long, hence truncated. '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:50:36 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[6],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 22:50:35 on modular [Seed = 499646999] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 84 2 [ Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 24 over Rational Field ] >> qEigenform(D[6],12);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Set of sequences over Power Structure of ModSym Total time: 0.470 seconds, Total memory usage: 4.61MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:50:11 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[6],12);Parent)$1); Output: Magma V2.11-10 Tue Dec 6 2005 22:50:10 on modular [Seed = 381882085] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 84 2 [ Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 24 over Rational Field ] >> qEigenform(D[6],12);Parent)$1); ^ Runtime error in '[]': Sequence element 6 not defined >> qEigenform(D[6],12);Parent)$1); ^ User error: bad syntax Total time: 0.470 seconds, Total memory usage: 4.61MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:48:09 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:48:08 on modular [Seed = 232265076] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 84 2 [ Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 24 over Rational Field ] Total time: 0.470 seconds, Total memory usage: 4.61MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:47:59 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:47:58 on modular [Seed = 114499177] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 21 2 [] Total time: 0.220 seconds, Total memory usage: 4.37MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:47:48 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:47:48 on modular [Seed = 13965413] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 28 2 [] Total time: 0.210 seconds, Total memory usage: 4.36MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:47:36 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:47:35 on modular [Seed = 4264289963] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 84, weight 3, character $.3, and dimension 2 over Rational Field ] Total time: 0.780 seconds, Total memory usage: 4.65MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:47:28 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:47:27 on modular [Seed = 4112575928] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 12 2 [] Total time: 0.230 seconds, Total memory usage: 4.43MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:43:42 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:43:41 on modular [Seed = 3896373639] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 84, weight 3, character $.2, and dimension 4 over Rational Field ] Total time: 0.780 seconds, Total memory usage: 4.70MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:43:34 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:43:33 on modular [Seed = 3862167425] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 84, weight 3, character $.2, and dimension 4 over Rational Field ] Total time: 0.790 seconds, Total memory usage: 4.70MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:43:23 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:43:22 on modular [Seed = 3708339333] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 4 2 [ Modular symbols space of level 84, weight 3, character $.1, and dimension 12 over Rational Field ] Total time: 0.750 seconds, Total memory usage: 4.72MB '213.78.' ************** MAGMA ***************** Host 213.78.42.15 (213.78.42.15) Time: Tue Dec 6 22:20:13 2005 Input: 12+2 Output: Magma V2.11-10 Tue Dec 6 2005 22:20:12 on modular [Seed = 2976163423] ------------------------------------- 14 Total time: 0.190 seconds, Total memory usage: 3.24MB '213.78.' ************** MAGMA ***************** Host 213.78.42.15 (213.78.42.15) Time: Tue Dec 6 22:08:27 2005 Input: 10^2 Output: Magma V2.11-10 Tue Dec 6 2005 22:08:26 on modular [Seed = 1824095556] ------------------------------------- 100 Total time: 0.190 seconds, Total memory usage: 3.24MB '213.78.' ************** MAGMA ***************** Host 213.78.42.15 (213.78.42.15) Time: Tue Dec 6 22:08:12 2005 Input: 10^2 Output: Magma V2.11-10 Tue Dec 6 2005 22:08:12 on modular [Seed = 1234982999] ------------------------------------- 100 Total time: 0.190 seconds, Total memory usage: 3.24MB '213.78.' ************** MAGMA ***************** Host 213.78.42.15 (213.78.42.15) Time: Tue Dec 6 22:08:06 2005 Input: 10^2 Output: Magma V2.11-10 Tue Dec 6 2005 22:08:05 on modular [Seed = 1150375251] ------------------------------------- 100 Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:40:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:40:57 on modular [Seed = 3556891758] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.240 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:40:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:40:46 on modular [Seed = 3506755964] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:40:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:40:38 on modular [Seed = 3457926763] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.240 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:40:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:40:21 on modular [Seed = 3407790941] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:40:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:40:11 on modular [Seed = 3222653000] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:21:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:21:20 on modular [Seed = 2877206408] ------------------------------------- Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x - 6, 0, 1) @} Total time: 0.470 seconds, Total memory usage: 38.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:21:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:21:06 on modular [Seed = 2692064433] ------------------------------------- Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 0.310 seconds, Total memory usage: 38.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:20:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:20:25 on modular [Seed = 2638780535] ------------------------------------- [] >> P:=J!B[1];//Q:=J![x+2,0]; ^ Runtime error in '[]': Sequence element 1 not defined >> Chabauty(P,11); ^ User error: Identifier 'P' has not been declared or assigned Total time: 0.660 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:11:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:11:33 on modular [Seed = 2537721921] ------------------------------------- [ (x - 10, 316, 1) ] {@ <10, 1, 5, 1> @} Total time: 1.139 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:11:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:11:22 on modular [Seed = 2354678597] ------------------------------------- [ (x - 10, 316, 1) ] {@ <10, 1, 4, 1>, <381, 1, 4, 1> @} Total time: 1.179 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:10:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:10:56 on modular [Seed = 2304544314] ------------------------------------- [] >> P:=J!B[1];//Q:=J![x+2,0]; ^ Runtime error in '[]': Sequence element 1 not defined >> Chabauty(P,7); ^ User error: Identifier 'P' has not been declared or assigned Total time: 0.670 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:12:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:12:11 on modular [Seed = 670192335] ------------------------------------- [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] {@ @} Total time: 1.209 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:12:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:12:04 on modular [Seed = 499673338] ------------------------------------- [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] {@ <23738, 1, 4, 1>, <66775, 1, 4, 1>, <79574, 1, 4, 1>, <44999, 1, 4, 1>, <0, 1, 4, 1>, <49648, 1, 4, 1>, <7813, 1, 4, 1>, <44823, 1, 4, 1>, <25980, 1, 4, 1> @} Total time: 2.169 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:11:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:11:55 on modular [Seed = 449802178] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 162 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 162 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 162, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (661695299665*2^3 + O(2^53))*$.1^2 +... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] {@ <23738, 1, 4, 1>, <66775, 1, 4, 1>, <79574, 1, 4, 1>, <44999, 1, 4, 1>, <0, 1, 4, 1>, <49648, 1, 4, 1>, <7813, 1, 4, 1>, <44823, 1, 4, 1>, <25980, 1, 4, 1> @} Total time: 2.569 seconds, Total memory usage: 39.03MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:40 on modular [Seed = 398615172] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ @} Total time: 1.980 seconds, Total memory usage: 39.00MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:31 on modular [Seed = 215579016] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <274218, 1, 4, 1>, <271994, 1, 4, 1>, <34871, 1, 4, 1>, <0, 1, 4, 1>, <119449, 1, 4, 1>, <222506, 1, 4, 1>, <276781, 1, 4, 1>, <208779, 1, 4, 1>, <256392, 1, 4, 1>, <2850, 1, 4, 1> @} Total time: 3.290 seconds, Total memory usage: 38.97MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:24 on modular [Seed = 165444746] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <100876, 1, 4, 1>, <25523, 1, 4, 1> @} Total time: 2.060 seconds, Total memory usage: 38.96MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:16 on modular [Seed = 114520957] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <61454, 1, 4, 1>, <52643, 1, 4, 1>, <71747, 1, 4, 1>, <38993, 1, 4, 1>, <54970, 1, 4, 1>, <29585, 1, 4, 1>, <51208, 1, 4, 1>, <33843, 1, 4, 1> @} Total time: 2.790 seconds, Total memory usage: 38.97MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:08 on modular [Seed = 64386672] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <27266, 1, 4, 1>, <8341, 1, 4, 1>, <14210, 1, 4, 1>, <15232, 1, 4, 1>, <0, 1, 4, 1>, <7733, 1, 4, 1> @} Total time: 2.370 seconds, Total memory usage: 38.98MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:02 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:00 on modular [Seed = 2270360211] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <0, 1, 4, 1> @} Total time: 1.879 seconds, Total memory usage: 38.97MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:09:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:09:37 on modular [Seed = 2321547175] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <1458, 1, 4, 1>, <1350, 1, 4, 1>, <607, 1, 4, 1> @} Total time: 2.020 seconds, Total memory usage: 39.69MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:00:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:58 on modular [Seed = 3812005009] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 2.069 seconds, Total memory usage: 40.90MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:56 on modular [Seed = 3862139286] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 10368... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 10368 over Ration..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 10368, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: (1 + O(2^44))*$.1^3 + O(2^50)*$.1^2 - (3053654638673*2^3 + O... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 1.149 seconds, Total memory usage: 39.75MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:52 on modular [Seed = 3913064093] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 10368... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 10368 over Ration..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 10368, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (511427965222993*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 1.159 seconds, Total memory usage: 39.74MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:38 on modular [Seed = 3963198570] ------------------------------------- 3 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 2, 16, 1), (x + 2, -16, 1), (x - 1, 17, 1), (x - 1, -17, 1), (x^2 - 8, 2*x + 16, 2), (x^2 - 8, -2*x - 16, 2) @} Total time: 0.760 seconds, Total memory usage: 39.02MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:33 on modular [Seed = 4146235262] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 2, 8, 1), (x + 2, -8, 1) @} Total time: 0.770 seconds, Total memory usage: 39.02MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:20 on modular [Seed = 4197419630] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 0.700 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:08 on modular [Seed = 4247293277] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 2, 4, 1), (x + 2, -4, 1), (x - 1, 7, 1), (x - 1, -7, 1), (x^2 + x - 2, x + 6, 2), (x^2 + x - 2, -x - 6, 2) @} Total time: 0.810 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:54 on modular [Seed = 13988018] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 0.790 seconds, Total memory usage: 39.44MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:45 on modular [Seed = 199130034] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x^2 - 2, x + 2, 2), (x^2 - 2, -x - 2, 2) @} Total time: 0.690 seconds, Total memory usage: 39.17MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:42 on modular [Seed = 249265840] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 6 ove... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 6 over Rational F..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 6, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (600047619*2^3 + O(2^53))*$.1^2 + O(... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x^2 - 2, x + 2, 2), (x^2 - 2, -x - 2, 2) @} Total time: 0.630 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:31 on modular [Seed = 298082778] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 1, 1, 1), (x + 1, -1, 1), (x^2 + 2, x + 2, 2), (x^2 + 2, -x - 2, 2), (x^2 - x - 1, x + 2, 2), (x^2 - x - 1, -x - 2, 2) @} Total time: 0.610 seconds, Total memory usage: 39.19MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:28 on modular [Seed = 348218588] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 2 ove... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 2 over Rational F..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 2, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (8257011711*2^3 + O(2^53))*$.1^2 + O... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 1, 1, 1), (x + 1, -1, 1), (x^2 + 2, x + 2, 2), (x^2 + 2, -x - 2, 2), (x^2 - x - 1, x + 2, 2), (x^2 - x - 1, -x - 2, 2) @} Total time: 0.640 seconds, Total memory usage: 39.56MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:24 on modular [Seed = 533360595] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 2 ove... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 2 over Rational F..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 2, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreePart( F: $.1^4 + O(2^50)*$.1^3 + (4239300493311*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/loclib.m", line 120, column 5: >> assert R eq 0; ^ Runtime error in assert: Assertion failed >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 1, 1, 1), (x + 1, -1, 1), (x^2 + 2, x + 2, 2), (x^2 + 2, -x - 2, 2), (x^2 - x - 1, x + 2, 2), (x^2 - x - 1, -x - 2, 2) @} Total time: 0.460 seconds, Total memory usage: 39.18MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:57:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:57:14 on modular [Seed = 586633780] ------------------------------------- 2 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x^2, 9, 2), (x^2, -9, 2), (x + 2, 7, 1), (x + 2, -7, 1), (x, 9, 1), (x, -9, 1), (x - 3, 18, 1), (x - 3, -18, 1), (x^2 + 2*x, x + 9, 2), (x^2 + 2*x, -x - 9, 2), (x^2 + 2*x + 6, 2*x - 3, 2), (x^2 + 2*x + 6, -2*x + 3, 2) @} Total time: 0.680 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:39:48 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:39:46 on modular [Seed = 804930305] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 1.360 seconds, Total memory usage: 40.01MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:28:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:28:36 on modular [Seed = 1100230281] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x - 1, 2, 1), (x - 1, -2, 1) @} Total time: 0.630 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:28:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); Order(P1);Order(P2); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:28:19 on modular [Seed = 1150364571] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> P1:=J![x+1,0];P2:=J![x,1]; ^ Runtime error in '!': Points specified by second polynomial are not on the curve >> P1:=J![x+1,0];P2:=J![x,1]; ^ Runtime error in '!': Points specified by second polynomial are not on the curve {@ (1, 0, 0), (x - 1, 2, 1), (x - 1, -2, 1) @} >> Order(P1);Order(P2); ^ Runtime error in 'Order': Bad argument types Argument types given: Intrinsic >> Order(P1);Order(P2); ^ Runtime error in 'Order': Bad argument types Argument types given: Intrinsic Total time: 0.620 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:15:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); Order(P1);Order(P2); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:15:42 on modular [Seed = 1468153221] ------------------------------------- 0 Abelian Group isomorphic to Z/10 Defined on 1 generator Relations: 10*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/10 Defined on 1 generator Relations: 10*P[1] = 0 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x^2, 1, 2), (x^2, -1, 2), (x + 1, 0, 1), (x, 1, 1), (x, -1, 1), (x^2 + x, x + 1, 2), (x^2 + x, -x - 1, 2), (x^2 - 2*x + 2, 2*x - 3, 2), (x^2 - 2*x + 2, -2*x + 3, 2) @} 2 5 Total time: 0.610 seconds, Total memory usage: 39.65MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:14:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:14:29 on modular [Seed = 1519333443] ------------------------------------- 0 Abelian Group isomorphic to Z/10 Defined on 1 generator Relations: 10*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/10 Defined on 1 generator Relations: 10*P[1] = 0 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x^2, 1, 2), (x^2, -1, 2), (x + 1, 0, 1), (x, 1, 1), (x, -1, 1), (x^2 + x, x + 1, 2), (x^2 + x, -x - 1, 2), (x^2 - 2*x + 2, 2*x - 3, 2), (x^2 - 2*x + 2, -2*x + 3, 2) @} Total time: 0.610 seconds, Total memory usage: 39.64MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:45:14 2005 Input: 5*0.5; Output: Magma V2.11-10 Tue Dec 6 2005 16:45:14 on modular [Seed = 788228921] ------------------------------------- 2.500000000000000000000000000000000000000 Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:44:10 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Tue Dec 6 2005 16:44:10 on modular [Seed = 870741604] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.180 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:43:21 2005 Input: 174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009*0.5; Output: Magma V2.11-10 Tue Dec 6 2005 16:43:21 on modular [Seed = 887190192] ------------------------------------- >> 88974928763633474431825 923127812764804103556846639940324358013520109641 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:42:45 2005 Input: 174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009/2; Output: Magma V2.11-10 Tue Dec 6 2005 16:42:45 on modular [Seed = 1133905387] ------------------------------------- >> 88974928763633474431825 923127812764804103556846639940324358013520109641 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:42:17 2005 Input: 174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009*0.5; Output: Magma V2.11-10 Tue Dec 6 2005 16:42:17 on modular [Seed = 1150354048] ------------------------------------- >> 88974928763633474431825 923127812764804103556846639940324358013520109641 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:41:34 2005 Input: 2*3; Output: Magma V2.11-10 Tue Dec 6 2005 16:41:34 on modular [Seed = 1318783165] ------------------------------------- 6 Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:27:48 2005 Input: 0.5*174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009 ; Output: Magma V2.11-10 Tue Dec 6 2005 16:27:48 on modular [Seed = 1824075077] ------------------------------------- >> 917788974928763633474431825 92312781276480410355684663994032435801352010 ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:27:21 2005 Input: 0.5*174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009; Output: Magma V2.11-10 Tue Dec 6 2005 16:27:21 on modular [Seed = 1906585653] ------------------------------------- >> 917788974928763633474431825 92312781276480410355684663994032435801352010 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:27:17 2005 Input: 0.5*174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009; Output: Magma V2.11-10 Tue Dec 6 2005 16:27:17 on modular [Seed = 2058304306] ------------------------------------- >> 917788974928763633474431825 92312781276480410355684663994032435801352010 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:27:11 2005 Input: 0.5*174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009 Output: Magma V2.11-10 Tue Dec 6 2005 16:27:11 on modular [Seed = 2141863469] ------------------------------------- >> 917788974928763633474431825 92312781276480410355684663994032435801352010 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 15:54:15 2005 Input: A := matrix[1,2,3,4]; Output: Magma V2.11-10 Tue Dec 6 2005 15:54:14 on modular [Seed = 3691668948] ------------------------------------- >> A := matrix[1,2,3,4]; ^ User error: Identifier 'matrix' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:52:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:52:47 on modular [Seed = 3845348484] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ @} Total time: 2.319 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:52:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:52:38 on modular [Seed = 3896398208] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ <74649, 1, 4, 1>, <249979, 1, 4, 1>, <36233, 1, 4, 1>, <33970, 1, 4, 1>, <158613, 1, 4, 1>, <0, 1, 4, 1>, <87568, 1, 4, 1>, <200366, 1, 4, 1>, <103472, 1, 4, 1>, <107491, 1, 4, 1>, <129032, 1, 4, 1> @} Total time: 3.740 seconds, Total memory usage: 39.46MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:52:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:52:29 on modular [Seed = 4013512348] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ <22706, 1, 4, 1>, <26230, 1, 4, 1> @} Total time: 2.399 seconds, Total memory usage: 39.29MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:52:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:52:20 on modular [Seed = 4062464904] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ <4933, 1, 4, 1>, <76884, 1, 4, 1>, <12222, 1, 4, 1>, <78542, 1, 4, 1>, <74569, 1, 4, 1>, <2337, 1, 4, 1>, <58863, 1, 4, 1> @} Total time: 3.140 seconds, Total memory usage: 39.31MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:52:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:52:10 on modular [Seed = 4180624253] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ <11470, 1, 4, 1>, <20108, 1, 4, 1>, <17396, 1, 4, 1>, <24070, 1, 4, 1>, <18897, 1, 4, 1> @} Total time: 2.710 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:51:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:51:55 on modular [Seed = 4280894826] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ <10632, 1, 4, 1>, <4110, 1, 4, 1>, <1421, 1, 4, 1>, <5279, 1, 4, 1> @} Total time: 2.600 seconds, Total memory usage: 39.28MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 15:51:33 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to 2 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DD:=Points(Jac)[ii]; if HasOrder(DD,8) then count:=count+1; //print DD, "has order 8, a=", a," b= ", b; //2*DD; //(F1.1^2*x^5+F1.1*x^3+x-x*(F1.1^2*x+F1.1^29)-(F1.1^2*x+F1.1^29)^2)/DD[1]^2; end if; end for; count; end if; end if; end for; end for; print "done"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 6 2005 15:51:12 on modular [Seed = 47495357] ------------------------------------- 4 Errors: /bin/sh: line 1: 15740 Alarm clock nice -n 19 /usr/local/bin/magma '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 15:50:34 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to 1 do a:=F1.1^i; for j:=1 to 2 do //#Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DD:=Points(Jac)[ii]; if HasOrder(DD,8) then count:=count+1; //print DD, "has order 8, a=", a," b= ", b; //2*DD; //(F1.1^2*x^5+F1.1*x^3+x-x*(F1.1^2*x+F1.1^29)-(F1.1^2*x+F1.1^29)^2)/DD[1]^2; end if; end for; count; end if; end if; end for; end for; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 15:50:34 on modular [Seed = 165655533] ------------------------------------- done Total time: 0.220 seconds, Total memory usage: 3.43MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 15:50:07 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to 1 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DD:=Points(Jac)[ii]; if HasOrder(DD,8) then count:=count+1; //print DD, "has order 8, a=", a," b= ", b; //2*DD; //(F1.1^2*x^5+F1.1*x^3+x-x*(F1.1^2*x+F1.1^29)-(F1.1^2*x+F1.1^29)^2)/DD[1]^2; end if; end for; count; end if; end if; end for; end for; print "done"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 6 2005 15:49:47 on modular [Seed = 215660057] ------------------------------------- 4 Errors: /bin/sh: line 1: 15730 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:42:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:42:44 on modular [Seed = 314877608] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 96, 20*x - 48, 2) ] 2.808141398815367927531209718024871984466 10.484172230478017379230331171078121089143 72128.720041723759240639205711309210490074464 {@ @} Total time: 2.049 seconds, Total memory usage: 39.63MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:42:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:42:35 on modular [Seed = 433040860] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 96, 20*x - 48, 2) ] 2.808141398815367927531209718024871984466 10.484172230478017379230331171078121089143 72128.720041723759240639205711309210490074464 {@ <12, 1, 4, 1> @} Total time: 2.040 seconds, Total memory usage: 39.62MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:38:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:37:59 on modular [Seed = 771463412] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 96, 20*x - 48, 2) ] 2.808141398815367927531209718024871984466 10.484172230478017379230331171078121089143 72128.720041723759240639205711309210490074464 Total time: 1.700 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:37:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:37:16 on modular [Seed = 820413746] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <0, 1, 4, 1> @} Total time: 3.930 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:37:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:37:05 on modular [Seed = 937525816] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <261417, 1, 4, 1>, <44441, 1, 4, 1>, <0, 1, 4, 1>, <35100, 1, 4, 1>, <31761, 1, 4, 1>, <181934, 1, 4, 1>, <115431, 1, 4, 1>, <194062, 1, 4, 1>, <154635, 1, 4, 1>, <60434, 1, 4, 1> @} Total time: 5.070 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:36:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:36:52 on modular [Seed = 1038845991] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <51030, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 3.930 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:36:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:36:38 on modular [Seed = 1083635048] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <7881, 1, 4, 1>, <0, 1, 4, 1>, <42540, 1, 4, 1>, <2153, 1, 4, 1>, <21258, 1, 4, 1>, <30750, 1, 4, 1>, <71880, 1, 4, 1> @} Total time: 4.589 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:36:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:36:27 on modular [Seed = 1200747122] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <21308, 1, 4, 1>, <0, 1, 4, 1>, <24889, 1, 4, 1>, <9389, 1, 4, 1>, <3447, 1, 4, 1>, <2598, 1, 4, 1> @} Total time: 4.160 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:36:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:36:17 on modular [Seed = 1251802997] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <5942, 1, 4, 1>, <5847, 1, 4, 1>, <0, 1, 4, 1>, <6498, 1, 4, 1>, <6885, 1, 4, 1>, <4110, 1, 4, 1> @} Total time: 4.160 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:36:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:36:00 on modular [Seed = 1367860255] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <1912, 1, 4, 1>, <0, 1, 4, 1>, <943, 1, 4, 1>, <0, 1, 0, 6> @} Total time: 6.250 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:35:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=11000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 6 2005 15:35:16 on modular [Seed = 1502604323] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] Errors: /bin/sh: line 1: 15655 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:32:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:32:56 on modular [Seed = 1552607834] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 Total time: 3.350 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:28:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:28:13 on modular [Seed = 1656018241] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 Total time: 3.359 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:55 on modular [Seed = 1757339417] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ @} Total time: 2.220 seconds, Total memory usage: 39.60MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:44 on modular [Seed = 1874450453] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <114991, 1, 4, 1>, <70244, 1, 4, 1>, <195507, 1, 4, 1>, <13960, 1, 4, 1>, <138074, 1, 4, 1>, <29547, 1, 4, 1>, <0, 1, 4, 1>, <82450, 1, 4, 1>, <202201, 1, 4, 1>, <148362, 1, 4, 1>, <232543, 1, 4, 1> @} Total time: 3.600 seconds, Total memory usage: 39.57MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:35 on modular [Seed = 1923400439] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <128766, 1, 4, 1>, <2520, 1, 4, 1> @} Total time: 2.310 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:26 on modular [Seed = 2041563637] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <33619, 1, 4, 1>, <15634, 1, 4, 1>, <79085, 1, 4, 1>, <41020, 1, 4, 1>, <22301, 1, 4, 1>, <71556, 1, 4, 1>, <57819, 1, 4, 1>, <73062, 1, 4, 1> @} Total time: 3.089 seconds, Total memory usage: 39.59MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:19 on modular [Seed = 2091568361] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <5116, 1, 4, 1>, <3542, 1, 4, 1>, <20049, 1, 4, 1>, <0, 1, 4, 1>, <3726, 1, 4, 1>, <22, 1, 4, 1> @} Total time: 2.549 seconds, Total memory usage: 39.56MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:12 on modular [Seed = 2186555929] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <0, 1, 4, 1> @} Total time: 1.940 seconds, Total memory usage: 39.53MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:04 on modular [Seed = 2270246692] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <475, 1, 4, 1>, <2015, 1, 4, 1>, <488, 1, 4, 1>, <2354, 1, 4, 1> @} Total time: 2.200 seconds, Total memory usage: 39.57MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:24:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:24:38 on modular [Seed = 2354723957] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 Total time: 1.669 seconds, Total memory usage: 39.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:23:52 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:23:50 on modular [Seed = 2437358008] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 Total time: 1.659 seconds, Total memory usage: 39.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:22:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:22:39 on modular [Seed = 2555259112] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ @} Total time: 1.540 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:22:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:22:27 on modular [Seed = 2638949857] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <392168, 1, 4, 1> @} Total time: 1.649 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:22:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:22:18 on modular [Seed = 2725516071] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <51383, 1, 4, 1>, <171963, 1, 4, 1>, <50185, 1, 4, 1>, <0, 1, 4, 1>, <96713, 1, 4, 1>, <155355, 1, 4, 1>, <180270, 1, 4, 1>, <128425, 1, 4, 1>, <31518, 1, 4, 1>, <106257, 1, 4, 1> @} Total time: 2.819 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:22:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:22:10 on modular [Seed = 2809206314] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <111296, 1, 4, 1> @} Total time: 1.540 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:22:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:22:03 on modular [Seed = 2893684024] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <17507, 1, 4, 1>, <76475, 1, 4, 1>, <28388, 1, 4, 1>, <30994, 1, 4, 1>, <76221, 1, 4, 1>, <35202, 1, 4, 1>, <73827, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.450 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:21:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:21:57 on modular [Seed = 2992898856] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <16251, 1, 4, 1> @} Total time: 1.429 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:21:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:21:49 on modular [Seed = 3076589077] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <1850, 1, 4, 1>, <0, 1, 4, 1>, <801, 1, 4, 1> @} Total time: 1.520 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:21:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=4000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:21:02 on modular [Seed = 3161066861] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 4.389 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:38 on modular [Seed = 3239538916] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.940 seconds, Total memory usage: 39.92MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:33 on modular [Seed = 3322968044] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (187785903341577*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.730 seconds, Total memory usage: 39.93MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:28 on modular [Seed = 3407706852] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (522236902506487*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.919 seconds, Total memory usage: 40.35MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:24 on modular [Seed = 3490079203] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: (1 + O(2^44))*$.1^3 + O(2^50)*$.1^2 + (4827400634359*2^3 + O... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.279 seconds, Total memory usage: 39.34MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:19 on modular [Seed = 3573769453] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (414826611343369*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.660 seconds, Total memory usage: 39.85MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:15 on modular [Seed = 3658247142] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (139579446788105*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.279 seconds, Total memory usage: 39.36MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:04 on modular [Seed = 3741937304] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (121347758161911*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.660 seconds, Total memory usage: 39.84MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:59 on modular [Seed = 3828503976] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (96911345975287*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.160 seconds, Total memory usage: 39.18MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:40 on modular [Seed = 3913242806] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (306779403452407*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.200 seconds, Total memory usage: 39.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:36 on modular [Seed = 3996671885] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (505458911543305*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.589 seconds, Total memory usage: 39.78MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:32 on modular [Seed = 4079305364] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (305167809028105*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.310 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:28 on modular [Seed = 4163783059] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (18461674176521*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 3.080 seconds, Total memory usage: 40.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:24 on modular [Seed = 4247473298] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreePart( F: $.1^4 + O(2^50)*$.1^3 + (382136867618807*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/loclib.m", line 120, column 5: >> assert R eq 0; ^ Runtime error in assert: Assertion failed >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.149 seconds, Total memory usage: 39.16MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:20 on modular [Seed = 47494001] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (220979510771703*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.930 seconds, Total memory usage: 40.35MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:15 on modular [Seed = 131184255] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (207781347065865*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 3.069 seconds, Total memory usage: 40.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:01 on modular [Seed = 215662218] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (40731028881399*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 3.109 seconds, Total memory usage: 40.63MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:17:58 on modular [Seed = 298296216] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (396582569639927*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.180 seconds, Total memory usage: 39.21MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:17:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:17:54 on modular [Seed = 381723803] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (127222226747383*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.149 seconds, Total memory usage: 39.18MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:55 on modular [Seed = 533310815] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 2.040 seconds, Total memory usage: 39.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:51 on modular [Seed = 619877984] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (201481855696893*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 2.140 seconds, Total memory usage: 40.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:47 on modular [Seed = 704617313] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (489467683536899*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 2.700 seconds, Total memory usage: 40.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:43 on modular [Seed = 788044901] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (102843116552189*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 1.889 seconds, Total memory usage: 39.77MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:40 on modular [Seed = 870678908] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: (1 + O(2^44))*$.1^3 + O(2^50)*$.1^2 + (1735166787581*2^3 + O... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 2.180 seconds, Total memory usage: 40.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:37 on modular [Seed = 955157098] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (478662650494973*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 1.870 seconds, Total memory usage: 39.75MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:34 on modular [Seed = 1038847848] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (95639888986115*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 2.020 seconds, Total memory usage: 39.92MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:30 on modular [Seed = 1117057051] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (403373980778499*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 1.899 seconds, Total memory usage: 39.79MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:13:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:13:28 on modular [Seed = 1200747728] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 1.449 seconds, Total memory usage: 38.33MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:13:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:13:09 on modular [Seed = 1285226470] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ <2034, 1, 4, 1>, <0, 1, 4, 1>, <132, 1, 4, 1> @} Total time: 2.960 seconds, Total memory usage: 40.59MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:13:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:13:04 on modular [Seed = 1367859966] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (420621710327809*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ <2034, 1, 4, 1>, <0, 1, 4, 1>, <132, 1, 4, 1> @} Total time: 2.540 seconds, Total memory usage: 40.09MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:13:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:12:59 on modular [Seed = 1451288034] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (369089060511745*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ <2034, 1, 4, 1>, <0, 1, 4, 1>, <132, 1, 4, 1> @} Total time: 2.390 seconds, Total memory usage: 39.92MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:12:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:12:53 on modular [Seed = 1536026849] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (534561969995777*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ <2034, 1, 4, 1>, <0, 1, 4, 1>, <132, 1, 4, 1> @} Total time: 2.740 seconds, Total memory usage: 40.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:10:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:10:15 on modular [Seed = 3094219751] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 2.950 seconds, Total memory usage: 40.67MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:10:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:10:12 on modular [Seed = 3691671424] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (487120654958593*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 1.840 seconds, Total memory usage: 39.29MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:10:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:10:08 on modular [Seed = 3861927358] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (108548325900289*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 2.049 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:10:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:10:01 on modular [Seed = 4197206427] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (70932133183489*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 1.899 seconds, Total memory usage: 39.35MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:18:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:18:21 on modular [Seed = 2725511726] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 1.520 seconds, Total memory usage: 38.33MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:18:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:18:15 on modular [Seed = 3211337529] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <111296, 1, 4, 1> @} Total time: 1.550 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:18:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:18:09 on modular [Seed = 3094222904] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ <0, 1, 4, 1> @} Total time: 1.530 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:18:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:18:04 on modular [Seed = 3043169576] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 1.530 seconds, Total memory usage: 38.33MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:16:41 on modular [Seed = 3339806511] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ @} Total time: 1.620 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:16:28 on modular [Seed = 3222691901] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ @} Total time: 1.649 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:16:21 on modular [Seed = 3590354466] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ <15944, 1, 4, 1> @} Total time: 1.700 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:16:13 on modular [Seed = 3540349919] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ <8251, 1, 4, 1>, <0, 1, 4, 1>, <11146, 1, 4, 1>, <448, 1, 4, 1>, <4373, 1, 4, 1>, <82, 1, 4, 1>, <11205, 1, 4, 1>, <14480, 1, 4, 1>, <7203, 1, 4, 1>, <6052, 1, 4, 1> @} Total time: 2.649 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:08 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:16:06 on modular [Seed = 3963243755] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ <1865, 1, 4, 1>, <577, 1, 4, 1>, <440, 1, 4, 1>, <981, 1, 4, 1> @} Total time: 1.730 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:15:59 on modular [Seed = 3913239031] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 13996... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 139968 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 139968, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (226206511859851*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ <1865, 1, 4, 1>, <577, 1, 4, 1>, <440, 1, 4, 1>, <981, 1, 4, 1> @} Total time: 2.080 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:14:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:14:05 on modular [Seed = 3795075950] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 Total time: 1.700 seconds, Total memory usage: 39.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:14:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:14:02 on modular [Seed = 4280901236] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 13996... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 139968 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 139968, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (68700777678987*2^2 + O(2^52))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 Total time: 1.540 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:14:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:13:57 on modular [Seed = 4163787132] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 13996... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 139968 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 139968, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (113686586849141*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 Total time: 2.069 seconds, Total memory usage: 40.18MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:13:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:13:53 on modular [Seed = 265924239] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 13996... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 139968 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 139968, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (137517039482741*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 Total time: 1.830 seconds, Total memory usage: 39.85MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:03:08 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:03:04 on modular [Seed = 365147660] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 {@ <0, 1, 4, 1>, <30278, 1, 4, 1>, <123900, 1, 4, 1>, <12, 1, 4, 1>, <108970, 1, 4, 1> @} Total time: 4.129 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:02:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:02:53 on modular [Seed = 788040962] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 {@ <0, 1, 4, 1>, <18171, 1, 4, 1>, <12, 1, 4, 1>, <4493, 1, 4, 1>, <5609, 1, 4, 1>, <67229, 1, 4, 1>, <17513, 1, 4, 1>, <58828, 1, 4, 1> @} Total time: 4.429 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:02:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:02:38 on modular [Seed = 738035732] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 {@ <0, 1, 4, 1>, <16447, 1, 4, 1>, <12, 1, 4, 1>, <17053, 1, 4, 1>, <25317, 1, 4, 1>, <26687, 1, 4, 1> @} Total time: 4.000 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:02:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:02:26 on modular [Seed = 619874071] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 {@ <0, 1, 4, 1>, <12, 1, 4, 1>, <11287, 1, 4, 1> @} Total time: 3.649 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:01:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:01:55 on modular [Seed = 1055432044] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 {@ <0, 1, 4, 1>, <0, 1, 0, 1> @} Total time: 3.520 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:59:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:59:17 on modular [Seed = 1639447957] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 Total time: 3.180 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:59:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:59:05 on modular [Seed = 2058151538] ------------------------------------- [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 Total time: 2.229 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:58:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:58:29 on modular [Seed = 2007100243] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <27313, 1, 4, 1>, <14097, 1, 4, 1>, <4, 1, 4, 1>, <14724, 1, 4, 1>, <12362, 1, 4, 1>, <19554, 1, 4, 1> @} Total time: 4.809 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:58:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:58:03 on modular [Seed = 1889987642] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <12469, 1, 4, 1>, <2007, 1, 4, 1>, <11375, 1, 4, 1>, <4, 1, 4, 1>, <8660, 1, 4, 1>, <3067, 1, 4, 1>, <8037, 1, 4, 1>, <3970, 1, 4, 1>, <8725, 1, 4, 1> @} Total time: 5.280 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:57:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:57:47 on modular [Seed = 2388136663] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <125, 1, 4, 1>, <4, 1, 4, 1>, <0, 1, 0, 6> @} Total time: 6.790 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:49:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,43); //Order(Q);Order(P); //P+Q; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 6 2005 13:49:35 on modular [Seed = 2169970636] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 Errors: /bin/sh: line 1: 12787 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:49:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,41); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:49:22 on modular [Seed = 2622096092] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <2597267, 1, 4, 1>, <1697915, 1, 4, 1>, <1774761, 1, 4, 1>, <2447684, 1, 4, 1>, <2657729, 1, 4, 1>, <1379556, 1, 4, 1>, <2659667, 1, 4, 1>, <1439973, 1, 4, 1>, <471083, 1, 4, 1>, <2507694, 1, 4, 1>, <1346880, 1, 4, 1>, <2522497, 1, 4, 1>, <661046, 1, 4, 1>, <1249728, 1, 4, 1>, <1545728, 1, 4, 1>, <4, 1, 4, 1>, <899637, 1, 4, 1>, <2090612, 1, 4, 1>, <1028885, 1, 4, 1> @} Total time: 7.490 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:49:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,37); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:49:08 on modular [Seed = 2605516237] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <60442, 1, 4, 1>, <990905, 1, 4, 1>, <777413, 1, 4, 1>, <784817, 1, 4, 1>, <1593893, 1, 4, 1>, <1634301, 1, 4, 1>, <4, 1, 4, 1>, <1813900, 1, 4, 1>, <905869, 1, 4, 1>, <666141, 1, 4, 1>, <1092020, 1, 4, 1>, <714265, 1, 4, 1>, <837924, 1, 4, 1>, <1520237, 1, 4, 1>, <146155, 1, 4, 1>, <777551, 1, 4, 1> @} Total time: 6.799 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:49:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:48:57 on modular [Seed = 2520779006] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <4, 1, 4, 1>, <474037, 1, 4, 1>, <784897, 1, 4, 1>, <267935, 1, 4, 1>, <386199, 1, 4, 1> @} Total time: 5.179 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:48:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:48:41 on modular [Seed = 2910532566] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <405390, 1, 4, 1>, <628452, 1, 4, 1>, <4, 1, 4, 1> @} Total time: 4.950 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:48:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:48:30 on modular [Seed = 2826843940] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <19419, 1, 4, 1>, <4, 1, 4, 1>, <51761, 1, 4, 1>, <109357, 1, 4, 1>, <148391, 1, 4, 1>, <98436, 1, 4, 1>, <39998, 1, 4, 1>, <272237, 1, 4, 1>, <129795, 1, 4, 1>, <135415, 1, 4, 1> @} Total time: 5.769 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:48:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:48:18 on modular [Seed = 2809215015] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 4.990 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:48:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:48:07 on modular [Seed = 2725526839] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <4, 1, 4, 1>, <13890, 1, 4, 1>, <4601, 1, 4, 1>, <5685, 1, 4, 1>, <1506, 1, 4, 1>, <62657, 1, 4, 1>, <25978, 1, 4, 1> @} Total time: 5.240 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:47:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:47:54 on modular [Seed = 3111069201] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <27313, 1, 4, 1>, <14097, 1, 4, 1>, <4, 1, 4, 1>, <14724, 1, 4, 1>, <12362, 1, 4, 1>, <19554, 1, 4, 1> @} Total time: 4.809 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:47:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:47:39 on modular [Seed = 3026332429] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <125, 1, 4, 1>, <4, 1, 4, 1>, <0, 1, 0, 6> @} Total time: 6.700 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:47:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:47:16 on modular [Seed = 2992909684] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <12469, 1, 4, 1>, <2007, 1, 4, 1>, <11375, 1, 4, 1>, <4, 1, 4, 1>, <8660, 1, 4, 1>, <3067, 1, 4, 1>, <8037, 1, 4, 1>, <3970, 1, 4, 1>, <8725, 1, 4, 1> @} Total time: 6.320 seconds, Total memory usage: 39.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:47:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:46:57 on modular [Seed = 3357431579] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 5.960 seconds, Total memory usage: 39.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:46:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=2500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:46:36 on modular [Seed = 3339802681] ------------------------------------- Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 5.929 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:45:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:45:19 on modular [Seed = 3256114575] ------------------------------------- Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 5.269 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:44:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); Exp(5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:44:20 on modular [Seed = 3641657007] ------------------------------------- Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 5.392650408063836341891309004159701491924 148.413159102576603421115580040552279623351 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 5.309 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:41:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:41:44 on modular [Seed = 3607187405] ------------------------------------- Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 5.392650408063836341891309004159701491924 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 5.309 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:40:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:40:26 on modular [Seed = 3523496634] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ @} Total time: 1.510 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:40:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:40:17 on modular [Seed = 3913249162] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ <19799, 1, 4, 1>, <69794, 1, 4, 1>, <76881, 1, 4, 1>, <24561, 1, 4, 1>, <6404, 1, 4, 1>, <46483, 1, 4, 1>, <27624, 1, 4, 1>, <14777, 1, 4, 1> @} Total time: 2.419 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:40:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:40:02 on modular [Seed = 3828513940] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ <22652, 1, 4, 1>, <6356, 1, 4, 1>, <13101, 1, 4, 1>, <0, 1, 4, 1>, <8760, 1, 4, 1>, <14809, 1, 4, 1> @} Total time: 2.009 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:39:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:39:54 on modular [Seed = 3811932779] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ <7960, 1, 4, 1>, <0, 1, 4, 1>, <52, 1, 4, 1>, <4578, 1, 4, 1>, <2294, 1, 4, 1> @} Total time: 2.040 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:39:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:39:46 on modular [Seed = 4264060798] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (510006934241283*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ <7960, 1, 4, 1>, <0, 1, 4, 1>, <52, 1, 4, 1>, <4578, 1, 4, 1>, <2294, 1, 4, 1> @} Total time: 2.480 seconds, Total memory usage: 39.75MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:39:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:39:35 on modular [Seed = 4112739424] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ <876, 1, 4, 1>, <2267, 1, 4, 1>, <458, 1, 4, 1> @} Total time: 2.859 seconds, Total memory usage: 40.79MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:37:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:37:09 on modular [Seed = 114335521] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.590 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:37:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:37:05 on modular [Seed = 499876926] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (126189686554627*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.790 seconds, Total memory usage: 40.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:37:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:37:01 on modular [Seed = 466453028] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (205148595421187*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.720 seconds, Total memory usage: 39.93MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:36:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:36:58 on modular [Seed = 381718318] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (533042781290493*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.560 seconds, Total memory usage: 39.72MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:36:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:36:54 on modular [Seed = 771470361] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (28027162198013*2^2 + O(2^52))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.429 seconds, Total memory usage: 39.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:36:52 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:36:50 on modular [Seed = 687780120] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: (1 + O(2^46))*$.1^3 + O(2^50)*$.1^2 - (10673799036931*2^2 + ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.740 seconds, Total memory usage: 39.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:36:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:36:43 on modular [Seed = 670154350] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (405529284706307*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.530 seconds, Total memory usage: 39.68MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:33:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,61); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:33:20 on modular [Seed = 586463264] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <1501825, 1, 4, 1>, <2735217, 1, 4, 1>, <1248183, 1, 4, 1>, <13168104, 1, 4, 1>, <6926292, 1, 4, 1>, <5473814, 1, 4, 1>, <11483073, 1, 4, 1>, <78936, 1, 4, 1>, <5601040, 1, 4, 1>, <9052154, 1, 4, 1>, <13602106, 1, 4, 1>, <11931435, 1, 4, 1>, <5826426, 1, 4, 1>, <10829460, 1, 4, 1>, <10412941, 1, 4, 1>, <6629779, 1, 4, 1>, <798658, 1, 4, 1>, <1503626, 1, 4, 1>, <1197885, 1, 4, 1>, <7465266, 1, 4, 1>, <11451148, 1, 4, 1>, <7160797, 1, 4, 1>, <10985804, 1, 4, 1>, <3096037, 1, 4, 1>, <10632461, 1, 4, 1>, <9349970, 1, 4, 1>, <7280407, 1, 4, 1>, <10751359, 1, 4, 1>, <9497510, 1, 4, 1>, <1050706, 1, 4, 1> @} Total time: 7.799 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:33:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,47); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:33:04 on modular [Seed = 972006187] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <18553, 1, 4, 1>, <2154130, 1, 4, 1>, <954111, 1, 4, 1>, <1916482, 1, 4, 1>, <2198198, 1, 4, 1>, <2186846, 1, 4, 1>, <3655441, 1, 4, 1>, <1347416, 1, 4, 1>, <1125178, 1, 4, 1>, <2414931, 1, 4, 1>, <3563582, 1, 4, 1>, <0, 1, 4, 1>, <3742107, 1, 4, 1>, <2076665, 1, 4, 1>, <2066139, 1, 4, 1>, <2001988, 1, 4, 1>, <3036467, 1, 4, 1>, <3616228, 1, 4, 1>, <1073009, 1, 4, 1>, <4230318, 1, 4, 1>, <758107, 1, 4, 1>, <1457617, 1, 4, 1> @} Total time: 6.280 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:32:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,43); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:32:55 on modular [Seed = 887270929] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <2769263, 1, 4, 1>, <1802899, 1, 4, 1>, <2073720, 1, 4, 1>, <436913, 1, 4, 1>, <1103776, 1, 4, 1>, <977490, 1, 4, 1>, <1077728, 1, 4, 1> @} Total time: 3.339 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:32:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,41); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:32:38 on modular [Seed = 1302058873] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <285375, 1, 4, 1>, <648401, 1, 4, 1>, <1191607, 1, 4, 1>, <2640996, 1, 4, 1>, <2667085, 1, 4, 1> @} Total time: 3.069 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:32:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,37); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:32:27 on modular [Seed = 1218368001] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <1705268, 1, 4, 1>, <97004, 1, 4, 1>, <109642, 1, 4, 1>, <21290, 1, 4, 1> @} Total time: 2.990 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:32:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:32:18 on modular [Seed = 1200741639] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <693692, 1, 4, 1>, <37303, 1, 4, 1>, <749589, 1, 4, 1>, <512357, 1, 4, 1>, <620638, 1, 4, 1> @} Total time: 2.910 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:32:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:32:04 on modular [Seed = 1117050900] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <336140, 1, 4, 1> @} Total time: 2.390 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:55 on modular [Seed = 1485750833] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <85580, 1, 4, 1>, <232404, 1, 4, 1>, <59226, 1, 4, 1>, <139180, 1, 4, 1>, <0, 1, 4, 1>, <11333, 1, 4, 1>, <20295, 1, 4, 1>, <254229, 1, 4, 1>, <255518, 1, 4, 1>, <251564, 1, 4, 1> @} Total time: 3.740 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:44 on modular [Seed = 1468124469] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <108284, 1, 4, 1> @} Total time: 2.359 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:35 on modular [Seed = 1384433703] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <52562, 1, 4, 1>, <80301, 1, 4, 1>, <72852, 1, 4, 1>, <16320, 1, 4, 1>, <0, 1, 4, 1>, <8676, 1, 4, 1>, <43621, 1, 4, 1>, <32832, 1, 4, 1>, <7697, 1, 4, 1> @} Total time: 3.220 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:26 on modular [Seed = 1774187237] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <27252, 1, 4, 1> @} Total time: 2.209 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:18 on modular [Seed = 1689452512] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <325, 1, 4, 1>, <9792, 1, 4, 1>, <3109, 1, 4, 1>, <12273, 1, 4, 1>, <11846, 1, 4, 1> @} Total time: 2.620 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:10 on modular [Seed = 1672870143] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <684, 1, 4, 1>, <0, 1, 4, 1>, <351, 1, 4, 1> @} Total time: 2.259 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:28:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:28:18 on modular [Seed = 2074992715] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 Total time: 1.790 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:27:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,67); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:27:28 on modular [Seed = 1923410192] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <11197197, 1, 4, 1>, <3005276, 1, 4, 1>, <7014446, 1, 4, 1>, <15971564, 1, 4, 1>, <17044899, 1, 4, 1>, <1864961, 1, 4, 1>, <12575718, 1, 4, 1>, <7013876, 1, 4, 1>, <1506713, 1, 4, 1>, <269252, 1, 4, 1>, <14859615, 1, 4, 1>, <6565980, 1, 4, 1>, <7207186, 1, 4, 1>, <9024818, 1, 4, 1>, <0, 1, 4, 1>, <7340076, 1, 4, 1>, <5335980, 1, 4, 1>, <4780394, 1, 4, 1>, <15417502, 1, 4, 1>, <11231438, 1, 4, 1>, <9369949, 1, 4, 1>, <4531156, 1, 4, 1>, <11596801, 1, 4, 1>, <19892731, 1, 4, 1>, <194286, 1, 4, 1>, <13346200, 1, 4, 1>, <17158751, 1, 4, 1>, <18649052, 1, 4, 1>, <19865104, 1, 4, 1>, <2283723, 1, 4, 1>, <20021136, 1, 4, 1>, <12434465, 1, 4, 1>, <3079662, 1, 4, 1> @} Total time: 9.070 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:27:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,61); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:27:12 on modular [Seed = 2287866875] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <7404317, 1, 4, 1>, <11140375, 1, 4, 1>, <9649939, 1, 4, 1>, <7165503, 1, 4, 1>, <7103941, 1, 4, 1>, <1746578, 1, 4, 1>, <8125851, 1, 4, 1>, <8082873, 1, 4, 1>, <3826247, 1, 4, 1>, <511199, 1, 4, 1>, <6540532, 1, 4, 1>, <9013869, 1, 4, 1>, <11599251, 1, 4, 1>, <3294978, 1, 4, 1>, <58529, 1, 4, 1>, <2332353, 1, 4, 1>, <8852801, 1, 4, 1>, <3326549, 1, 4, 1>, <5261737, 1, 4, 1>, <9348995, 1, 4, 1>, <8457519, 1, 4, 1>, <245557, 1, 4, 1>, <6001473, 1, 4, 1>, <7770011, 1, 4, 1>, <7062636, 1, 4, 1>, <2777906, 1, 4, 1>, <1484683, 1, 4, 1>, <1439110, 1, 4, 1>, <2275404, 1, 4, 1>, <9047087, 1, 4, 1> @} Total time: 7.650 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:26:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,59); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:26:21 on modular [Seed = 2253394488] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <7214339, 1, 4, 1> @} Total time: 2.399 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:26:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,53); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:26:11 on modular [Seed = 2203391795] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <5277684, 1, 4, 1>, <4487334, 1, 4, 1>, <3155679, 1, 4, 1> @} Total time: 2.750 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:26:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,47); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:55 on modular [Seed = 2555251800] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <865641, 1, 4, 1>, <2039568, 1, 4, 1>, <335647, 1, 4, 1>, <4187170, 1, 4, 1>, <3965865, 1, 4, 1>, <3143254, 1, 4, 1>, <2527018, 1, 4, 1>, <0, 1, 4, 1>, <3041163, 1, 4, 1>, <2395431, 1, 4, 1>, <1358670, 1, 4, 1>, <349407, 1, 4, 1>, <3670730, 1, 4, 1>, <4339743, 1, 4, 1>, <1470315, 1, 4, 1>, <1904195, 1, 4, 1>, <3883581, 1, 4, 1>, <2555510, 1, 4, 1>, <3054728, 1, 4, 1>, <4230330, 1, 4, 1>, <1395649, 1, 4, 1>, <707559, 1, 4, 1> @} Total time: 6.089 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:25:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,43); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:41 on modular [Seed = 2504196419] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <84385, 1, 4, 1>, <1261711, 1, 4, 1>, <2763557, 1, 4, 1>, <351559, 1, 4, 1>, <1051948, 1, 4, 1>, <2904115, 1, 4, 1>, <956780, 1, 4, 1>, <3094272, 1, 4, 1>, <327840, 1, 4, 1>, <0, 1, 4, 1>, <1641952, 1, 4, 1>, <2424089, 1, 4, 1>, <152365, 1, 4, 1>, <1926920, 1, 4, 1>, <93440, 1, 4, 1>, <1368710, 1, 4, 1>, <1467527, 1, 4, 1>, <1661588, 1, 4, 1>, <1301014, 1, 4, 1>, <3117856, 1, 4, 1>, <2372065, 1, 4, 1> @} Total time: 5.389 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:25:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,41); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:31 on modular [Seed = 2453931642] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <1715400, 1, 4, 1>, <2614156, 1, 4, 1>, <2628725, 1, 4, 1>, <865815, 1, 4, 1>, <2608615, 1, 4, 1>, <2064431, 1, 4, 1>, <804, 1, 4, 1>, <1106307, 1, 4, 1>, <1734890, 1, 4, 1>, <1944940, 1, 4, 1> @} Total time: 3.490 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:25:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,5); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:21 on modular [Seed = 2943957837] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 >> Chabauty(P,5); ^ Runtime error in 'Chabauty': The curve must have good reduction at p. Total time: 1.629 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:25:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,37); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:09 on modular [Seed = 2758946898] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <273808, 1, 4, 1>, <615796, 1, 4, 1>, <1393398, 1, 4, 1>, <1321033, 1, 4, 1>, <1658043, 1, 4, 1>, <488625, 1, 4, 1>, <1611907, 1, 4, 1>, <174526, 1, 4, 1>, <869901, 1, 4, 1>, <910815, 1, 4, 1>, <1760878, 1, 4, 1>, <1240452, 1, 4, 1>, <585687, 1, 4, 1>, <892909, 1, 4, 1>, <1346326, 1, 4, 1>, <1673351, 1, 4, 1>, <1337456, 1, 4, 1> @} Total time: 5.019 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:25:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:00 on modular [Seed = 2708944302] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <190679, 1, 4, 1>, <383865, 1, 4, 1>, <39695, 1, 4, 1>, <556144, 1, 4, 1>, <217957, 1, 4, 1>, <920606, 1, 4, 1>, <350099, 1, 4, 1>, <436388, 1, 4, 1>, <650884, 1, 4, 1> @} Total time: 3.419 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:53 on modular [Seed = 3194497780] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <0, 1, 4, 1> @} Total time: 2.250 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:44 on modular [Seed = 3144495096] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <49292, 1, 4, 1>, <174996, 1, 4, 1>, <222417, 1, 4, 1>, <259938, 1, 4, 1>, <150597, 1, 4, 1>, <174407, 1, 4, 1>, <196353, 1, 4, 1>, <181489, 1, 4, 1>, <0, 1, 4, 1>, <113422, 1, 4, 1> @} Total time: 3.540 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:32 on modular [Seed = 2959484135] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <5335, 1, 4, 1> @} Total time: 2.120 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:22 on modular [Seed = 3441121700] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <45158, 1, 4, 1>, <18984, 1, 4, 1>, <16432, 1, 4, 1>, <71595, 1, 4, 1>, <37302, 1, 4, 1>, <9848, 1, 4, 1>, <20928, 1, 4, 1> @} Total time: 2.960 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:14 on modular [Seed = 3390856875] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <3374, 1, 4, 1>, <11705, 1, 4, 1>, <4444, 1, 4, 1>, <4094, 1, 4, 1>, <2759, 1, 4, 1>, <20875, 1, 4, 1> @} Total time: 2.520 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:06 on modular [Seed = 3339801534] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <8143, 1, 4, 1>, <12281, 1, 4, 1>, <8787, 1, 4, 1>, <1619, 1, 4, 1>, <11906, 1, 4, 1>, <2329, 1, 4, 1>, <340, 1, 4, 1>, <10996, 1, 4, 1>, <12684, 1, 4, 1> @} Total time: 2.970 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:22:58 on modular [Seed = 3725085359] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <1864, 1, 4, 1>, <138, 1, 4, 1>, <2204, 1, 4, 1> @} Total time: 2.120 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:20:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:20:23 on modular [Seed = 3556917696] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 Total time: 1.620 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:09:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:09:54 on modular [Seed = 3996679791] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 {@ @} Total time: 1.929 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:09:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:09:45 on modular [Seed = 3946675057] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 {@ <24622, 1, 4, 1>, <66269, 1, 4, 1>, <8671, 1, 4, 1>, <51446, 1, 4, 1>, <48474, 1, 4, 1>, <76752, 1, 4, 1>, <14483, 1, 4, 1>, <81071, 1, 4, 1> @} Total time: 2.960 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:09:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:09:37 on modular [Seed = 3761664114] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 {@ <16291, 1, 4, 1>, <19040, 1, 4, 1>, <2542, 1, 4, 1>, <20245, 1, 4, 1>, <11372, 1, 4, 1> @} Total time: 2.379 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:09:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:09:30 on modular [Seed = 4247481684] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 {@ <10911, 1, 4, 1>, <7542, 1, 4, 1>, <11767, 1, 4, 1>, <2646, 1, 4, 1>, <2070, 1, 4, 1> @} Total time: 2.359 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:09:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:09:22 on modular [Seed = 4197214811] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 {@ <1935, 1, 4, 1>, <2274, 1, 4, 1>, <540, 1, 4, 1> @} Total time: 2.000 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:08:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=4000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:08:15 on modular [Seed = 199072759] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 Total time: 4.120 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:06:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:06:19 on modular [Seed = 64328555] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 Total time: 1.480 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:06:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:06:09 on modular [Seed = 416188538] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 Total time: 1.449 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:02:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:02:22 on modular [Seed = 365134920] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 5, 1>, <32702, 1, 5, 1> @} Total time: 2.229 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:02:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,101); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:02:10 on modular [Seed = 314868547] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <27012122, 1, 4, 1>, <56559795, 1, 4, 1>, <99616790, 1, 4, 1>, <28246568, 1, 4, 1>, <30642484, 1, 4, 1>, <40333758, 1, 4, 1>, <9564867, 1, 4, 1>, <38013215, 1, 4, 1>, <2, 1, 4, 1>, <43796927, 1, 4, 1>, <9774641, 1, 4, 1>, <38047577, 1, 4, 1> @} Total time: 4.400 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:02:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,97); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:01:47 on modular [Seed = 687782224] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <71795493, 1, 4, 1>, <36374146, 1, 4, 1>, <45174985, 1, 4, 1>, <36345198, 1, 4, 1>, <56254183, 1, 4, 1>, <2717480, 1, 4, 1>, <85025356, 1, 4, 1>, <78179222, 1, 4, 1>, <4109767, 1, 4, 1>, <46528428, 1, 4, 1>, <66044578, 1, 4, 1>, <17744430, 1, 4, 1>, <74555086, 1, 4, 1>, <82216558, 1, 4, 1>, <83103622, 1, 4, 1>, <39892898, 1, 4, 1>, <26067587, 1, 4, 1>, <66094937, 1, 4, 1>, <34142239, 1, 4, 1>, <68850693, 1, 4, 1>, <45456511, 1, 4, 1>, <74967757, 1, 4, 1>, <0, 1, 4, 1>, <65450611, 1, 4, 1>, <6021947, 1, 4, 1>, <64373942, 1, 4, 1>, <7256233, 1, 4, 1>, <50207547, 1, 4, 1>, <48021213, 1, 4, 1>, <17290430, 1, 4, 1>, <77710172, 1, 4, 1>, <41538690, 1, 4, 1>, <51844808, 1, 4, 1>, <11365604, 1, 4, 1>, <4724195, 1, 4, 1>, <62726853, 1, 4, 1>, <32037834, 1, 4, 1>, <10981780, 1, 4, 1>, <63646517, 1, 4, 1>, <37558129, 1, 4, 1>, <17899769, 1, 4, 1>, <4567421, 1, 4, 1>, <23122583, 1, 4, 1>, <26809605, 1, 4, 1>, <56574871, 1, 4, 1>, <14718989, 1, 4, 1> @} Total time: 13.300 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:01:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,89); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:01:25 on modular [Seed = 636729251] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <24247596, 1, 4, 1> @} Total time: 2.839 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:01:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,83); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:01:01 on modular [Seed = 586461324] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <2021557, 1, 4, 1>, <8664646, 1, 4, 1>, <11863020, 1, 4, 1>, <29433056, 1, 4, 1>, <13807776, 1, 4, 1>, <13089905, 1, 4, 1>, <38676359, 1, 4, 1>, <16496936, 1, 4, 1>, <7688636, 1, 4, 1>, <19040011, 1, 4, 1>, <20888026, 1, 4, 1>, <21039143, 1, 4, 1>, <6866840, 1, 4, 1>, <26238789, 1, 4, 1>, <7273001, 1, 4, 1>, <28512238, 1, 4, 1>, <30588134, 1, 4, 1>, <15681343, 1, 4, 1>, <34879780, 1, 4, 1>, <10256907, 1, 4, 1>, <22381330, 1, 4, 1>, <3506172, 1, 4, 1>, <17966338, 1, 4, 1>, <17338558, 1, 4, 1>, <21725622, 1, 4, 1>, <25899387, 1, 4, 1>, <330457, 1, 4, 1>, <46188223, 1, 4, 1>, <39542733, 1, 4, 1>, <37416054, 1, 4, 1>, <6578097, 1, 4, 1>, <7209682, 1, 4, 1>, <21209936, 1, 4, 1>, <22232558, 1, 4, 1>, <31086576, 1, 4, 1>, <12403862, 1, 4, 1>, <25319803, 1, 4, 1>, <19001234, 1, 4, 1>, <20288888, 1, 4, 1> @} Total time: 10.900 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:00:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,73); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:00:36 on modular [Seed = 1072278920] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <10480108, 1, 4, 1>, <1286378, 1, 4, 1>, <10037775, 1, 4, 1>, <19888360, 1, 4, 1>, <13103320, 1, 4, 1>, <9262653, 1, 4, 1>, <4447586, 1, 4, 1>, <24792296, 1, 4, 1>, <21057293, 1, 4, 1>, <7770948, 1, 4, 1>, <28231748, 1, 4, 1>, <2452023, 1, 4, 1>, <25084182, 1, 4, 1>, <11151608, 1, 4, 1>, <10185541, 1, 4, 1>, <20878892, 1, 4, 1>, <22213178, 1, 4, 1>, <7446635, 1, 4, 1>, <26164840, 1, 4, 1>, <6523878, 1, 4, 1>, <7365467, 1, 4, 1>, <4805262, 1, 4, 1>, <17879037, 1, 4, 1>, <20681855, 1, 4, 1>, <9239797, 1, 4, 1>, <23016988, 1, 4, 1>, <11429561, 1, 4, 1>, <4417018, 1, 4, 1>, <21422353, 1, 4, 1>, <0, 1, 4, 1>, <27643269, 1, 4, 1>, <49607, 1, 4, 1>, <14177976, 1, 4, 1>, <14060588, 1, 4, 1>, <25094947, 1, 4, 1> @} Total time: 9.230 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:00:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,79); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:00:22 on modular [Seed = 887269105] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <12329804, 1, 4, 1> @} Total time: 2.879 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:00:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,73); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:00:03 on modular [Seed = 853844194] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <10480108, 1, 4, 1>, <1286378, 1, 4, 1>, <10037775, 1, 4, 1>, <19888360, 1, 4, 1>, <13103320, 1, 4, 1>, <9262653, 1, 4, 1>, <4447586, 1, 4, 1>, <24792296, 1, 4, 1>, <21057293, 1, 4, 1>, <7770948, 1, 4, 1>, <28231748, 1, 4, 1>, <2452023, 1, 4, 1>, <25084182, 1, 4, 1>, <11151608, 1, 4, 1>, <10185541, 1, 4, 1>, <20878892, 1, 4, 1>, <22213178, 1, 4, 1>, <7446635, 1, 4, 1>, <26164840, 1, 4, 1>, <6523878, 1, 4, 1>, <7365467, 1, 4, 1>, <4805262, 1, 4, 1>, <17879037, 1, 4, 1>, <20681855, 1, 4, 1>, <9239797, 1, 4, 1>, <23016988, 1, 4, 1>, <11429561, 1, 4, 1>, <4417018, 1, 4, 1>, <21422353, 1, 4, 1>, <0, 1, 4, 1>, <27643269, 1, 4, 1>, <49607, 1, 4, 1>, <14177976, 1, 4, 1>, <14060588, 1, 4, 1>, <25094947, 1, 4, 1> @} Total time: 9.230 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:59:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,71); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:59:41 on modular [Seed = 1218370230] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <1413031, 1, 4, 1>, <5185722, 1, 4, 1>, <19457557, 1, 4, 1>, <7357455, 1, 4, 1>, <21138811, 1, 4, 1>, <21842205, 1, 4, 1>, <7523086, 1, 4, 1>, <16168209, 1, 4, 1>, <13862217, 1, 4, 1>, <14310106, 1, 4, 1>, <21052861, 1, 4, 1>, <7717318, 1, 4, 1>, <2309298, 1, 4, 1>, <10027565, 1, 4, 1>, <0, 1, 4, 1>, <4955708, 1, 4, 1>, <20492067, 1, 4, 1>, <3555447, 1, 4, 1>, <14780959, 1, 4, 1>, <18964423, 1, 4, 1>, <11674927, 1, 4, 1>, <14548571, 1, 4, 1>, <14945119, 1, 4, 1>, <11773912, 1, 4, 1>, <8971958, 1, 4, 1>, <12730745, 1, 4, 1>, <15502653, 1, 4, 1>, <836768, 1, 4, 1>, <15949237, 1, 4, 1>, <17632988, 1, 4, 1> @} Total time: 7.469 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:59:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,67); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:59:21 on modular [Seed = 1167316815] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <7699685, 1, 4, 1>, <13602056, 1, 4, 1>, <4610343, 1, 4, 1>, <5459277, 1, 4, 1>, <9612913, 1, 4, 1>, <4606819, 1, 4, 1>, <8326665, 1, 4, 1>, <17870986, 1, 4, 1>, <792199, 1, 4, 1>, <13558230, 1, 4, 1>, <13335759, 1, 4, 1>, <17055999, 1, 4, 1>, <12810586, 1, 4, 1>, <2829083, 1, 4, 1>, <12618534, 1, 4, 1>, <15563003, 1, 4, 1>, <6806210, 1, 4, 1>, <2433608, 1, 4, 1>, <16723006, 1, 4, 1>, <14449298, 1, 4, 1>, <12219126, 1, 4, 1>, <18183838, 1, 4, 1>, <2081372, 1, 4, 1>, <5134728, 1, 4, 1>, <12447858, 1, 4, 1>, <5318474, 1, 4, 1>, <2860131, 1, 4, 1>, <11339111, 1, 4, 1>, <17572280, 1, 4, 1>, <3109435, 1, 4, 1>, <15440538, 1, 4, 1>, <7892848, 1, 4, 1> @} Total time: 8.980 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:59:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,61); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:59:10 on modular [Seed = 1117048927] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <6115384, 1, 4, 1>, <4897665, 1, 4, 1>, <11935065, 1, 4, 1>, <13385356, 1, 4, 1>, <107975, 1, 4, 1>, <13692166, 1, 4, 1>, <7468882, 1, 4, 1>, <449555, 1, 4, 1>, <12105611, 1, 4, 1>, <682609, 1, 4, 1>, <4407031, 1, 4, 1>, <2694933, 1, 4, 1>, <2404111, 1, 4, 1>, <3452130, 1, 4, 1>, <12998416, 1, 4, 1>, <12047364, 1, 4, 1>, <12416374, 1, 4, 1>, <7869665, 1, 4, 1> @} Total time: 5.059 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:58:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,59); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:58:54 on modular [Seed = 1602866547] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <4543845, 1, 4, 1> @} Total time: 2.740 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:58:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,53); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:58:34 on modular [Seed = 1417856621] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <5120360, 1, 4, 1>, <4704672, 1, 4, 1>, <406634, 1, 4, 1>, <6031745, 1, 4, 1>, <1736850, 1, 4, 1>, <4614562, 1, 4, 1>, <3065420, 1, 4, 1>, <1669026, 1, 4, 1>, <5352906, 1, 4, 1>, <1574013, 1, 4, 1>, <440055, 1, 4, 1>, <5893703, 1, 4, 1>, <1807695, 1, 4, 1>, <3672395, 1, 4, 1>, <7409625, 1, 4, 1>, <1793342, 1, 4, 1>, <288960, 1, 4, 1>, <4301686, 1, 4, 1>, <2796213, 1, 4, 1>, <3524899, 1, 4, 1>, <4685631, 1, 4, 1>, <7093616, 1, 4, 1>, <476980, 1, 4, 1>, <5546670, 1, 4, 1>, <6073579, 1, 4, 1> @} Total time: 6.719 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:58:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,47); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:58:19 on modular [Seed = 1367851801] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <17975, 1, 4, 1>, <782586, 1, 4, 1>, <1600382, 1, 4, 1>, <2115842, 1, 4, 1>, <2717370, 1, 4, 1>, <1231405, 1, 4, 1>, <1632457, 1, 4, 1>, <4793661, 1, 4, 1>, <100360, 1, 4, 1>, <3138765, 1, 4, 1>, <1590302, 1, 4, 1>, <3086344, 1, 4, 1>, <1220364, 1, 4, 1>, <3822439, 1, 4, 1>, <1218985, 1, 4, 1>, <2915915, 1, 4, 1>, <2441318, 1, 4, 1>, <3667531, 1, 4, 1>, <0, 1, 4, 1>, <2754918, 1, 4, 1>, <160337, 1, 4, 1>, <3166042, 1, 4, 1> @} Total time: 6.179 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:58:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,43); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:58:07 on modular [Seed = 1857616946] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <2511769, 1, 4, 1>, <1801855, 1, 4, 1>, <3062749, 1, 4, 1>, <1636020, 1, 4, 1>, <1528725, 1, 4, 1>, <1530864, 1, 4, 1> @} Total time: 3.379 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:58:01 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; Roots(x^2-t); P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:57:59 on modular [Seed = 1706293041] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 >> Roots(x^2-t); ^ User error: Identifier 't' has not been declared or assigned done Total time: 0.200 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:58:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,41); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:57:55 on modular [Seed = 1622602288] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <1827852, 1, 4, 1>, <1930548, 1, 4, 1>, <1154948, 1, 4, 1>, <1100631, 1, 4, 1>, <1621336, 1, 4, 1>, <1974720, 1, 4, 1>, <2653637, 1, 4, 1>, <2812782, 1, 4, 1>, <2504748, 1, 4, 1>, <889326, 1, 4, 1>, <0, 1, 4, 1>, <231922, 1, 4, 1>, <2712831, 1, 4, 1>, <531580, 1, 4, 1>, <1132337, 1, 4, 1>, <2266332, 1, 4, 1>, <2492533, 1, 4, 1>, <2739753, 1, 4, 1>, <2680070, 1, 4, 1>, <2382434, 1, 4, 1>, <367077, 1, 4, 1>, <398225, 1, 4, 1>, <2304265, 1, 4, 1>, <1572938, 1, 4, 1> @} Total time: 5.839 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:57:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,37); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:57:40 on modular [Seed = 2108419866] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <219670, 1, 4, 1>, <527290, 1, 4, 1>, <58100, 1, 4, 1>, <558655, 1, 4, 1>, <966095, 1, 4, 1>, <239803, 1, 4, 1>, <1271528, 1, 4, 1>, <768807, 1, 4, 1>, <1233186, 1, 4, 1>, <868458, 1, 4, 1>, <251903, 1, 4, 1>, <97833, 1, 4, 1>, <734224, 1, 4, 1>, <479277, 1, 4, 1>, <1958, 1, 4, 1>, <651779, 1, 4, 1>, <626724, 1, 4, 1> @} Total time: 5.139 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:57:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:57:29 on modular [Seed = 2058151936] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <291824, 1, 4, 1>, <651910, 1, 4, 1>, <30464, 1, 4, 1>, <547473, 1, 4, 1>, <822159, 1, 4, 1>, <876805, 1, 4, 1>, <696689, 1, 4, 1>, <0, 1, 4, 1>, <715713, 1, 4, 1>, <48457, 1, 4, 1> @} Total time: 3.560 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:57:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:57:11 on modular [Seed = 2487355151] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <468048, 1, 4, 1> @} Total time: 2.580 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:57:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:59 on modular [Seed = 2236814850] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <254187, 1, 4, 1>, <246453, 1, 4, 1>, <5918, 1, 4, 1>, <69818, 1, 4, 1>, <5443, 1, 4, 1>, <141219, 1, 4, 1>, <225826, 1, 4, 1>, <0, 1, 4, 1>, <258262, 1, 4, 1>, <148240, 1, 4, 1> @} Total time: 3.740 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:56:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:49 on modular [Seed = 2287868677] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <104488, 1, 4, 1>, <58245, 1, 4, 1>, <2, 1, 4, 1> @} Total time: 2.569 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:56:45 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; Roots(x^2-a); P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:44 on modular [Seed = 3009748995] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 [ ] done Total time: 0.200 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:56:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:39 on modular [Seed = 3059754253] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <104488, 1, 4, 1>, <58245, 1, 4, 1>, <2, 1, 4, 1> @} Total time: 2.580 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:56:27 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; Roots(x^2-1); P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; Points(Jacobian(C)); print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:25 on modular [Seed = 3194498331] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 [ <1, 2> ] {@ (1, 0, 0), (x^2 + F1.1^39*x + F1.1^26, F1.1^41*x + F1.1^45, 2), (x^2 + F1.1^51*x + F1.1^35, F1.1^22*x + F1.1^47, 2), (x^2 + F1.1^55*x + F1.1^5, F1.1^59*x + F1.1^22, 2), (x^2 + F1.1^16*x + F1.1^22, F1.1^7*x + F1.1^46, 2), (x^2 + F1.1^40*x + F1.1^48, F1.1^3*x + F1.1^26, 2), (x^2 + F1.1^59*x + F1.1^31, F1.1^45*x + F1.1^48, 2), (x^2 + F1.1^4*x + F1.1, F1.1^17*x + F1.1^18, 2), (x^2 + F1.1^31*x + F1.1^58, F1.1^55*x + F1.1^59, 2), (x^2 + F1.1*x + F1.1^5, F1.1^22*x + F1.1^14, 2), (x^2 + F1.1^26*x + F1.1^22, F1.1^41*x + F1.1^50, 2), (x^2 + F1.1^57*x + F1.1^51, F1.1^14*x + F1.1^33, 2), (x^2 + F1.1^59*x, F1.1^17*x, 2), (x^2 + F1.1^47*x + F1.1^26, F1.1^27*x + F1.1^58, 2), (x^2 + F1.1^59*x + F1.1^19, F1.1^30*x + F1.1^10, 2), (x^2 + F1.1^36*x + F1.1^52, F1.1^29*x + F1.1^33, 2), (x^2 + F1.1^60*x + F1.1^42, F1.1^21, 2), (x^2 + F1.1^11*x + F1.1^45, F1.1^43*x + F1.1^16, 2), (x^2 + F1.1^28*x + 1, F1.1^55*x + F1.1^12, 2), (x^2 + F1.1^21*x + F1.1^59, F1.1^21*x + F1.1^60, 2), (x^2 + F1.1^15*x + F1.1^42, F1.1^45*x + F1.1^17, 2), (x^2 + F1.1^60*x + F1.1^39, F1.1^56*x + F1.1^61, 2), (x^2 + F1.1^10*x + F1.1^9, F1.1^50*x + F1.1^53, 2), (x^2 + F1.1^42*x + F1.1^50, F1.1^61*x + F1.1^48, 2), (x^2 + F1.1^55, F1.1^34*x + F1.1^52, 2), (x^2 + F1.1^3*x + F1.1^18, F1.1^28*x + F1.1^24, 2), (x^2 + F1.1^62*x + F1.1^46, F1.1^12*x + F1.1^22, 2), (x^2 + F1.1^34*x + F1.1^51, F1.1^57*x + F1.1^11, 2), (x^2 + F1.1^37*x + F1.1^32, F1.1^11*x + F1.1^26, 2), (x^2 + F1.1^25*x + F1.1^13, F1.1^61*x + F1.1^28, 2), (x^2 + F1.1^14*x + F1.1^14, F1.1^44*x + F1.1^29, 2), (x^2 + F1.1^6*x + F1.1^32, F1.1^29*x + F1.1^42, 2), (x^2 + F1.1^22*x + F1.1^29, F1.1^17*x + F1.1^26, 2), (x^2 + F1.1^32*x + F1.1^53, F1.1^24*x + F1.1^59, 2), (x^2 + F1.1^54*x + F1.1^27, F1.1^15*x + F1.1^29, 2), (x^2 + F1.1^49*x + F1.1^30, F1.1^24*x + F1.1^44, 2), (x^2 + F1.1^16*x + F1.1, F1.1^9*x + F1.1^62, 2), (x^2 + F1.1^56*x + F1.1, F1.1^43*x + F1.1^14, 2), (x^2 + F1.1^60*x + F1.1^6, F1.1^34*x + F1.1^4, 2), (x^2 + F1.1^61*x + F1.1^9, F1.1*x + F1.1^31, 2), (x^2 + F1.1^38*x + F1.1^45, F1.1^28*x + F1.1^35, 2), (x^2 + F1.1^52*x + F1.1^19, F1.1^20*x + F1.1, 2), (x^2 + F1.1^39*x + F1.1^51, F1.1^37*x + F1.1^28, 2), (x^2 + F1.1^25*x + F1.1^55, F1.1^26*x + F1.1^17, 2), (x^2 + F1.1^14*x + F1.1^28, F1.1^33*x + F1.1^6, 2), (x^2 + x + F1.1^37, F1.1^18*x + F1.1, 2), (x^2 + F1.1^9*x + F1.1^37, F1.1^32*x + F1.1^42, 2), (x^2 + F1.1^30*x + F1.1^17, F1.1^26*x + F1.1^54, 2), (x^2 + F1.1^13*x + F1.1^3, F1.1^32*x + F1.1^33, 2), (x^2 + F1.1^62*x + F1.1^9, F1.1^12*x + F1.1^35, 2), (x^2 + F1.1^40*x + F1.1^26, F1.1^45*x + F1.1^7, 2), (x^2 + F1.1^62*x + F1.1^15, F1.1^31*x + F1.1^20, 2), (x^2 + F1.1^34*x + F1.1^14, F1.1^2*x + F1.1^61, 2), (x^2 + F1.1^6*x + F1.1^8, F1.1^9*x + F1.1^3, 2), (x^2 + F1.1^35*x + F1.1^34, F1.1^31*x + F1.1^53, 2), (x^2 + F1.1^34*x + F1.1^8, F1.1^28*x + F1.1^14, 2), (x^2 + F1.1^25*x + F1.1^58, F1.1^23*x + F1.1^27, 2), (x^2 + F1.1^42*x + F1.1^20, F1.1^6*x + F1.1^39, 2), (x^2 + F1.1^4*x + F1.1^58, F1.1^61*x + F1.1^34, 2), (x^2 + F1.1^55*x + F1.1^26, F1.1^3*x, 2), (x^2 + F1.1^4*x + 1, F1.1^61*x + F1.1^5, 2), (x^2 + F1.1^11*x + F1.1^52, F1.1^61*x + F1.1^47, 2), (x^2 + F1.1^24*x + F1.1^57, F1.1^3*x + F1.1^40, 2), (x^2 + F1.1^56*x + F1.1^58, F1.1^56*x + F1.1^23, 2), (x^2 + F1.1^13*x + F1.1^16, F1.1^2*x + F1.1^25, 2), (x^2 + F1.1^14*x + F1.1^30, F1.1^62*x + F1.1^18, 2), (x^2 + F1.1^40*x + F1.1^54, F1.1^14*x + F1.1^45, 2), (x^2 + F1.1^21*x + F1.1^23, F1.1^36*x + F1.1^11, 2), (x^2 + F1.1^56*x + F1.1^62, F1.1^25*x + F1.1^51, 2), (x^2 + F1.1^36*x + F1.1^55, F1.1^16*x + F1.1^28, 2), (x^2 + F1.1^8*x + F1.1^41, F1.1^49*x + F1.1^15, 2), (x^2 + F1.1^34*x + F1.1^12, F1.1^35*x + F1.1^8, 2), (x^2 + F1.1^9*x + F1.1^32, F1.1^32*x + F1.1^8, 2), (x^2 + F1.1^39*x + F1.1^8, F1.1^39*x + F1.1^60, 2), (x^2 + F1.1^12*x + F1.1^35, F1.1^43*x + F1.1^61, 2), (x + F1.1^8, F1.1^27, 1), (x^2 + F1.1^9*x + F1.1^2, F1.1^41*x + F1.1^43, 2), (x^2 + F1.1^29*x + F1.1^44, F1.1^23*x + F1.1^27, 2), (x^2 + F1.1^39*x + F1.1^53, F1.1^34*x + F1.1^23, 2), (x^2 + F1.1^50*x + F1.1^2, F1.1^46*x + F1.1^57, 2), (x^2 + F1.1^41*x + F1.1^14, F1.1^59*x + F1.1^56, 2), (x^2 + F1.1^23*x + F1.1^12, F1.1^33*x + F1.1^35, 2), (x^2 + F1.1^10*x + F1.1^5, F1.1^29*x + F1.1^34, 2), (x^2 + F1.1^44*x + F1.1^54, F1.1^22*x + F1.1^22, 2), (x^2 + F1.1^30*x + F1.1^29, F1.1^41*x + F1.1^39, 2), (x^2 + F1.1^50*x + F1.1^4, F1.1^19, 2), (x^2 + F1.1^27*x + F1.1^17, F1.1^40*x + F1.1^11, 2), (x^2 + F1.1^38*x + F1.1^58, F1.1^7*x + F1.1^13, 2), (x^2 + F1.1^26*x + F1.1^9, F1.1^52*x + F1.1^54, 2), (x^2 + F1.1^37*x + F1.1^6, F1.1^21*x + F1.1^57, 2), (x^2 + F1.1^19*x + F1.1^48, F1.1^17*x + F1.1^24, 2), (x^2 + F1.1^39*x + F1.1^10, F1.1^42*x + F1.1^15, 2), (x^2 + F1.1^44*x + F1.1^10, F1.1^47*x + F1.1^38, 2), (x^2 + x + F1.1^21, F1.1^24*x + F1.1^11, 2), (x^2 + F1.1^42*x + F1.1^15, F1.1^37*x + F1.1^37, 2), (x^2 + F1.1^50*x + F1.1^31, F1.1^28*x + F1.1^4, 2), (x^2 + F1.1^7*x + F1.1^16, F1.1^6*x + F1.1^11, 2), (x^2 + F1.1^18*x + F1.1^48, F1.1^26*x + F1.1^8, 2), (x^2 + F1.1^23*x + F1.1^37, F1.1^47*x + F1.1^48, 2), (x^2 + F1.1^2*x + F1.1^44, F1.1^53*x + F1.1^36, 2), (x^2 + F1.1^14*x + F1.1^61, F1.1^52*x + F1.1^20, 2), (x^2 + F1.1^37*x + F1.1^56, F1.1^19*x + F1.1^44, 2), (x^2 + F1.1^44*x + F1.1^5, F1.1^62*x + F1.1^17, 2), (x^2 + F1.1^52*x + F1.1^11, F1.1^23*x + F1.1^26, 2), (x^2 + F1.1^2*x + F1.1^61, F1.1^23*x + F1.1^27, 2), (x^2 + F1.1^2*x + F1.1^28, F1.1^56*x + F1.1^48, 2), (x^2 + F1.1^21*x + F1.1^19, F1.1^3*x + F1.1^47, 2), (x^2 + F1.1^53*x + F1.1^3, F1.1^12*x + F1.1^28, 2), (x^2 + F1.1^51*x + F1.1^34, F1.1^61*x + F1.1^20, 2), (x^2 + F1.1*x + F1.1^43, F1.1^22*x + F1.1^33, 2), (x^2 + F1.1^40*x + F1.1^38, x + F1.1^47, 2), (x^2 + x + F1.1^5, F1.1^9*x + F1.1^10, 2), (x^2 + F1.1^62*x + F1.1^44, F1.1^21*x + F1.1^2, 2), (x^2 + F1.1^22*x + F1.1^16, F1.1^4*x + F1.1^19, 2), (x^2 + F1.1^51*x + F1.1^58, F1.1^39*x + F1.1^28, 2), (x^2 + F1.1^45*x + F1.1^8, F1.1^36*x + F1.1^15, 2), (x^2 + F1.1^34*x + F1.1^16, F1.1^56*x + F1.1^13, 2), (x^2 + F1.1^9*x + F1.1^50, F1.1^37*x + F1.1^21, 2), (x + F1.1^47, F1.1^18, 1), (x^2 + F1.1^4*x + F1.1^56, F1.1^34*x + F1.1^37, 2), (x^2 + F1.1^44*x + F1.1^29, F1.1^19*x + F1.1^45, 2), (x^2 + F1.1^40*x + F1.1^2, F1.1^19*x + F1.1^9, 2), (x^2 + F1.1^21*x + F1.1^21, F1.1^33*x + F1.1^23, 2), (x^2 + F1.1^43*x + F1.1^18, F1.1^29*x + F1.1^41, 2), (x^2 + F1.1^6*x + F1.1^41, F1.1^42*x + F1.1^50, 2), (x^2 + F1.1^27*x + F1.1^47, F1.1^23*x + F1.1^27, 2), (x^2 + F1.1^15*x + F1.1^60, F1.1^26*x + F1.1^5, 2), (x^2 + F1.1^62*x + F1.1^49, F1.1*x + F1.1^41, 2), (x^2 + F1.1^28*x + F1.1^8, F1.1^56*x + F1.1^3, 2), (x^2 + F1.1^48*x + F1.1^17, F1.1^61*x + F1.1^50, 2), (x^2 + F1.1^49*x + F1.1^19, F1.1^48*x + F1.1^39, 2), (x^2 + F1.1^19*x + F1.1^27, F1.1^21*x + F1.1^10, 2), (x^2 + F1.1^37*x + F1.1^35, F1.1^10*x + F1.1^43, 2), (x^2 + F1.1^61*x + F1.1^8, F1.1^41*x + F1.1^42, 2), (x^2 + F1.1^49*x + F1.1^61, F1.1^55*x + F1.1^52, 2), (x^2 + F1.1^58*x + F1.1^35, F1.1^32*x + F1.1^42, 2), (x^2 + F1.1^45*x + F1.1^30, F1.1^5*x + F1.1^28, 2), (x^2 + F1.1^61*x + F1.1^37, F1.1^36*x + 1, 2), (x^2 + F1.1^46*x + F1.1^36, F1.1^40*x + F1.1^38, 2), (x^2 + F1.1^15*x + F1.1^52, F1.1^11*x + F1.1^16, 2), (x^2 + F1.1^42*x + F1.1^10, F1.1^44*x + F1.1^26, 2), (x^2 + F1.1^26*x + F1.1^48, F1.1^49*x + F1.1^6, 2), (x^2 + F1.1^17*x + F1.1^45, F1.1^51*x + F1.1^27, 2), (x^2 + F1.1^6*x + F1.1^38, F1.1^15*x + F1.1^30, 2), (x^2 + F1.1^34*x + F1.1^4, F1.1^44*x + F1.1^62, 2), (x^2 + F1.1^25*x + F1.1^14, F1.1^27*x + F1.1^57, 2), (x^2 + F1.1^45*x + F1.1^43, F1.1^13*x + F1.1^51, 2), (x^2 + F1.1^14*x + F1.1^25, F1.1*x + F1.1^39, 2), (x^2 + F1.1^23*x + F1.1^29, F1.1^51*x + F1.1^16, 2), (x^2 + F1.1^11*x + F1.1^55, F1.1^27*x + F1.1^38, 2), (x^2 + F1.1^16, F1.1^38*x + F1.1^12, 2), (x^2 + F1.1^24*x + F1.1^32, F1.1^44*x + F1.1^29, 2), (x^2 + F1.1^26*x + F1.1^18, F1.1^3*x + F1.1^23, 2), (x^2 + F1.1^8*x + F1.1^27, F1.1^42*x + F1.1^28, 2), (x^2 + F1.1^6*x + F1.1^55, F1.1^5*x + F1.1^29, 2), (x^2 + F1.1^55*x + F1.1^13, F1.1^46, 2), (x^2 + F1.1^58*x + F1.1^9, F1.1^40*x + F1.1^33, 2), (x^2 + F1.1^33*x + F1.1^30, F1.1^3*x + F1.1^38, 2), (x^2 + F1.1^33*x + F1.1^25, F1.1^24*x + F1.1^6, 2), (x^2 + F1.1^27*x + F1.1^16, F1.1^37*x + F1.1^44, 2), (x^2 + F1.1^22*x + F1.1^18, F1.1^33*x + F1.1^38, 2), (x^2 + F1.1^12*x + F1.1^14, F1.1^2*x + F1.1^54, 2), (x^2 + F1.1^35*x + 1, F1.1^16*x + F1.1^34, 2), (x^2 + F1.1^20*x + F1.1^61, F1.1^11*x + F1.1^31, 2), (x^2 + F1.1^11*x + F1.1^46, F1.1^43*x + F1.1^48, 2), (x^2 + F1.1^18*x + F1.1^62, F1.1^61*x + F1.1, 2), (x^2 + F1.1^3*x + F1.1, F1.1^41*x + F1.1^52, 2), (x^2 + F1.1^16*x + F1.1^29, F1.1^37*x + F1.1^28, 2), (x^2 + F1.1^24*x + F1.1^27, F1.1^40*x + F1.1^43, 2), (x^2 + F1.1^13*x + F1.1^55, F1.1^20*x + F1.1^50, 2), (x^2 + F1.1^4*x + F1.1^49, F1.1^52*x + F1.1^58, 2), (x^2 + F1.1^37*x + F1.1^15, F1.1^33*x + F1.1^59, 2), (x^2 + F1.1^25*x + F1.1^61, x + F1.1^45, 2), (x^2 + F1.1^27*x + F1.1^38, F1.1^30*x + F1.1^51, 2), (x^2 + F1.1^55*x + F1.1^37, F1.1^29*x + F1.1^5, 2), (x^2 + F1.1^45*x + F1.1^49, F1.1^21*x + F1.1^18, 2), (x^2 + F1.1^10*x + F1.1^24, F1.1^38*x + F1.1^22, 2), (x^2 + F1.1^62*x + F1.1^40, F1.1^4*x + F1.1^10, 2), (x^2 + F1.1^49*x + F1.1^42, F1.1^49*x + F1.1^32, 2), (x^2 + F1.1^43*x + F1.1^14, F1.1^45*x + F1.1^46, 2), (x^2 + F1.1^34*x + F1.1^30, F1.1^59*x + F1.1^43, 2), (x + F1.1^11, F1.1^52, 1), (x^2 + F1.1^44*x + F1.1^44, F1.1^52*x + F1.1^28, 2), (x^2 + F1.1^39*x + F1.1^56, x + F1.1^25, 2), (x^2 + F1.1^15*x + F1.1^15, F1.1^58*x + F1.1^ ** WARNING: Output too long, hence truncated. '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:56:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:23 on modular [Seed = 2742366216] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <199, 1, 4, 1> @} Total time: 2.509 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:56:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:14 on modular [Seed = 2775791223] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <27094, 1, 4, 1>, <13160, 1, 4, 1>, <19042, 1, 4, 1>, <19507, 1, 4, 1>, <9810, 1, 4, 1>, <25736, 1, 4, 1> @} Total time: 2.839 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:56:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:05 on modular [Seed = 2826845041] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 5, 1>, <32702, 1, 5, 1> @} Total time: 2.240 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:55:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:55:54 on modular [Seed = 3573761555] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <1347, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.350 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:55:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P1,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:55:37 on modular [Seed = 3624815405] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.899 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:55:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P1,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:55:29 on modular [Seed = 3272953918] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.899 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:55:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P1,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:55:21 on modular [Seed = 3322959139] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.899 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:55:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:55:13 on modular [Seed = 3374275108] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.919 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:52:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P1,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:52:40 on modular [Seed = 4062477942] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.909 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:52:33 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; Points(Jacobian(C)); print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:52:31 on modular [Seed = 4112742726] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 {@ (1, 0, 0), (x^2 + F1.1^55*x + F1.1^5, F1.1^59*x + F1.1^22, 2), (x^2 + F1.1^59*x + F1.1^31, F1.1^45*x + F1.1^48, 2), (x^2 + F1.1*x + F1.1^5, F1.1^22*x + F1.1^14, 2), (x^2 + F1.1^59*x, F1.1^17*x, 2), (x^2 + F1.1^36*x + F1.1^52, F1.1^29*x + F1.1^33, 2), (x^2 + F1.1^28*x + 1, F1.1^55*x + F1.1^12, 2), (x^2 + F1.1^60*x + F1.1^39, F1.1^56*x + F1.1^61, 2), (x^2 + F1.1^55, F1.1^34*x + F1.1^52, 2), (x^2 + F1.1^34*x + F1.1^51, F1.1^57*x + F1.1^11, 2), (x^2 + F1.1^14*x + F1.1^14, F1.1^44*x + F1.1^29, 2), (x^2 + F1.1^32*x + F1.1^53, F1.1^24*x + F1.1^59, 2), (x^2 + F1.1^16*x + F1.1, F1.1^9*x + F1.1^62, 2), (x^2 + F1.1^61*x + F1.1^9, F1.1*x + F1.1^31, 2), (x^2 + F1.1^39*x + F1.1^51, F1.1^37*x + F1.1^28, 2), (x^2 + x + F1.1^37, F1.1^18*x + F1.1, 2), (x^2 + F1.1^13*x + F1.1^3, F1.1^32*x + F1.1^33, 2), (x^2 + F1.1^62*x + F1.1^15, F1.1^31*x + F1.1^20, 2), (x^2 + F1.1^35*x + F1.1^34, F1.1^31*x + F1.1^53, 2), (x^2 + F1.1^42*x + F1.1^20, F1.1^6*x + F1.1^39, 2), (x^2 + F1.1^4*x + 1, F1.1^61*x + F1.1^5, 2), (x^2 + F1.1^56*x + F1.1^58, F1.1^56*x + F1.1^23, 2), (x^2 + F1.1^40*x + F1.1^54, F1.1^14*x + F1.1^45, 2), (x^2 + F1.1^36*x + F1.1^55, F1.1^16*x + F1.1^28, 2), (x^2 + F1.1^9*x + F1.1^32, F1.1^32*x + F1.1^8, 2), (x + F1.1^8, F1.1^27, 1), (x^2 + F1.1^39*x + F1.1^53, F1.1^34*x + F1.1^23, 2), (x^2 + F1.1^23*x + F1.1^12, F1.1^33*x + F1.1^35, 2), (x^2 + F1.1^30*x + F1.1^29, F1.1^41*x + F1.1^39, 2), (x^2 + F1.1^38*x + F1.1^58, F1.1^7*x + F1.1^13, 2), (x^2 + F1.1^19*x + F1.1^48, F1.1^17*x + F1.1^24, 2), (x^2 + x + F1.1^21, F1.1^24*x + F1.1^11, 2), (x^2 + F1.1^7*x + F1.1^16, F1.1^6*x + F1.1^11, 2), (x^2 + F1.1^2*x + F1.1^44, F1.1^53*x + F1.1^36, 2), (x^2 + F1.1^44*x + F1.1^5, F1.1^62*x + F1.1^17, 2), (x^2 + F1.1^2*x + F1.1^28, F1.1^56*x + F1.1^48, 2), (x^2 + F1.1^51*x + F1.1^34, F1.1^61*x + F1.1^20, 2), (x^2 + x + F1.1^5, F1.1^9*x + F1.1^10, 2), (x^2 + F1.1^51*x + F1.1^58, F1.1^39*x + F1.1^28, 2), (x^2 + F1.1^9*x + F1.1^50, F1.1^37*x + F1.1^21, 2), (x^2 + F1.1^44*x + F1.1^29, F1.1^19*x + F1.1^45, 2), (x^2 + F1.1^43*x + F1.1^18, F1.1^29*x + F1.1^41, 2), (x^2 + F1.1^15*x + F1.1^60, F1.1^26*x + F1.1^5, 2), (x^2 + F1.1^48*x + F1.1^17, F1.1^61*x + F1.1^50, 2), (x^2 + F1.1^37*x + F1.1^35, F1.1^10*x + F1.1^43, 2), (x^2 + F1.1^58*x + F1.1^35, F1.1^32*x + F1.1^42, 2), (x^2 + F1.1^46*x + F1.1^36, F1.1^40*x + F1.1^38, 2), (x^2 + F1.1^26*x + F1.1^48, F1.1^49*x + F1.1^6, 2), (x^2 + F1.1^34*x + F1.1^4, F1.1^44*x + F1.1^62, 2), (x^2 + F1.1^14*x + F1.1^25, F1.1*x + F1.1^39, 2), (x^2 + F1.1^16, F1.1^38*x + F1.1^12, 2), (x^2 + F1.1^8*x + F1.1^27, F1.1^42*x + F1.1^28, 2), (x^2 + F1.1^58*x + F1.1^9, F1.1^40*x + F1.1^33, 2), (x^2 + F1.1^27*x + F1.1^16, F1.1^37*x + F1.1^44, 2), (x^2 + F1.1^35*x + 1, F1.1^16*x + F1.1^34, 2), (x^2 + F1.1^18*x + F1.1^62, F1.1^61*x + F1.1, 2), (x^2 + F1.1^24*x + F1.1^27, F1.1^40*x + F1.1^43, 2), (x^2 + F1.1^37*x + F1.1^15, F1.1^33*x + F1.1^59, 2), (x^2 + F1.1^55*x + F1.1^37, F1.1^29*x + F1.1^5, 2), (x^2 + F1.1^62*x + F1.1^40, F1.1^4*x + F1.1^10, 2), (x^2 + F1.1^34*x + F1.1^30, F1.1^59*x + F1.1^43, 2), (x^2 + F1.1^39*x + F1.1^56, x + F1.1^25, 2), (x^2 + F1.1^41*x + F1.1^9, F1.1^15*x + F1.1^7, 2), (x^2 + F1.1^49*x + F1.1^23, F1.1^26*x + F1.1^35, 2), (x^2 + F1.1^21*x + F1.1, F1.1^57*x + F1.1^58, 2), (x^2 + F1.1^9*x + F1.1^60, F1.1^62*x + F1.1^41, 2), (x^2 + F1.1^33*x + F1.1^19, F1.1^17*x + F1.1^22, 2), (x^2 + F1.1^34*x + F1.1^3, F1.1^53*x + F1.1^51, 2), (x^2 + F1.1^16*x + F1.1^51, F1.1^25*x + F1.1^14, 2), (x^2 + F1.1^59*x + F1.1^15, F1.1^43*x + F1.1^12, 2), (x^2 + F1.1^47*x + F1.1^30, F1.1^51*x + F1.1^32, 2), (x^2 + F1.1*x + F1.1^33, F1.1^17*x + F1.1^51, 2), (x^2 + F1.1^57*x + F1.1^40, F1.1^58*x + F1.1^52, 2), (x^2 + F1.1^48*x + F1.1^36, F1.1^11*x + F1.1^18, 2), (x^2 + F1.1^44*x + F1.1^16, F1.1^14*x + F1.1^33, 2), (x^2 + F1.1^14*x + F1.1^36, F1.1^38*x + F1.1^10, 2), (x^2 + F1.1^58*x + F1.1^40, F1.1^54*x, 2), (x^2 + F1.1^62*x + F1.1^56, F1.1^21*x + F1.1^8, 2), (x^2 + F1.1^54*x + F1.1^19, F1.1^16*x + F1.1^24, 2), (x^2 + F1.1^28*x + F1.1^41, F1.1^37*x + F1.1^55, 2), (x^2 + x + F1.1^3, F1.1^2*x + F1.1^40, 2), (x^2 + F1.1^47*x + F1.1^52, F1.1^52*x + F1.1^20, 2), (x^2 + F1.1^35*x + F1.1^21, F1.1^50*x + F1.1^36, 2), (x^2 + F1.1^8*x + F1.1^47, F1.1^31*x + F1.1^25, 2), (x^2 + F1.1^32*x + F1.1^24, F1.1^45*x + F1.1^21, 2), (x^2 + F1.1^56*x + F1.1^44, F1.1^3*x + F1.1^22, 2), (x^2 + F1.1^58*x + F1.1^31, F1.1^4*x + F1.1^54, 2), (x^2 + F1.1^40*x + F1.1^3, F1.1^27*x + F1.1^48, 2), (x^2 + F1.1^32*x + F1.1^38, F1.1^6*x + F1.1^35, 2), (x^2 + F1.1^55*x + F1.1^30, F1.1^50*x + F1.1^12, 2), (x^2 + F1.1^36*x + F1.1^42, F1.1^36*x + F1.1^3, 2), (x^2 + F1.1^35*x + F1.1^22, F1.1^7*x + F1.1^28, 2), (x^2 + F1.1^56*x + F1.1^40, F1.1^6*x + F1.1^54, 2), (x^2 + F1.1^7*x + F1.1^37, F1.1^29*x + F1.1^48, 2), (x^2 + F1.1^43*x + F1.1^39, F1.1^57*x + F1.1^3, 2), (x^2 + F1.1^5*x + F1.1^56, F1.1^42*x + F1.1^4, 2), (x^2 + F1.1^21*x + F1.1^3, F1.1^29*x + F1.1^28, 2), (x^2 + F1.1^16*x + F1.1^58, F1.1^11*x + F1.1^33, 2), (x^2 + F1.1^8*x + F1.1^37, F1.1^4*x + F1.1^53, 2), (x^2 + F1.1^25*x + F1.1^19, F1.1^27*x + F1.1^28, 2), (x^2 + F1.1^17*x + F1.1^60, F1.1^51*x + F1.1^3, 2), (x^2 + F1.1^40*x + F1.1^51, F1.1^29*x + F1.1^10, 2), (x^2 + F1.1^60*x + F1.1^3, F1.1^37*x + F1.1^21, 2), (x^2 + F1.1*x + F1.1^2, F1.1^24*x + F1.1^12, 2), (x^2 + F1.1^31*x + F1.1^45, F1.1^30*x + F1.1^58, 2), (x^2 + F1.1^21*x + F1.1^8, F1.1^16*x + F1.1^7, 2), (x^2 + F1.1^15*x + F1.1^58, F1.1^38*x + F1.1^58, 2), (x^2 + F1.1^36*x + F1.1^24, F1.1^60*x + F1.1^32, 2), (x^2 + F1.1^54*x + F1.1^39, F1.1*x + F1.1, 2), (x^2 + F1.1^53*x + F1.1^41, F1.1^38*x + F1.1^8, 2), (x^2 + F1.1^56*x + F1.1^41, F1.1^13*x + F1.1^52, 2), (x^2 + F1.1^19*x + F1.1^30, F1.1^22*x + F1.1^33, 2), (x^2 + x + F1.1^6, F1.1^58*x + F1.1^3, 2), (x^2 + F1.1^8*x + F1.1^40, F1.1^33*x + F1.1^15, 2), (x^2 + F1.1^14*x + F1.1^38, F1.1^52*x + F1.1^40, 2), (x^2 + F1.1^40*x + F1.1^18, F1.1^45*x + F1.1^3, 2), (x^2 + F1.1^3*x + F1.1^32, F1.1^46*x + F1.1^21, 2), (x^2 + F1.1^22*x + F1.1^54, F1.1^52*x + F1.1, 2), (x^2 + F1.1^58*x + F1.1^22, F1.1^58*x + F1.1^27, 2), (x^2 + F1.1^23*x + F1.1^57, F1.1^54*x + F1.1^39, 2), (x^2 + F1.1^12*x + F1.1^45, F1.1^58*x + F1.1^16, 2), (x^2 + F1.1^38*x + F1.1^59, F1.1^6*x + F1.1^19, 2), (x^2 + F1.1^62*x + F1.1^55, F1.1^6*x + F1.1^54, 2), (x^2 + F1.1^33*x + F1.1^46, F1.1^54*x + F1.1^22, 2), (x^2 + F1.1^9*x + F1.1^23, F1.1^44*x + F1.1^42, 2), (x^2 + F1.1^20*x + F1.1^37, F1.1^42*x + F1.1^27, 2), (x^2 + F1.1^16*x + F1.1^28, F1.1^34*x + F1.1^23, 2), (x^2 + F1.1^56*x + F1.1^35, F1.1^34*x + F1.1^58, 2), (x^2 + F1.1^25*x + F1.1^48, F1.1^44*x + F1.1^53, 2), (x^2 + F1.1^44*x + F1.1^11, F1.1^35*x + F1.1^3, 2), (x^2 + F1.1^14*x + F1.1^6, F1.1^26*x + F1.1^51, 2), (x^2 + F1.1^61*x + F1.1^36, F1.1^25*x + F1.1^43, 2), (x^2 + F1.1^52*x + F1.1^47, F1.1^9*x + F1.1^23, 2), (x^2 + F1.1^15*x, F1.1^49*x, 2), (x^2 + F1.1^61*x + F1.1^7, F1.1^59*x + F1.1^9, 2), (x^2 + F1.1^62*x + F1.1^51, F1.1^31*x + F1.1^38, 2), (x^2 + F1.1^23*x + F1.1^22, F1.1^45*x + F1.1^7, 2), (x^2 + F1.1^44*x + F1.1^11, F1.1^9*x + F1.1^12, 2), (x^2 + F1.1^13*x + F1.1^37, F1.1^46*x + F1.1^36, 2), (x^2 + F1.1^5*x + F1.1^55, F1.1^48*x + F1.1^32, 2), (x^2 + F1.1^56*x + F1.1^26, F1.1^22*x + F1.1^29, 2), (x^2 + F1.1^12*x + F1.1^17, F1.1^57*x + F1.1^27, 2), (x^2 + F1.1^55*x + F1.1^58, F1.1^21*x + F1.1^36, 2), (x^2 + F1.1^3*x + F1.1^8, F1.1^36*x + F1.1, 2), (x^2 + F1.1^16*x + F1.1^43, F1.1^20*x, 2), (x^2 + F1.1^25*x + F1.1^30, F1.1^30*x + F1.1^11, 2), (x^2 + F1.1^10*x + F1.1^57, F1.1^58*x + F1.1, 2), (x^2 + F1.1^61*x + F1.1^19, F1.1^15*x + F1.1^11, 2), (x^2 + F1.1^59*x + F1.1^39, F1.1^17*x + F1.1^53, 2), (x^2 + F1.1^38*x + F1.1^60, x + F1.1^13, 2), (x^2 + F1.1^42*x + F1.1^26, F1.1^13*x + F1.1^44, 2), (x^2 + F1.1^2*x + F1.1^23, F1.1^41*x + F1.1^61, 2), (x + F1.1^27, F1.1^8, 1), (x^2 + F1.1^33*x + F1.1^37, F1.1^61*x + F1.1^18, 2), (x^2 + F1.1^4*x + F1.1^60, F1.1^15*x + F1.1^39, 2), (x^2 + F1.1^12*x + F1.1^52, F1.1^56*x + F1.1^21, 2), (x^2 + F1.1^44*x + F1.1^39, F1.1^36*x + F1.1^10, 2), (x^2 + F1.1^10*x + F1.1^20, F1.1^43*x + F1.1^37, 2), (x^2 + F1.1^8*x + F1.1^39, F1.1^48*x + F1.1^47, 2), (x^2 + F1.1^42*x + F1.1^52, F1.1^32*x + F1.1^5, 2), (x^2 + F1.1^42*x + F1.1^51, F1.1^40*x + F1.1^38, 2), (x^2 + F1.1^35*x + F1.1^6, F1.1^20*x + F1.1^11, 2), (x^2 + F1.1^3*x + F1.1^3, F1.1^37*x + F1.1^37, 2), (x^2 + F1.1^38*x + F1.1^26, F1.1^22*x + F1.1^36, 2), (x^2 + F1.1^4*x + F1.1^35, F1.1^13*x + F1.1^49, 2), (x^2 + F1.1^2*x + F1.1^22, F1.1^28*x + F1.1^38, 2), (x^2 + F1.1^26*x + F1.1^50, F1.1^44*x + F1.1^11, 2), (x^2 + F1.1^60*x + F1.1^31, F1.1^20*x + F1.1^55, 2), (x^2 + F1.1^13*x + F1.1^31, F1.1^28*x + F1.1^47, 2), (x^2 + F1.1^2*x + F1.1^40, F1.1^3*x + F1.1^56, 2), (x^2 + F1.1^26*x + F1.1^12, F1.1^19*x + F1.1^36, 2), (x^2 + F1.1^8*x + F1.1^9, F1.1^51*x + F1.1^57, 2), (x^2 + F1.1^35*x, F1.1^26*x, 2), (x^2 + F1.1^4*x + F1.1^21, F1.1^5*x + F1.1^7, 2), (x^2 + F1.1^10*x + F1.1^19, F1.1^54*x + F1.1^34, 2), (x^2 + F1.1^47*x + F1.1^34, F1.1^11*x + F1.1^31, 2), (x^2 + F1.1^13*x + F1.1^31, F1.1^19*x + F1.1^62, 2), (x^2 + F1.1^56*x + F1.1^35, x + F1.1^56, 2), (x^2 + F1.1^10*x + F1.1^43, F1.1^21*x + F1.1^58, 2), (x^2 + F1.1^54*x + F1.1^59, F1.1^44*x + F1.1^29, 2), (x^2 + F1.1^31*x + F1.1^38, F1.1^34*x + F1.1^10, 2), (x^2 + F1.1^43*x + F1.1^17, F1.1^19*x + F1.1^31, 2), (x^2 + F1.1^40*x + F1.1^4, F1.1^28*x + F1.1^16, 2), (x^2 + F1.1^34*x + F1.1^31, F1.1^48*x + F1.1^35, 2), (x^2 + F1.1^8*x + F1.1^35, F1.1^47*x + F1.1^36, 2), (x^2 + F1.1^43*x + F1.1^45, F1.1^52*x + F1.1 ** WARNING: Output too long, hence truncated. '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:52:01 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; Points(Jac(C)); print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:52:01 on modular [Seed = 3946669894] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 >> Points(Jac(C)); ^ User error: Identifier 'Jac' has not been declared or assigned done Total time: 0.200 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:51:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x+2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); P+Q; 2*J![x-2,8]; Output: Magma V2.11-10 Tue Dec 6 2005 12:51:09 on modular [Seed = 3980091660] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} (x - 2, 8, 1) (x^2 - 4*x + 4, 5*x - 2, 2) Total time: 1.899 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:51:08 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; Rationalpoints(C); print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:51:08 on modular [Seed = 281451760] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 >> Rationalpoints(C); ^ User error: Identifier 'Rationalpoints' has not been declared or assigned done Total time: 0.210 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:50:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x+2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); 2*P; 2*J![x-2,8]; Output: Magma V2.11-10 Tue Dec 6 2005 12:50:32 on modular [Seed = 466458399] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} (x^2 - 4*x + 4, 5*x - 2, 2) (x^2 - 4*x + 4, 5*x - 2, 2) Total time: 1.929 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:50:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x+2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; 2*J![x-2,8]; Output: Magma V2.11-10 Tue Dec 6 2005 12:50:09 on modular [Seed = 516461091] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} (x^2 - 4*x + 4, 5*x - 2, 2) Total time: 1.919 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:49:25 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:49:25 on modular [Seed = 14060617] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 done Total time: 0.210 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:49:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x+2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Order(Q); Output: Magma V2.11-10 Tue Dec 6 2005 12:49:03 on modular [Seed = 182224489] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} 2 Total time: 1.929 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:49:03 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; Mod(v^2+x*v-f,u); P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:49:03 on modular [Seed = 249070189] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 >> Mod(v^2+x*v-f,u); ^ Runtime error in procedure call: Attempting to call something that is not callable done Total time: 0.210 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:48:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-2,8]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Order(Q); Output: Magma V2.11-10 Tue Dec 6 2005 12:48:41 on modular [Seed = 904116384] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} 0 Total time: 1.919 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:48:36 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f/u; //Mod(v^2+x*v-f P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:48:36 on modular [Seed = 955168160] ------------------------------------- (F1.1*x^5 + F1.1^61*x^3 + F1.1^23*x^2 + F1.1^19*x + F1.1^54)/(x^2 + F1.1^27*x + F1.1^51) done Total time: 0.200 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:48:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:48:17 on modular [Seed = 603303598] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.899 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:47:53 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2-x*v-f/u; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:47:52 on modular [Seed = 619882880] ------------------------------------- (F1.1*x^5 + F1.1^61*x^3 + F1.1^23*x^2 + F1.1^19*x + F1.1^54)/(x^2 + F1.1^27*x + F1.1^51) done Total time: 0.200 seconds, Total memory usage: 3.43MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:47:38 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2-x*v-f; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:47:38 on modular [Seed = 1519179660] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 done Total time: 0.200 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:46:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:46:37 on modular [Seed = 1552601421] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 Total time: 1.899 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:46:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:46:20 on modular [Seed = 1200736839] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 Total time: 1.919 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:44:13 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:44:13 on modular [Seed = 1251787979] ------------------------------------- done Total time: 0.210 seconds, Total memory usage: 3.43MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:42:05 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u(x):=x^2+F1.1^27*x+F1.1^51; v(x):=F1.1^39+F1.1^59; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:42:04 on modular [Seed = 1302053709] ------------------------------------- >> u(x):=x^2+F1.1^27*x+F1.1^51; ^ User error: Illegal left hand side of an assignment statement >> v(x):=F1.1^39+F1.1^59; ^ User error: Illegal left hand side of an assignment statement done Total time: 0.210 seconds, Total memory usage: 3.43MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:41:19 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; u(x):=x^2+F1.1^27*x+F1.1^51; v(x):=F1.1^39+F1.1^59; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:41:19 on modular [Seed = 2007103780] ------------------------------------- >> u(x):=x^2+F1.1^27*x+F1.1^51; ^ User error: Identifier 'x' has not been declared or assigned >> v(x):=F1.1^39+F1.1^59; ^ User error: Illegal left hand side of an assignment statement done Total time: 0.210 seconds, Total memory usage: 3.43MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:41:03 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1^i; b:=F1.1^j; a1:=F1.1^55; a2:=F1.1^59; u(x):=x^2+F1.1^27*x+F1.1^51; v(x):=F1.1^39+F1.1^59; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:41:03 on modular [Seed = 2058155065] ------------------------------------- >> a:=F1.1^i; ^ User error: Identifier 'i' has not been declared or assigned >> b:=F1.1^j; ^ User error: Identifier 'j' has not been declared or assigned >> u(x):=x^2+F1.1^27*x+F1.1^51; ^ User error: Identifier 'x' has not been declared or assigned >> v(x):=F1.1^39+F1.1^59; ^ User error: Illegal left hand side of an assignment statement >> C:=HyperellipticCurve(a*x^5+b*x^3+x,x); ^ User error: Identifier 'a' has not been declared or assigned >> N1:=#C; //over Fq ^ User error: Identifier 'C' has not been declared or assigned >> CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); ^ User error: Identifier 'a' has not been declared or assigned >> N2:=#CC; //over Fq^2; ^ User error: Identifier 'CC' has not been declared or assigned done Total time: 0.200 seconds, Total memory usage: 3.34MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:18:02 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := x+y^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > P; > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:18:02 on modular [Seed = 3272951652] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x + y^2 {@ @} false 0 1 Curve over Rational Field defined by $.1*$.3 + $.2^2 Curve over Rational Field defined by x*y + 1 Curve over Rational Field defined by x*y + 1 >> IsAnalyticallyIrreducible(D2,Origin(A)); ^ Runtime error in 'IsAnalyticallyIrreducible': Second argument must be a point of the first >> g := ResolutionGraph(D2,Origin(A)); ^ Runtime error in 'ResolutionGraph': Second argument must be a rational point on the first argument >> g; ^ User error: Identifier 'g' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.53MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:17:45 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := x+y^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:17:45 on modular [Seed = 3289530384] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x + y^2 {@ @} false 0 1 Curve over Rational Field defined by x*y + 1 Curve over Rational Field defined by x*y + 1 >> IsAnalyticallyIrreducible(D2,Origin(A)); ^ Runtime error in 'IsAnalyticallyIrreducible': Second argument must be a point of the first >> g := ResolutionGraph(D2,Origin(A)); ^ Runtime error in 'ResolutionGraph': Second argument must be a rational point on the first argument >> g; ^ User error: Identifier 'g' has not been declared or assigned Total time: 0.210 seconds, Total memory usage: 3.53MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:16:57 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := x+y^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > P; > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:16:56 on modular [Seed = 3441117480] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x + y^2 {@ @} false 0 1 Curve over Rational Field defined by $.1*$.3 + $.2^2 Curve over Rational Field defined by x*y + 1 Curve over Rational Field defined by x*y + 1 >> IsAnalyticallyIrreducible(D2,Origin(A)); ^ Runtime error in 'IsAnalyticallyIrreducible': Second argument must be a point of the first >> g := ResolutionGraph(D2,Origin(A)); ^ Runtime error in 'ResolutionGraph': Second argument must be a rational point on the first argument >> g; ^ User error: Identifier 'g' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.53MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:16:10 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1*x+0*y+0*(-x^3-y^2)+(0*x+1*y+1*(-x^3-y^2))^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > P; > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:16:09 on modular [Seed = 4129321666] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x^6 + 2*x^3*y^2 - 2*x^3*y + x + y^4 - 2*y^3 + y^2 {@ @} false 2 1 Curve over Rational Field defined by $.1^6 + 2*$.1^3*$.2^2*$.3 - 2*$.1^3*$.2*$.3^2 + $.1*$.3^5 + $.2^4*$.3^2 - 2*$.2^3*$.3^3 + $.2^2*$.3^4 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -2, 6, 3, 0 ]) 2 4 ; 2 ([ -2, 4, 2, 0 ]) 3 ; 3 ([ -2, 2, 1, 0 ]) ; 4 ([ -2, 8, 4, 0 ]) 5 ; 5 ([ -2, 10, 5, 0 ]) 6 ; 6 ([ -2, 12, 6, 0 ]) 7 ; 7 ([ -2, 14, 7, 0 ]) 8 ; 8 ([ -1, 16, 8, 2 ]) ; Total time: 0.320 seconds, Total memory usage: 4.26MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:15:51 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1*x+0*y+0*(-x^3-y^2)+(0*x+1*y+1*(-x^3-y^2))^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:15:51 on modular [Seed = 4214063057] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x^6 + 2*x^3*y^2 - 2*x^3*y + x + y^4 - 2*y^3 + y^2 {@ @} false 2 1 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -2, 6, 3, 0 ]) 2 4 ; 2 ([ -2, 4, 2, 0 ]) 3 ; 3 ([ -2, 2, 1, 0 ]) ; 4 ([ -2, 8, 4, 0 ]) 5 ; 5 ([ -2, 10, 5, 0 ]) 6 ; 6 ([ -2, 12, 6, 0 ]) 7 ; 7 ([ -2, 14, 7, 0 ]) 8 ; 8 ([ -1, 16, 8, 2 ]) ; Total time: 0.320 seconds, Total memory usage: 4.26MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:04:53 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1*x+0*y+0*(-x^3-y^2)+(0*x+1*y+1*(-x^3-y^2))^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:04:53 on modular [Seed = 182226386] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x^6 + 2*x^3*y^2 - 2*x^3*y + x + y^4 - 2*y^3 + y^2 {@ @} false 2 1 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -2, 6, 3, 0 ]) 2 4 ; 2 ([ -2, 4, 2, 0 ]) 3 ; 3 ([ -2, 2, 1, 0 ]) ; 4 ([ -2, 8, 4, 0 ]) 5 ; 5 ([ -2, 10, 5, 0 ]) 6 ; 6 ([ -2, 12, 6, 0 ]) 7 ; 7 ([ -2, 14, 7, 0 ]) 8 ; 8 ([ -1, 16, 8, 2 ]) ; Total time: 0.330 seconds, Total memory usage: 4.26MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 11:58:04 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := x; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 11:58:03 on modular [Seed = 2487358689] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x {@ @} false 0 1 Curve over Rational Field defined by x Curve over Rational Field defined by x true >> g := ResolutionGraph(D2,Origin(A)); ^ Runtime error in 'ResolutionGraph': Point is not singular on curve >> g; ^ User error: Identifier 'g' has not been declared or assigned Total time: 0.220 seconds, Total memory usage: 3.53MB '129.162' ************** MAGMA ***************** Host 129.162.1.32 (129.162.1.32) Time: Tue Dec 6 10:49:56 2005 Input: 5+3 Output: Magma V2.11-10 Tue Dec 6 2005 10:49:56 on modular [Seed = 4197190320] ------------------------------------- 8 Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:18:40 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f :=[-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]), f7, f8, f9]; Q:=Matrix(S,3,3,f); print (Q*A*Transpose(Q)-A); for i in [1..9] do d:=Evaluate(Derivative(f[i]),0); print i; print Factorization(Numerator(d)); print Factorization(Denominator(d)); end for; Output: Magma V2.11-10 Tue Dec 6 2005 10:18:38 on modular [Seed = 887282836] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] 1 [ , , ] [ , , ] 2 [ , , ] [ , ] 3 [ , , ] [ , ] 4 [ , , ] [ , , ] 5 [ , , ] [ , , ] 6 [ , , ] [ , , ] 7 [ , , ] [ , , ] 8 [ , ] [ , , ] 9 [ , ] [ , ] Total time: 1.679 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:18:22 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f :=[-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]), f7, f8, f9]; Q:=Matrix(S,3,3,f); print (Q*A*Transpose(Q)-A); for i in [1..9] do: d:=Evaluate(Derivative(f[1]),0); print i; print Factorization(Numerator(d)); print Factorization(Denominator(d)); end for; Output: Magma V2.11-10 Tue Dec 6 2005 10:18:20 on modular [Seed = 1005442025] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> for i in [1..9] do: ^ User error: bad syntax >> print i; ^ User error: Identifier 'i' has not been declared or assigned [ , , ] [ , , ] >> end for; ^ User error: bad syntax Total time: 1.649 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:16:24 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f :=[-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]), f7, f8, f9]; Q:=Matrix(S,3,3,f); print (Q*A*Transpose(Q)-A); d1:=Evaluate(Derivative(f[1]),0); print d1; print Factorization(Numerator(d1)); print Factorization(Denominator(d1)); Output: Magma V2.11-10 Tue Dec 6 2005 10:16:22 on modular [Seed = 586467850] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) [ , , ] [ , , ] Total time: 1.649 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:13:57 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); d1:=Evaluate(Derivative(f1),0); print d1; print Factorization(Numerator(d1)); print Factorization(Denominator(d1)); Output: Magma V2.11-10 Tue Dec 6 2005 10:13:55 on modular [Seed = 704626932] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) [ , , ] [ , , ] Total time: 1.649 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:13:04 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); d1:=Evaluate(Derivative(f1),0); print d1; print Factorization(Nominator(d1)); print Factorization(Denominator(d1)); Output: Magma V2.11-10 Tue Dec 6 2005 10:13:02 on modular [Seed = 754631996] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) >> print Factorization(Nominator(d1)); ^ User error: Identifier 'Nominator' has not been declared or assigned [ , , ] Total time: 1.659 seconds, Total memory usage: 4.60MB '219.166' ************** MAGMA ***************** Host 219.166.29.134 (219.166.29.134) Time: Tue Dec 6 10:11:29 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+2*y+2*(-1*x*y-x^3)+(2+3*x+0*y+2*(-1*x*y-x^3))^2+(1+4*x+2*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:11:28 on modular [Seed = 1384430364] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 + 8*x^4*y - 12*x^4 + 54*x^3 + 4*x^2*y^2 + 84*x^2*y + 57*x^2 + 48*x*y^2 + 38*x*y + 25*x + 8*y^3 + 12*y^2 + 8*y + 6 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 19/2*x*y^4 + 12*x*y^3 + 3/2*y^6 + 2*y^5 + 3*y^4 + 2*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 19/2*x*y^4 + 12*x*y^3 + 3/2*y^6 + 2*y^5 + 3*y^4 + 2*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 1.070 seconds, Total memory usage: 3.91MB '219.166' ************** MAGMA ***************** Host 219.166.29.134 (219.166.29.134) Time: Tue Dec 6 10:10:58 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+2*y+2*(-1*x*y-x^3)+(2+3*x+1*y+2*(-1*x*y-x^3))^2+(1+4*x+2*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:10:57 on modular [Seed = 1502588982] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 + 8*x^4*y - 12*x^4 - 4*x^3*y + 54*x^3 + 4*x^2*y^2 + 84*x^2*y + 57*x^2 + 44*x*y^2 + 44*x*y + 25*x + 8*y^3 + 13*y^2 + 12*y + 6 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 11*x*y^4 + 11*x*y^3 + 3/2*y^6 + 3*y^5 + 13/4*y^4 + 2*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 11*x*y^4 + 11*x*y^3 + 3/2*y^6 + 3*y^5 + 13/4*y^4 + 2*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 1.350 seconds, Total memory usage: 4.01MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:10:56 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); L:=PolynomialRing(RationalField(),3); d1:=Evaluate(Derivative(f1),0); print d1; print Factorization(d1); Output: Magma V2.11-10 Tue Dec 6 2005 10:10:54 on modular [Seed = 1552594636] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) >> print Factorization(d1); ^ Runtime error in 'Factorization': Bad argument types Argument types given: FldFunRatMElt Total time: 1.669 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:10:40 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); L:=PolynomialRing(RationalField(),3); d1:=Evaluate(Derivative(f1),0); print d1; print L!d1; Output: Magma V2.11-10 Tue Dec 6 2005 10:10:38 on modular [Seed = 1133881795] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) >> print L!d1; ^ Runtime error in '!': Illegal coercion Total time: 1.649 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:09:17 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); L:=PolynomialRing(RationalField(),3); print (L!(Evaluate(Derivative(f1),0))); Output: Magma V2.11-10 Tue Dec 6 2005 10:09:15 on modular [Seed = 1218355901] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print (L!(Evaluate(Derivative(f1),0))); ^ Runtime error in '!': Illegal coercion Total time: 1.649 seconds, Total memory usage: 4.60MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 10:09:01 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+1*y+2*(-1*x*y-x^3)+(2+3*x+1*y+2*(-1*x*y-x^3))^2+(1+4*x+2*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:09:00 on modular [Seed = 2074972569] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 + 8*x^4*y - 12*x^4 - 4*x^3*y + 54*x^3 + 4*x^2*y^2 + 84*x^2*y + 57*x^2 + 44*x*y^2 + 44*x*y + 25*x + 8*y^3 + 13*y^2 + 11*y + 6 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 11*x*y^4 + 11*x*y^3 + 3/2*y^6 + 11/4*y^5 + 13/4*y^4 + 2*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 11*x*y^4 + 11*x*y^3 + 3/2*y^6 + 11/4*y^5 + 13/4*y^4 + 2*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 0.930 seconds, Total memory usage: 3.91MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:08:25 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); print Evaluate(Derivative(f1),0); Output: Magma V2.11-10 Tue Dec 6 2005 10:08:23 on modular [Seed = 2124977271] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) Total time: 1.649 seconds, Total memory usage: 4.60MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 10:08:03 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+1*y+2*(-0*x*y-x^3)+(2+3*x+1*y+2*(-0*x*y-x^3))^2+(1+4*x+2*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:08:02 on modular [Seed = 1706266457] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 - 12*x^4 - 4*x^3*y + 54*x^3 + 96*x^2*y + 57*x^2 + 48*x*y^2 + 54*x*y + 25*x + 8*y^3 + 13*y^2 + 11*y + 6 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 24*x^2*y^3 + 25/4*x*y^5 + 27/2*x*y^4 + 12*x*y^3 + 3/2*y^6 + 11/4*y^5 + 13/4*y^4 + 2*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 24*x^2*y^3 + 25/4*x*y^5 + 27/2*x*y^4 + 12*x*y^3 + 3/2*y^6 + 11/4*y^5 + 13/4*y^4 + 2*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 0.870 seconds, Total memory usage: 4.26MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:07:21 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); Output: Magma V2.11-10 Tue Dec 6 2005 10:07:20 on modular [Seed = 1824163370] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] Total time: 1.649 seconds, Total memory usage: 4.60MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 10:07:21 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+0*y+2*(-0*x*y-x^3)+(2+3*x+1*y+2*(-0*x*y-x^3))^2+(2+4*x+3*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:07:19 on modular [Seed = 2420538072] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 - 12*x^4 - 4*x^3*y + 54*x^3 + 144*x^2*y + 105*x^2 + 108*x*y^2 + 150*x*y + 61*x + 27*y^3 + 55*y^2 + 40*y + 13 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 105/4*x^2*y^4 + 36*x^2*y^3 + 61/4*x*y^5 + 75/2*x*y^4 + 27*x*y^3 + 13/4*y^6 + 10*y^5 + 55/4*y^4 + 27/4*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 105/4*x^2*y^4 + 36*x^2*y^3 + 61/4*x*y^5 + 75/2*x*y^4 + 27*x*y^3 + 13/4*y^6 + 10*y^5 + 55/4*y^4 + 27/4*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 2.189 seconds, Total memory usage: 4.24MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 10:06:38 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+1*y+2*(-0*x*y-x^3)+(2+3*x+1*y+2*(-0*x*y-x^3))^2+(2+4*x+3*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:06:36 on modular [Seed = 2605543432] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 - 12*x^4 - 4*x^3*y + 54*x^3 + 144*x^2*y + 105*x^2 + 108*x*y^2 + 150*x*y + 61*x + 27*y^3 + 55*y^2 + 41*y + 13 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 105/4*x^2*y^4 + 36*x^2*y^3 + 61/4*x*y^5 + 75/2*x*y^4 + 27*x*y^3 + 13/4*y^6 + 41/4*y^5 + 55/4*y^4 + 27/4*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 105/4*x^2*y^4 + 36*x^2*y^3 + 61/4*x*y^5 + 75/2*x*y^4 + 27*x*y^3 + 13/4*y^6 + 41/4*y^5 + 55/4*y^4 + 27/4*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 1.149 seconds, Total memory usage: 4.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:06:38 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); Output: Magma V2.11-10 Tue Dec 6 2005 10:06:36 on modular [Seed = 2555278091] ------------------------------------- [ 0 0 -2] [ 0 0 -2] [ 0 0 0] Total time: 1.659 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:06:21 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); Output: Magma V2.11-10 Tue Dec 6 2005 10:06:19 on modular [Seed = 2153147140] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] Total time: 1.649 seconds, Total memory usage: 4.60MB '219.166' ************** MAGMA ***************** Host 219.166.29.134 (219.166.29.134) Time: Tue Dec 6 10:06:00 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+1*y+2*(-0*x*y-x^3)+(2+3*x+1*y+2*(-0*x*y-x^3))^2+(2+4*x+3*y)^3; > C := Curve(A,f > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:05:59 on modular [Seed = 2270260287] ------------------------------------- Affine Space of dimension 2 Variables : x, y >> C; ^ User error: bad syntax >> SingularPoints(C); ^ User error: Identifier 'C' has not been declared or assigned >> HasSingularPointsOverExtension(C); ^ User error: Identifier 'C' has not been declared or assigned >> Genus(C); ^ User error: Identifier 'C' has not been declared or assigned >> NumberOfPunctures(C); ^ User error: Identifier 'C' has not been declared or assigned >> P:=ProjectiveClosure(C); ^ User error: Identifier 'C' has not been declared or assigned >> D:=AffinePatch(P,2); ^ User error: Identifier 'P' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> D2:=Curve(A,DefiningPolynomial(D)); ^ User error: Identifier 'D' has not been declared or assigned >> D2; ^ User error: Identifier 'D2' has not been declared or assigned >> IsAnalyticallyIrreducible(D2,Origin(A)); ^ User error: Identifier 'D2' has not been declared or assigned >> g := ResolutionGraph(D2,Origin(A)); ^ User error: Identifier 'D2' has not been declared or assigned >> g; ^ User error: Identifier 'g' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:06:00 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); Output: Magma V2.11-10 Tue Dec 6 2005 10:05:58 on modular [Seed = 2321310003] ------------------------------------- >> Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); ^ Runtime error in 'Matrix': Can't coerce sequence element 1 into the coefficient ring >> print (Q*A*Transpose(Q)-A); ^ User error: Identifier 'Q' has not been declared or assigned Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:04:57 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); Output: Magma V2.11-10 Tue Dec 6 2005 10:04:55 on modular [Seed = 2405000311] ------------------------------------- Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:03:58 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 10:03:57 on modular [Seed = 3026345862] ------------------------------------- ((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))/(t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) Total time: 1.439 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:02:46 2005 Input: K:=PolynomialRing(RationalField(),3); print Factorization((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)); print Factorization((a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)); print Factorization((2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)); print Factorization((a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)); print Factorization((-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)); print Factorization((a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)); Output: Magma V2.11-10 Tue Dec 6 2005 10:02:46 on modular [Seed = 3177928388] ------------------------------------- [ ] [ , , ] [ ] [ , , ] [ , , ] [ , , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:00:13 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Factorization((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)); print Factorization((t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))); Output: Magma V2.11-10 Tue Dec 6 2005 10:00:12 on modular [Seed = 1874430844] ------------------------------------- ((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))/(t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) [ ] [ ] Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 09:59:46 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Factorization((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)); print Factorization((t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)); Output: Magma V2.11-10 Tue Dec 6 2005 09:59:44 on modular [Seed = 365162781] ------------------------------------- ((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))/(t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) [ ] >> a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)); ^ User error: bad syntax Total time: 1.470 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 09:57:45 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 09:57:44 on modular [Seed = 670155430] ------------------------------------- ((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))/(t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 09:56:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 09:56:31 on modular [Seed = 4264041227] ------------------------------------- 0 0 0 0 Total time: 2.919 seconds, Total memory usage: 5.67MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 09:51:02 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); t0:=t*a*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4); print Evaluate(px/q,t0); print Evaluate(py/q,t0); print Evaluate(pz/q,t0); Output: Magma V2.11-10 Tue Dec 6 2005 09:51:00 on modular [Seed = 870700290] ------------------------------------- (-2*a + 1)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)*t/(t^2 - 2/(a^4*c + a^3*b*c + a^3*c^2 - 5/2*a^3*c + a^2*b*c^2 - 3/2*a^2*b*c - 3/2*a^2*c^2 + 2*a^2*c - 1/2*a*b*c^2 + 1/2*a*b*c + 1/2*a*c^2 - 1/2*a*c)*t + 1/(a^6*b*c^3 - 1/4*a^6*c^2 + a^5*b^2*c^3 + a^5*b*c^4 - 3*a^5*b*c^3 - 1/2*a^5*b*c^2 - 1/2*a^5*c^3 + a^5*c^2 + a^4*b^2*c^4 - 2*a^4*b^2*c^3 - 1/4*a^4*b^2*c^2 - 2*a^4*b*c^4 + 5/2*a^4*b*c^3 + 3/2*a^4*b*c^2 - 1/4*a^4*c^4 + 3/2*a^4*c^3 - 25/16*a^4*c^2 - a^3*b^2*c^4 + a^3*b^2*c^3 + 1/2*a^3*b^2*c^2 + a^3*b*c^4 - 13/8*a^3*b*c^2 + 1/2*a^3*c^4 - 13/8*a^3*c^3 + 19/16*a^3*c^2 + 1/4*a^2*b^2*c^4 - 5/16*a^2*b^2*c^2 - 11/16*a^2*b*c^3 + 3/4*a^2*b*c^2 - 5/16*a^2*c^4 + 3/4*a^2*c^3 - 7/16*a^2*c^2 - 1/16*a*b^2*c^3 + 1/16*a*b^2*c^2 - 1/16*a*b*c^4 + 3/16*a*b*c^3 - 1/8*a*b*c^2 + 1/16*a*c^4 - 1/8*a*c^3 + 1/16*a*c^2)) ((-a + c)/(a + c - 1)*t^2 + (a^2 - 2*a*b*c + a*b - a + 1/2*c)/(a^5*b*c^2 - 1/4*a^5*c + a^4*b^2*c^2 + a^4*b*c^3 - 5/2*a^4*b*c^2 - 1/2*a^4*b*c - 1/2*a^4*c^2 + 7/8*a^4*c + a^3*b^2*c^3 - 3/2*a^3*b^2*c^2 - 1/4*a^3*b^2*c - 3/2*a^3*b*c^3 + 5/4*a^3*b*c^2 + 5/4*a^3*b*c - 1/4*a^3*c^3 + 5/4*a^3*c^2 - 9/8*a^3*c - 1/2*a^2*b^2*c^3 + 1/4*a^2*b^2*c^2 + 3/8*a^2*b^2*c + 1/4*a^2*b*c^3 + 5/8*a^2*b*c^2 - a^2*b*c + 3/8*a^2*c^3 - a^2*c^2 + 5/8*a^2*c + 1/8*a*b^2*c^2 - 1/8*a*b^2*c + 1/8*a*b*c^3 - 3/8*a*b*c^2 + 1/4*a*b*c - 1/8*a*c^3 + 1/4*a*c^2 - 1/8*a*c)*t + 1/(a^6*b*c^3 - 1/4*a^6*c^2 + a^5*b^2*c^3 + a^5*b*c^4 - 3*a^5*b*c^3 - 1/2*a^5*b*c^2 - 1/2*a^5*c^3 + a^5*c^2 + a^4*b^2*c^4 - 2*a^4*b^2*c^3 - 1/4*a^4*b^2*c^2 - 2*a^4*b*c^4 + 5/2*a^4*b*c^3 + 3/2*a^4*b*c^2 - 1/4*a^4*c^4 + 3/2*a^4*c^3 - 25/16*a^4*c^2 - a^3*b^2*c^4 + a^3*b^2*c^3 + 1/2*a^3*b^2*c^2 + a^3*b*c^4 - 13/8*a^3*b*c^2 + 1/2*a^3*c^4 - 13/8*a^3*c^3 + 19/16*a^3*c^2 + 1/4*a^2*b^2*c^4 - 5/16*a^2*b^2*c^2 - 11/16*a^2*b*c^3 + 3/4*a^2*b*c^2 - 5/16*a^2*c^4 + 3/4*a^2*c^3 - 7/16*a^2*c^2 - 1/16*a*b^2*c^3 + 1/16*a*b^2*c^2 - 1/16*a*b*c^4 + 3/16*a*b*c^3 - 1/8*a*b*c^2 + 1/16*a*c^4 - 1/8*a*c^3 + 1/16*a*c^2))/(t^2 - 2/(a^4*c + a^3*b*c + a^3*c^2 - 5/2*a^3*c + a^2*b*c^2 - 3/2*a^2*b*c - 3/2*a^2*c^2 + 2*a^2*c - 1/2*a*b*c^2 + 1/2*a*b*c + 1/2*a*c^2 - 1/2*a*c)*t + 1/(a^6*b*c^3 - 1/4*a^6*c^2 + a^5*b^2*c^3 + a^5*b*c^4 - 3*a^5*b*c^3 - 1/2*a^5*b*c^2 - 1/2*a^5*c^3 + a^5*c^2 + a^4*b^2*c^4 - 2*a^4*b^2*c^3 - 1/4*a^4*b^2*c^2 - 2*a^4*b*c^4 + 5/2*a^4*b*c^3 + 3/2*a^4*b*c^2 - 1/4*a^4*c^4 + 3/2*a^4*c^3 - 25/16*a^4*c^2 - a^3*b^2*c^4 + a^3*b^2*c^3 + 1/2*a^3*b^2*c^2 + a^3*b*c^4 - 13/8*a^3*b*c^2 + 1/2*a^3*c^4 - 13/8*a^3*c^3 + 19/16*a^3*c^2 + 1/4*a^2*b^2*c^4 - 5/16*a^2*b^2*c^2 - 11/16*a^2*b*c^3 + 3/4*a^2*b*c^2 - 5/16*a^2*c^4 + 3/4*a^2*c^3 - 7/16*a^2*c^2 - 1/16*a*b^2*c^3 + 1/16*a*b^2*c^2 - 1/16*a*b*c^4 + 3/16*a*b*c^3 - 1/8*a*b*c^2 + 1/16*a*c^4 - 1/8*a*c^3 + 1/16*a*c^2)) ((2*a*c - c)/(a^2 + a*c - a)*t^2 + (-2*b + 1)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)*t)/(t^2 - 2/(a^4*c + a^3*b*c + a^3*c^2 - 5/2*a^3*c + a^2*b*c^2 - 3/2*a^2*b*c - 3/2*a^2*c^2 + 2*a^2*c - 1/2*a*b*c^2 + 1/2*a*b*c + 1/2*a*c^2 - 1/2*a*c)*t + 1/(a^6*b*c^3 - 1/4*a^6*c^2 + a^5*b^2*c^3 + a^5*b*c^4 - 3*a^5*b*c^3 - 1/2*a^5*b*c^2 - 1/2*a^5*c^3 + a^5*c^2 + a^4*b^2*c^4 - 2*a^4*b^2*c^3 - 1/4*a^4*b^2*c^2 - 2*a^4*b*c^4 + 5/2*a^4*b*c^3 + 3/2*a^4*b*c^2 - 1/4*a^4*c^4 + 3/2*a^4*c^3 - 25/16*a^4*c^2 - a^3*b^2*c^4 + a^3*b^2*c^3 + 1/2*a^3*b^2*c^2 + a^3*b*c^4 - 13/8*a^3*b*c^2 + 1/2*a^3*c^4 - 13/8*a^3*c^3 + 19/16*a^3*c^2 + 1/4*a^2*b^2*c^4 - 5/16*a^2*b^2*c^2 - 11/16*a^2*b*c^3 + 3/4*a^2*b*c^2 - 5/16*a^2*c^4 + 3/4*a^2*c^3 - 7/16*a^2*c^2 - 1/16*a*b^2*c^3 + 1/16*a*b^2*c^2 - 1/16*a*b*c^4 + 3/16*a*b*c^3 - 1/8*a*b*c^2 + 1/16*a*c^4 - 1/8*a*c^3 + 1/16*a*c^2)) Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 09:48:06 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); t0:=t*a*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/((a-1/2)^2*c); print Evaluate(px/q,t0); print Evaluate(py/q,t0); print Evaluate(pz/q,t0); Output: Magma V2.11-10 Tue Dec 6 2005 09:48:05 on modular [Seed = 4163772218] ------------------------------------- (-2*a^3*c + 3*a^2*c - 3/2*a*c + 1/4*c)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)*t/(t^2 + (-2*a + 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t + (a^2 - a + 1/4)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)) ((-a + c)/(a + c - 1)*t^2 + (a^3 - 2*a^2*b*c + a^2*b - 3/2*a^2 + a*b*c - 1/2*a*b + 1/2*a*c + 1/2*a - 1/4*c)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)*t + (a^2 - a + 1/4)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a))/(t^2 + (-2*a + 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t + (a^2 - a + 1/4)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)) ((2*a*c - c)/(a^2 + a*c - a)*t^2 + (-2*a^2*b*c + a^2*c + 2*a*b*c - a*c - 1/2*b*c + 1/4*c)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)*t)/(t^2 + (-2*a + 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t + (a^2 - a + 1/4)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)) Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 09:47:27 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); t0:=t*a*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/((a-1/2)^2*c); print Evaluate(p1,t0); print Evaluate(p2,t0); print Evaluate(py,t0); print Evaluate(q,t0); Output: Magma V2.11-10 Tue Dec 6 2005 09:47:26 on modular [Seed = 1351006025] ------------------------------------- (2*a^3*b*c - 1/2*a^3 + 2*a^2*b^2*c - 2*a^2*b*c - a^2*b - 1/2*a^2*c + a^2 - 1/2*a*b^2 - 1/2*a*b*c + a*b + 1/2*a*c - 1/2*a)/(a - 1/2)*t (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^2 - a + 1/4)*t + (-a*b + 1/2*a)/(a - 1/2) (-a^8*b^2*c^2 + 1/2*a^8*b*c - 1/16*a^8 - 2*a^7*b^3*c^2 + a^7*b^2*c^3 + 2*a^7*b^2*c^2 + 3/2*a^7*b^2*c - 3/2*a^7*b*c - 1/4*a^7*b - 1/16*a^7*c + 1/4*a^7 - a^6*b^4*c^2 + 2*a^6*b^3*c^3 + 2*a^6*b^3*c^2 + 3/2*a^6*b^3*c - 2*a^6*b^2*c^3 - 3/2*a^6*b^2*c^2 - 3*a^6*b^2*c - 3/8*a^6*b^2 - 1/2*a^6*b*c^3 + 1/2*a^6*b*c^2 + 11/8*a^6*b*c + 3/4*a^6*b + 1/16*a^6*c^2 + 1/8*a^6*c - 3/8*a^6 + a^5*b^4*c^3 + 1/2*a^5*b^4*c - 2*a^5*b^3*c^3 - a^5*b^3*c^2 - 3/2*a^5*b^3*c - 1/4*a^5*b^3 + 2*a^5*b^2*c^2 + 3/2*a^5*b^2*c + 3/4*a^5*b^2 + a^5*b*c^3 - 3/4*a^5*b*c^2 - 1/2*a^5*b*c - 3/4*a^5*b + 1/16*a^5*c^3 - 1/4*a^5*c^2 + 1/4*a^5 - 1/2*a^4*b^4*c^2 - 1/16*a^4*b^4 - 1/2*a^4*b^3*c^3 + 3/2*a^4*b^3*c^2 + 1/8*a^4*b^3*c + 1/4*a^4*b^3 + a^4*b^2*c^3 - 19/16*a^4*b^2*c^2 - 3/8*a^4*b^2*c - 3/8*a^4*b^2 - 3/8*a^4*b*c^3 - 1/8*a^4*b*c^2 + 3/8*a^4*b*c + 1/4*a^4*b - 1/8*a^4*c^3 + 5/16*a^4*c^2 - 1/8*a^4*c - 1/16*a^4 + 1/16*a^3*b^4*c + 1/8*a^3*b^3*c^2 - 1/4*a^3*b^3*c + 1/16*a^3*b^2*c^3 - 3/8*a^3*b^2*c^2 + 3/8*a^3*b^2*c - 1/8*a^3*b*c^3 + 3/8*a^3*b*c^2 - 1/4*a^3*b*c + 1/16*a^3*c^3 - 1/8*a^3*c^2 + 1/16*a^3*c)/(a^4*c - 2*a^3*c + 3/2*a^2*c - 1/2*a*c + 1/16*c)*t^2 + (a^6*b*c - 1/4*a^6 - 2*a^5*b^2*c^2 + 2*a^5*b^2*c - 3/2*a^5*b*c - 3/4*a^5*b - 1/4*a^5*c + 3/4*a^5 - 2*a^4*b^3*c^2 + a^4*b^3*c + 2*a^4*b^2*c^2 - a^4*b^2*c - 3/4*a^4*b^2 + a^4*b*c^2 - 1/2*a^4*b*c + 3/2*a^4*b + 3/8*a^4*c - 3/4*a^4 + 1/2*a^3*b^3*c - 1/4*a^3*b^3 + a^3*b^2*c^2 - 5/4*a^3*b^2*c + 3/4*a^3*b^2 - a^3*b*c^2 + 3/4*a^3*b*c - 3/4*a^3*b - 1/8*a^3*c^2 + 1/4*a^3 - 1/8*a^2*b^2*c - 1/8*a^2*b*c^2 + 1/4*a^2*b*c + 1/8*a^2*c^2 - 1/8*a^2*c)/(a^3*c - 3/2*a^2*c + 3/4*a*c - 1/8*c)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) (a^8*b^2*c^2 - 1/2*a^8*b*c + 1/16*a^8 + 2*a^7*b^3*c^2 + a^7*b^2*c^3 - 3*a^7*b^2*c^2 - 3/2*a^7*b^2*c - a^7*b*c^2 + 2*a^7*b*c + 1/4*a^7*b + 3/16*a^7*c - 5/16*a^7 + a^6*b^4*c^2 + 2*a^6*b^3*c^3 - 4*a^6*b^3*c^2 - 3/2*a^6*b^3*c - 2*a^6*b^2*c^3 + 1/2*a^6*b^2*c^2 + 9/2*a^6*b^2*c + 3/8*a^6*b^2 - 1/2*a^6*b*c^3 + 3*a^6*b*c^2 - 19/8*a^6*b*c - a^6*b + 3/16*a^6*c^2 - 3/4*a^6*c + 5/8*a^6 + a^5*b^4*c^3 - a^5*b^4*c^2 - 1/2*a^5*b^4*c - 2*a^5*b^3*c^3 + 3*a^5*b^3*c + 1/4*a^5*b^3 + 4*a^5*b^2*c^2 - 15/4*a^5*b^2*c - 9/8*a^5*b^2 + a^5*b*c^3 - 5/2*a^5*b*c^2 + 1/8*a^5*b*c + 3/2*a^5*b + 1/16*a^5*c^3 - 9/16*a^5*c^2 + 9/8*a^5*c - 5/8*a^5 - 1/2*a^4*b^4*c^2 + 1/2*a^4*b^4*c + 1/16*a^4*b^4 - 1/2*a^4*b^3*c^3 + 2*a^4*b^3*c^2 - 9/8*a^4*b^3*c - 1/2*a^4*b^3 + a^4*b^2*c^3 - 33/16*a^4*b^2*c^2 + 9/8*a^4*b^2 - 3/8*a^4*b*c^3 + 11/8*a^4*b*c - a^4*b - 1/8*a^4*c^3 + 9/16*a^4*c^2 - 3/4*a^4*c + 5/16*a^4 + 1/16*a^3*b^4*c - 1/16*a^3*b^4 + 1/8*a^3*b^3*c^2 - 3/8*a^3*b^3*c + 1/4*a^3*b^3 + 1/16*a^3*b^2*c^3 - 7/16*a^3*b^2*c^2 + 3/4*a^3*b^2*c - 3/8*a^3*b^2 - 1/8*a^3*b*c^3 + 1/2*a^3*b*c^2 - 5/8*a^3*b*c + 1/4*a^3*b + 1/16*a^3*c^3 - 3/16*a^3*c^2 + 3/16*a^3*c - 1/16*a^3)/(a^4*c - 2*a^3*c + 3/2*a^2*c - 1/2*a*c + 1/16*c)*t^2 + (-2*a^5*b^2*c^2 + a^5*b*c - 1/8*a^5 - 2*a^4*b^3*c^2 + 2*a^4*b^2*c^2 + 2*a^4*b^2*c + a^4*b*c^2 - 2*a^4*b*c - 3/8*a^4*b - 1/4*a^4*c + 3/8*a^4 + a^3*b^3*c + a^3*b^2*c^2 - 2*a^3*b^2*c - 3/8*a^3*b^2 - a^3*b*c^2 + 1/2*a^3*b*c + 3/4*a^3*b - 1/8*a^3*c^2 + 1/2*a^3*c - 3/8*a^3 - 1/8*a^2*b^3 - 1/4*a^2*b^2*c + 3/8*a^2*b^2 - 1/8*a^2*b*c^2 + 1/2*a^2*b*c - 3/8*a^2*b + 1/8*a^2*c^2 - 1/4*a^2*c + 1/8*a^2)/(a^3*c - 3/2*a^2*c + 3/4*a*c - 1/8*c)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 09:02:41 2005 Input: K:=PolynomialRing(RationalField(),3); print Factorization((-2*a^2*b*c^2 + a^2*c^2 + 2*a*b*c^2 - a*c^2 - 1/2*b*c^2 + 1/4*c^2)); print Factorization((a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)); print Factorization((-2*a^3*c^2 + 3*a^2*c^2 - 3/2*a*c^2 + 1/4*c^2)); print Factorization((a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)); print Factorization((a^2*c - a*c + 1/4*c)); print Factorization((a^2*b*c - 1/4*a^2 - 1/4*a*b - 1/4*a*c + 1/4*a)); Output: Magma V2.11-10 Tue Dec 6 2005 09:02:40 on modular [Seed = 2809233133] ------------------------------------- [ , , ] [ , , ] [ , ] [ , , ] [ , ] [ , ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 09:00:32 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); print S; p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(Derivative(pz/q),0); print Evaluate(Derivative(px/q),0); print Evaluate(Derivative(py/q),0); Output: Magma V2.11-10 Tue Dec 6 2005 09:00:30 on modular [Seed = 1906813765] ------------------------------------- Univariate rational function field over Multivariate Rational function field of rank 3 over Rational Field Variables: t (-2*a^2*b*c^2 + a^2*c^2 + 2*a*b*c^2 - a*c^2 - 1/2*b*c^2 + 1/4*c^2)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) (-2*a^3*c^2 + 3*a^2*c^2 - 3/2*a*c^2 + 1/4*c^2)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) (a^2*c - a*c + 1/4*c)/(a^2*b*c - 1/4*a^2 - 1/4*a*b - 1/4*a*c + 1/4*a) Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:59:59 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); print S; p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(p1); Output: Magma V2.11-10 Tue Dec 6 2005 08:59:58 on modular [Seed = 2091558504] ------------------------------------- Univariate rational function field over Multivariate Rational function field of rank 3 over Rational Field Variables: t 2*a*c - c Total time: 1.439 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:59:36 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); print S; p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(S!(px/q)); Output: Magma V2.11-10 Tue Dec 6 2005 08:59:35 on modular [Seed = 2074976630] ------------------------------------- Univariate rational function field over Multivariate Rational function field of rank 3 over Rational Field Variables: t ((2*a - 1)/(a + c - 1)*t^2 + (-2*a^3*b*c + 1/2*a^3 - 2*a^2*b^2*c + 2*a^2*b*c + a^2*b + 1/2*a^2*c - a^2 + 1/2*a*b^2 + 1/2*a*b*c - a*b - 1/2*a*c + 1/2*a)/(a^3*c^2 + 2*a^2*c^3 - 5/2*a^2*c^2 + a*c^4 - 3*a*c^3 + 2*a*c^2 - 1/2*c^4 + c^3 - 1/2*c^2))/(t^4 + (-4*a*b*c + a + b + c - 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t^3 + (2*a^4*b*c - 1/2*a^4 + 2*a^3*b^2*c + 2*a^3*b*c^2 - 4*a^3*b*c - a^3*b - a^3*c + 3/2*a^3 + 6*a^2*b^2*c^2 - 2*a^2*b^2*c - 1/2*a^2*b^2 - 2*a^2*b*c^2 - 3/2*a^2*b*c + 2*a^2*b - 1/2*a^2*c^2 + 2*a^2*c - 5/4*a^2 - 5/2*a*b^2*c + 1/2*a*b^2 - 5/2*a*b*c^2 + 7/2*a*b*c - 1/2*a*b + 1/2*a*c^2 - 1/2*a*c + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)*t^2 + (-4*a^4*b^2*c^2 + 2*a^4*b*c - 1/4*a^4 - 4*a^3*b^3*c^2 + 4*a^3*b^2*c^2 + 4*a^3*b^2*c + 2*a^3*b*c^2 - 4*a^3*b*c - 3/4*a^3*b - 1/2*a^3*c + 3/4*a^3 + 2*a^2*b^3*c + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 3/4*a^2*b^2 - 2*a^2*b*c^2 + a^2*b*c + 3/2*a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^3 - 1/2*a*b^2*c + 3/4*a*b^2 - 1/4*a*b*c^2 + a*b*c - 3/4*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)/(a^5*c^3 + 2*a^4*c^4 - 7/2*a^4*c^3 + a^3*c^5 - 5*a^3*c^4 + 19/4*a^3*c^3 - 3/2*a^2*c^5 + 9/2*a^2*c^4 - 25/8*a^2*c^3 + 3/4*a*c^5 - 7/4*a*c^4 + a*c^3 - 1/8*c^5 + 1/4*c^4 - 1/8*c^3)*t + (a^6*b^2*c^2 - 1/2*a^6*b*c + 1/16*a^6 + 2*a^5*b^3*c^2 - 2*a^5*b^2*c^2 - 3/2*a^5*b^2*c - 1/2*a^5*b*c^2 + 3/2*a^5*b*c + 1/4*a^5*b + 1/8*a^5*c - 1/4*a^5 + a^4*b^4*c^2 - 2*a^4*b^3*c^2 - 3/2*a^4*b^3*c + 3*a^4*b^2*c + 3/8*a^4*b^2 + a^4*b*c^2 - 9/8*a^4*b*c - 3/4*a^4*b + 1/16*a^4*c^2 - 3/8*a^4*c + 3/8*a^4 - 1/2*a^3*b^4*c - 1/2*a^3*b^3*c^2 + 3/2*a^3*b^3*c + 1/4*a^3*b^3 + a^3*b^2*c^2 - 9/8*a^3*b^2*c - 3/4*a^3*b^2 - 3/8*a^3*b*c^2 - 1/4*a^3*b*c + 3/4*a^3*b - 1/8*a^3*c^2 + 3/8*a^3*c - 1/4*a^3 + 1/16*a^2*b^4 + 1/8*a^2*b^3*c - 1/4*a^2*b^3 + 1/16*a^2*b^2*c^2 - 3/8*a^2*b^2*c + 3/8*a^2*b^2 - 1/8*a^2*b*c^2 + 3/8*a^2*b*c - 1/4*a^2*b + 1/16*a^2*c^2 - 1/8*a^2*c + 1/16*a^2)/(a^6*c^4 + 2*a^5*c^5 - 4*a^5*c^4 + a^4*c^6 - 6*a^4*c^5 + 13/2*a^4*c^4 - 2*a^3*c^6 + 7*a^3*c^5 - 11/2*a^3*c^4 + 3/2*a^2*c^6 - 4*a^2*c^5 + 41/16*a^2*c^4 - 1/2*a*c^6 + 9/8*a*c^5 - 5/8*a*c^4 + 1/16*c^6 - 1/8*c^5 + 1/16*c^4)) Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:58:51 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K,1); print S; p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(S!(px/q)); Output: Magma V2.11-10 Tue Dec 6 2005 08:58:50 on modular [Seed = 1722858550] ------------------------------------- Multivariate Rational function field of rank 1 over Multivariate Rational function field of rank 3 over Rational Field Variables: t >> print Derivative(S!(px/q)); ^ Runtime error in 'Derivative': Bad argument types Argument types given: FldFunRatMElt Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:58:39 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(S!(px/q)); Output: Magma V2.11-10 Tue Dec 6 2005 08:58:38 on modular [Seed = 1841018665] ------------------------------------- >> print Derivative(S!(px/q)); ^ Runtime error in 'Derivative': Bad argument types Argument types given: FldFunRatMElt Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:55:26 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(px/q); Output: Magma V2.11-10 Tue Dec 6 2005 08:55:24 on modular [Seed = 1807593581] ------------------------------------- >> print Derivative(px/q); ^ Runtime error in 'Derivative': Bad argument types Argument types given: FldFunRatMElt Total time: 1.439 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:52:47 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(p1); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:52:44 on modular [Seed = 1350993078] ------------------------------------- >> print Derivative(p1); ^ Runtime error in 'Derivative': Bad argument types Argument types given: FldFunRatMElt 0 0 0 0 Total time: 2.690 seconds, Total memory usage: 5.54MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:52:18 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(p1); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:52:15 on modular [Seed = 1602849690] ------------------------------------- >> print Derivative(p1); ^ Runtime error in 'Derivative': Bad argument types Argument types given: RngMPolElt 0 0 0 0 Total time: 2.690 seconds, Total memory usage: 5.50MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:52:01 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:51:58 on modular [Seed = 1519160942] ------------------------------------- 0 0 0 0 Total time: 2.680 seconds, Total memory usage: 5.76MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:45:47 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print G[1]; print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:45:45 on modular [Seed = 1133876736] ------------------------------------- $.1 + (-a^2 + a*c - b - c + 1)/(a*c + b*c - c)*$.7 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.8 - $.9 (-a^5*b*c - a^5*c^2*t^2 + 1/4*a^5 - 2*a^4*b^2*c + a^4*b*c^2*t^2 + 2*a^4*b*c^2*t + a^4*b*c*t + 2*a^4*b*c + 3/4*a^4*b + a^4*c^3*t^2 + a^4*c^2*t^2 + a^4*c^2*t - 3/2*a^4*c*t + 1/4*a^4*c - 3/4*a^4 - a^3*b^3*c + 2*a^3*b^2*c^2*t + a^3*b^2*c*t + 2*a^3*b^2*c + 3/4*a^3*b^2 - a^3*b*c^3*t^2 - 3*a^3*b*c^2*t^2 - 2*a^3*b*c^2*t - 4*a^3*b*c*t - 1/2*a^3*b*c - 3/2*a^3*b - 3*a^3*c^3*t^2 + 7/4*a^3*c^2*t^2 - 5/2*a^3*c^2*t + 13/4*a^3*c*t - 1/2*a^3*c + 3/4*a^3 + 1/4*a^2*b^3 - a^2*b^2*c^2*t - 3/2*a^2*b^2*c*t + 1/4*a^2*b^2*c - 3/4*a^2*b^2 + a^2*b*c^3*t^2 + 13/4*a^2*b*c^2*t^2 - 1/2*a^2*b*c^2*t + 15/4*a^2*b*c*t - 1/2*a^2*b*c + 3/4*a^2*b + 13/4*a^2*c^3*t^2 - 3*a^2*c^2*t^2 + 2*a^2*c^2*t - 9/4*a^2*c*t + 1/4*a^2*c - 1/4*a^2 + 1/2*a*b^2*c*t - 1/4*a*b*c^3*t^2 - 3/2*a*b*c^2*t^2 + 1/2*a*b*c^2*t - a*b*c*t - 3/2*a*c^3*t^2 + 3/2*a*c^2*t^2 - 1/2*a*c^2*t + 1/2*a*c*t + 1/4*b*c^2*t^2 + 1/4*c^3*t^2 - 1/4*c^2*t^2)/(a^5*b*c + a^5*c^2*t^2 - 1/4*a^5 + 2*a^4*b^2*c + a^4*b*c^2*t^2 - 2*a^4*b*c^2*t - 2*a^4*b*c - 3/4*a^4*b + a^4*c^3*t^2 - 3*a^4*c^2*t^2 + 1/2*a^4*c*t - 1/4*a^4*c + 3/4*a^4 + a^3*b^3*c - 2*a^3*b^2*c^2*t - 2*a^3*b^2*c - 3/4*a^3*b^2 + a^3*b*c^3*t^2 - 2*a^3*b*c^2*t^2 + 3*a^3*b*c^2*t + a^3*b*c*t + 1/2*a^3*b*c + 3/2*a^3*b - 2*a^3*c^3*t^2 + 13/4*a^3*c^2*t^2 + 1/2*a^3*c^2*t - 5/4*a^3*c*t + 1/2*a^3*c - 3/4*a^3 - 1/4*a^2*b^3 + a^2*b^2*c^2*t + 1/2*a^2*b^2*c*t - 1/4*a^2*b^2*c + 3/4*a^2*b^2 - a^2*b*c^3*t^2 + 5/4*a^2*b*c^2*t^2 - 1/2*a^2*b*c^2*t - 3/2*a^2*b*c*t + 1/2*a^2*b*c - 3/4*a^2*b + 5/4*a^2*c^3*t^2 - 3/2*a^2*c^2*t^2 - 3/4*a^2*c^2*t + a^2*c*t - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2*c*t + 1/4*a*b*c^3*t^2 - 1/4*a*b*c^2*t^2 - 1/4*a*b*c^2*t + 1/2*a*b*c*t - 1/4*a*c^3*t^2 + 1/4*a*c^2*t^2 + 1/4*a*c^2*t - 1/4*a*c*t) 0 0 0 0 Total time: 1.909 seconds, Total memory usage: 7.44MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:44:17 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print G[1]; print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:44:15 on modular [Seed = 1318623403] ------------------------------------- $.1 + (-a^2 + a*c - b - c + 1)/(a*c + b*c - c)*$.7 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.8 - $.9 ((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*$.1^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1 + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))/($.1^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1 + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) 0 0 0 0 Total time: 2.509 seconds, Total memory usage: 5.67MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:43:06 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(R,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[R.1,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:43:04 on modular [Seed = 3026346647] ------------------------------------- (($.1 + (-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a))*$.1^2 + ((-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c))*$.1 + ((a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)*$.1 + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)))/($.1^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1 + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) 0 0 0 0 Total time: 2.540 seconds, Total memory usage: 5.14MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:42:22 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[R.1,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:42:20 on modular [Seed = 2976344006] ------------------------------------- >> print Evaluate(G[1],[R.1,0,0,0,0,0,pz/q,px/q,py/q]); ^ Runtime error in [ ... ]: Could not find a valid universe 0 0 0 0 Total time: 2.490 seconds, Total memory usage: 5.79MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:40:46 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:40:43 on modular [Seed = 3211356207] ------------------------------------- 0 0 0 0 Total time: 2.500 seconds, Total memory usage: 5.68MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 08:40:33 2005 Input: G :=DirichletGroup(1680); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Tue Dec 6 2005 08:40:27 on modular [Seed = 2758964004] ------------------------------------- Group of Dirichlet characters of modulus 1680 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4, $.5, $.1*$.5, $.2*$.5, $.1*$.2*$.5, $.3*$.5, $.1*$.3*$.5, $.2*$.3*$.5, $.1*$.2*$.3*$.5, $.4*$.5, $.1*$.4*$.5, $.2*$.4*$.5, $.1*$.2*$.4*$.5, $.3*$.4*$.5, $.1*$.3*$.4*$.5, $.2*$.3*$.4*$.5, $.1*$.2*$.3*$.4*$.5 ] 4 2 Current total memory usage: 92.0MB, failed memory request: 80.9MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Integer Ring Total time: 5.690 seconds, Total memory usage: 92.02MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:37:51 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:37:48 on modular [Seed = 2537636080] ------------------------------------- (-4*a^3*b^2*c^2 + 4*a^3*b*c^2 + a^3*b*c - a^3*c - 4*a^2*b^3*c^2 + 10*a^2*b^2*c^2 + 6*a^2*b^2*c - 5*a^2*b*c^2 - 17/2*a^2*b*c - a^2*b - a^2*c^2 + 5/2*a^2*c + a^2 + 2*a*b^3*c^2 + 5*a*b^3*c - 3*a*b^2*c^2 - 12*a*b^2*c - 2*a*b^2 - 1/2*a*b*c^2 + 8*a*b*c + 4*a*b + 3/2*a*c^2 - a*c - 2*a - 1/2*b^3*c - b^3 - 1/2*b^2*c^2 + 1/2*b^2*c + 3*b^2 + b*c^2 + 1/2*b*c - 3*b - 1/2*c^2 - 1/2*c + 1)/(a^3*b*c - 1/2*a^3*c + 2*a^2*b*c^2 - 2*a^2*b*c - a^2*c^2 + a^2*c + a*b*c^3 - 2*a*b*c^2 + a*b*c - 1/2*a*c^3 + a*c^2 - 1/2*a*c)*$.1^2/($.1^4 + (-4*a*b*c + a + b + c - 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1^3 + (2*a^5*b*c^2 - 1/2*a^5*c + 2*a^4*b^2*c^2 + 2*a^4*b*c^3 - 5*a^4*b*c^2 - a^4*b*c - a^4*c^2 + 7/4*a^4*c + 2*a^3*b^2*c^3 - 3*a^3*b^2*c^2 - 1/2*a^3*b^2*c - 3*a^3*b*c^3 + 5/2*a^3*b*c^2 + 5/2*a^3*b*c - 1/2*a^3*c^3 + 5/2*a^3*c^2 - 9/4*a^3*c - a^2*b^2*c^3 + 9/2*a^2*b^2*c^2 + 3/4*a^2*b^2*c + 1/2*a^2*b*c^3 + 5/4*a^2*b*c^2 - 4*a^2*b*c + 3/4*a^2*c^3 - 2*a^2*c^2 + 5/4*a^2*c + 1/4*a^2 + 1/4*a*b^2*c^2 - 9/4*a*b^2*c + 1/4*a*b*c^3 - 11/4*a*b*c^2 + 5/2*a*b*c + 1/2*a*b - 1/4*a*c^3 + 1/2*a*c^2 + 1/4*a*c - 1/2*a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)*$.1^2 + (-4*a^4*b^2*c^2 + 2*a^4*b*c - 1/4*a^4 - 4*a^3*b^3*c^2 + 4*a^3*b^2*c^2 + 4*a^3*b^2*c + 2*a^3*b*c^2 - 4*a^3*b*c - 3/4*a^3*b - 1/2*a^3*c + 3/4*a^3 + 2*a^2*b^3*c + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 3/4*a^2*b^2 - 2*a^2*b*c^2 + a^2*b*c + 3/2*a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^3 - 1/2*a*b^2*c + 3/4*a*b^2 - 1/4*a*b*c^2 + a*b*c - 3/4*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)*$.1 + (a^6*b^2*c^2 - 1/2*a^6*b*c + 1/16*a^6 + 2*a^5*b^3*c^2 - 2*a^5*b^2*c^2 - 3/2*a^5*b^2*c - 1/2*a^5*b*c^2 + 3/2*a^5*b*c + 1/4*a^5*b + 1/8*a^5*c - 1/4*a^5 + a^4*b^4*c^2 - 2*a^4*b^3*c^2 - 3/2*a^4*b^3*c + 3*a^4*b^2*c + 3/8*a^4*b^2 + a^4*b*c^2 - 9/8*a^4*b*c - 3/4*a^4*b + 1/16*a^4*c^2 - 3/8*a^4*c + 3/8*a^4 - 1/2*a^3*b^4*c - 1/2*a^3*b^3*c^2 + 3/2*a^3*b^3*c + 1/4*a^3*b^3 + a^3*b^2*c^2 - 9/8*a^3*b^2*c - 3/4*a^3*b^2 - 3/8*a^3*b*c^2 - 1/4*a^3*b*c + 3/4*a^3*b - 1/8*a^3*c^2 + 3/8*a^3*c - 1/4*a^3 + 1/16*a^2*b^4 + 1/8*a^2*b^3*c - 1/4*a^2*b^3 + 1/16*a^2*b^2*c^2 - 3/8*a^2*b^2*c + 3/8*a^2*b^2 - 1/8*a^2*b*c^2 + 3/8*a^2*b*c - 1/4*a^2*b + 1/16*a^2*c^2 - 1/8*a^2*c + 1/16*a^2)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)) (-2*a^3*b*c^2 + 1/2*a^3*c - 2*a^2*b^2*c^2 + 3*a^2*b*c^2 + 3*a^2*b*c + 1/2*a^2*c^2 - 5/4*a^2*c - 1/2*a^2 + a*b^2*c^2 + 5/2*a*b^2*c - 1/2*a*b*c^2 - 7/2*a*b*c - a*b - 3/4*a*c^2 + 1/2*a*c + a - 1/4*b^2*c - 1/2*b^2 - 1/4*b*c^2 + b + 1/4*c^2 + 1/4*c - 1/2)/(a^2*b*c - 1/2*a^2*c + 2*a*b*c^2 - 2*a*b*c - a*c^2 + a*c + b*c^3 - 2*b*c^2 + b*c - 1/2*c^3 + c^2 - 1/2*c)*$.1^2/($.1^4 + (-4*a*b*c + a + b + c - 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1^3 + (2*a^5*b*c^2 - 1/2*a^5*c + 2*a^4*b^2*c^2 + 2*a^4*b*c^3 - 5*a^4*b*c^2 - a^4*b*c - a^4*c^2 + 7/4*a^4*c + 2*a^3*b^2*c^3 - 3*a^3*b^2*c^2 - 1/2*a^3*b^2*c - 3*a^3*b*c^3 + 5/2*a^3*b*c^2 + 5/2*a^3*b*c - 1/2*a^3*c^3 + 5/2*a^3*c^2 - 9/4*a^3*c - a^2*b^2*c^3 + 9/2*a^2*b^2*c^2 + 3/4*a^2*b^2*c + 1/2*a^2*b*c^3 + 5/4*a^2*b*c^2 - 4*a^2*b*c + 3/4*a^2*c^3 - 2*a^2*c^2 + 5/4*a^2*c + 1/4*a^2 + 1/4*a*b^2*c^2 - 9/4*a*b^2*c + 1/4*a*b*c^3 - 11/4*a*b*c^2 + 5/2*a*b*c + 1/2*a*b - 1/4*a*c^3 + 1/2*a*c^2 + 1/4*a*c - 1/2*a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)*$.1^2 + (-4*a^4*b^2*c^2 + 2*a^4*b*c - 1/4*a^4 - 4*a^3*b^3*c^2 + 4*a^3*b^2*c^2 + 4*a^3*b^2*c + 2*a^3*b*c^2 - 4*a^3*b*c - 3/4*a^3*b - 1/2*a^3*c + 3/4*a^3 + 2*a^2*b^3*c + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 3/4*a^2*b^2 - 2*a^2*b*c^2 + a^2*b*c + 3/2*a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^3 - 1/2*a*b^2*c + 3/4*a*b^2 - 1/4*a*b*c^2 + a*b*c - 3/4*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)*$.1 + (a^6*b^2*c^2 - 1/2*a^6*b*c + 1/16*a^6 + 2*a^5*b^3*c^2 - 2*a^5*b^2*c^2 - 3/2*a^5*b^2*c - 1/2*a^5*b*c^2 + 3/2*a^5*b*c + 1/4*a^5*b + 1/8*a^5*c - 1/4*a^5 + a^4*b^4*c^2 - 2*a^4*b^3*c^2 - 3/2*a^4*b^3*c + 3*a^4*b^2*c + 3/8*a^4*b^2 + a^4*b*c^2 - 9/8*a^4*b*c - 3/4*a^4*b + 1/16*a^4*c^2 - 3/8*a^4*c + 3/8*a^4 - 1/2*a^3*b^4*c - 1/2*a^3*b^3*c^2 + 3/2*a^3*b^3*c + 1/4*a^3*b^3 + a^3*b^2*c^2 - 9/8*a^3*b^2*c - 3/4*a^3*b^2 - 3/8*a^3*b*c^2 - 1/4*a^3*b*c + 3/4*a^3*b - 1/8*a^3*c^2 + 3/8*a^3*c - 1/4*a^3 + 1/16*a^2*b^4 + 1/8*a^2*b^3*c - 1/4*a^2*b^3 + 1/16*a^2*b^2*c^2 - 3/8*a^2*b^2*c + 3/8*a^2*b^2 - 1/8*a^2*b*c^2 + 3/8*a^2*b*c - 1/4*a^2*b + 1/16*a^2*c^2 - 1/8*a^2*c + 1/16*a^2)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)) ((-2*a^4*b*c^2 - a^4*b*c + a^4*c - 4*a^3*b^2*c^2 - 2*a^3*b^2*c + 2*a^3*b*c^3 + 6*a^3*b*c^2 + 13/2*a^3*b*c + a^3*b - a^3*c^2 - 5/2*a^3*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^3*c - 4*a^2*b^2*c^3 + 12*a^2*b^2*c^2 + 8*a^2*b^2*c + 2*a^2*b^2 + 3*a^2*b*c^3 - 29/2*a^2*b*c^2 - 10*a^2*b*c - 4*a^2*b - 2*a^2*c^3 + 7/2*a^2*c^2 + 3*a^2*c + 2*a^2 - 6*a*b^3*c^3 + 6*a*b^3*c^2 + 5/2*a*b^3*c + a*b^3 + 14*a*b^2*c^3 - 12*a*b^2*c^2 - 23/2*a*b^2*c - 3*a*b^2 - 10*a*b*c^3 + 9*a*b*c^2 + 23/2*a*b*c + 3*a*b + 3*a*c^3 - 3/2*a*c^2 - 7/2*a*c - a + 3*b^3*c^3 + 7/2*b^3*c^2 - 5*b^3*c - 6*b^2*c^3 - 13/2*b^2*c^2 + 11*b^2*c + 4*b*c^3 + 4*b*c^2 - 8*b*c - c^3 - c^2 + 2*c)/(a^3*b*c + a^3*c^2 - a^3*c + 3*a^2*b*c^2 - 3*a^2*b*c + 3*a^2*c^3 - 6*a^2*c^2 + 3*a^2*c + 3*a*b*c^3 - 6*a*b*c^2 + 3*a*b*c + 3*a*c^4 - 9*a*c^3 + 9*a*c^2 - 3*a*c + b*c^4 - 3*b*c^3 + 3*b*c^2 - b*c + c^5 - 4*c^4 + 6*c^3 - 4*c^2 + c)*$.1^4 + (-4*a^5*b^2*c^2 + 4*a^5*b*c^2 + a^5*b*c - a^5*c - 8*a^4*b^3*c^2 - 4*a^4*b^2*c^3 + 18*a^4*b^2*c^2 + 7*a^4*b^2*c - 8*a^4*b*c^2 - 21/2*a^4*b*c - a^4*b - a^4*c^2 + 7/2*a^4*c + a^4 - 4*a^3*b^4*c^2 + 8*a^3*b^3*c^3 + 16*a^3*b^3*c^2 + 11*a^3*b^3*c - 6*a^3*b^2*c^3 - 15*a^3*b^2*c^2 - 53/2*a^3*b^2*c - 3*a^3*b^2 + 5*a^3*b*c^3 + 4*a^3*b*c^2 + 18*a^3*b*c + 6*a^3*b + 3/2*a^3*c^2 - 7/2*a^3*c - 3*a^3 + 12*a^2*b^4*c^3 + 2*a^2*b^4*c^2 + 5*a^2*b^4*c - 28*a^2*b^3*c^3 - 18*a^2*b^3*c^2 - 35/2*a^2*b^3*c - 3*a^2*b^3 + 18*a^2*b^2*c^3 + 41/2*a^2*b^2*c^2 + 43/2*a^2*b^2*c + 9*a^2*b^2 - 9/2*a^2*b*c^3 - 19/2*a^2*b*c^2 - 19/2*a^2*b*c - 9*a^2*b - a^2*c^3 + 1/2*a^2*c^2 + 3/2*a^2*c + 3*a^2 - 6*a*b^4*c^3 - 15*a*b^4*c^2 - 1/2*a*b^4*c - a*b^4 + 9*a*b^3*c^3 + 35*a*b^3*c^2 + 6*a*b^3*c + 4*a*b^3 - a*b^2*c^3 - 55/2*a*b^2*c^2 - 10*a*b^2*c - 6*a*b^2 - 3*a*b*c^3 + 9*a*b*c^2 + 6*a*b*c + 4*a*b + 3/2*a*c^3 - 1/2*a*c^2 - 3/2*a*c - a + 3/2*b^4*c^2 + 3*b^4*c + 3/2*b^3*c^3 - 3/2*b^3*c^2 - 9*b^3*c - 3*b^2*c^3 - b^2*c^2 + 10*b^2*c + 2*b*c^3 + 3/2*b*c^2 - 5*b*c - 1/2*c^3 - 1/2*c^2 + c)/(a^4*b*c^2 + a^4*c^3 - a^4*c^2 + 3*a^3*b*c^3 - 7/2*a^3*b*c^2 + 3*a^3*c^4 - 13/2*a^3*c^3 + 7/2*a^3*c^2 + 3*a^2*b*c^4 - 15/2*a^2*b*c^3 + 9/2*a^2*b*c^2 + 3*a^2*c^5 - 21/2*a^2*c^4 + 12*a^2*c^3 - 9/2*a^2*c^2 + a*b*c^5 - 9/2*a*b*c^4 + 6*a*b*c^3 - 5/2*a*b*c^2 + a*c^6 - 11/2*a*c^5 + 21/2*a*c^4 - 17/2*a*c^3 + 5/2*a*c^2 - 1/2*b*c^5 + 3/2*b*c^4 - 3/2*b*c^3 + 1/2*b*c^2 - 1/2*c^6 + 2*c^5 - 3*c^4 + 2*c^3 - 1/2*c^2)*$.1^3 + (2*a^6*b^2*c^3 + a^6*b^2*c^2 - 3/2*a^6*b*c^2 - 1/4*a^6*b*c + 1/4*a^6*c - 2*a^5*b^3*c^3 + 3*a^5*b^3*c^2 + a^5*b^2*c^3 - 17/2*a^5*b^2*c^2 - 2*a^5*b^2*c - 5/2*a^5*b*c^3 + 7/2*a^5*b*c^2 + 29/8*a^5*b*c + 1/4*a^5*b + 3/4*a^5*c^2 - 9/8*a^5*c - 1/4*a^5 - 10*a^4*b^4*c^3 + 3*a^4*b^4*c^2 + 21*a^4*b^3*c^3 - 11/2*a^4*b^3*c^2 - 9/2*a^4*b^3*c - 29/2*a^4*b^2*c^3 + 13/4*a^4*b^2*c^2 + 11*a^4*b^2*c + a^4*b^2 + 19/4*a^4*b*c^3 + 33/8*a^4*b*c^2 - 33/4*a^4*b*c - 2*a^4*b + 1/2*a^4*c^3 - 21/8*a^4*c^2 + 5/4*a^4*c + a^4 - 6*a^3*b^5*c^3 + a^3*b^5*c^2 + 23*a^3*b^4*c^3 + 17/2*a^3*b^4*c^2 - 4*a^3*b^4*c - 55/2*a^3*b^3*c^3 - 95/4*a^3*b^3*c^2 + 43/4*a^3*b^3*c + 3/2*a^3*b^3 + 47/4*a^3*b^2*c^3 + 193/8*a^3*b^2*c^2 - 21/2*a^3*b^2*c - 9/2*a^3*b^2 - 1/4*a^3*b*c^3 - 53/4*a^3*b*c^2 + 13/4*a^3*b*c + 9/2*a^3*b - 5/4*a^3*c^3 + 23/8*a^3*c^2 + 1/2*a^3*c - 3/2*a^3 + 3*a^2*b^5*c^3 + 7*a^2*b^5*c^2 - 5/4*a^2*b^5*c - 15/2*a^2*b^4*c^3 - 97/4*a^2*b^4*c^2 + 7/4*a^2*b^4*c + a^2*b^4 + 17/4*a^2*b^3*c^3 + 251/8*a^2*b^3*c^2 + 9/4*a^2*b^3*c - 4*a^2*b^3 + 5/2*a^2*b^2*c^3 - 159/8*a^2*b^2*c^2 - 25/4*a^2*b^2*c + 6*a^2*b^2 - 13/4*a^2*b*c^3 + 53/8*a^2*b*c^2 + 5*a^2*b*c - 4*a^2*b + a^2*c^3 - 7/8*a^2*c^2 - 3/2*a^2*c + a^2 - 3/4*a*b^5*c^2 - 11/8*a*b^5*c + 1/4*a*b^5 - 3/4*a*b^4*c^3 + 13/8*a*b^4*c^2 + 45/8*a*b^4*c - 5/4*a*b^4 + 9/4*a*b^3*c^3 - 3/4*a*b^3*c^2 - 37/4*a*b^3*c + 5/2*a*b^3 - 5/2*a*b^2*c^3 - 1/2*a*b^2*c^2 + 31/4*a*b^2*c - 5/2*a*b^2 + 5/4*a*b*c^3 + 1/2*a*b*c^2 - 27/8*a*b*c + 5/4*a*b - 1/4*a*c^3 - 1/8*a*c^2 + 5/8*a*c - 1/4*a)/(a^4*b*c^2 + a^4*c^3 - a^4*c^2 + 3*a^3*b*c^3 - 7/2*a^3*b*c^2 + 3*a^3*c^4 - 13/2*a^3*c^3 + 7/2*a^3*c^2 + 3*a^2*b*c^4 - 15/2*a^2*b*c^3 + 9/2*a^2*b*c^2 + 3*a^2*c^5 - 21/2*a^2*c^4 + 12*a^2*c^3 - 9/2*a^2*c^2 + a*b*c^5 - 9/2*a*b*c^4 + 6*a*b*c^3 - 5/2*a*b*c^2 + a*c^6 - 11/2*a*c^5 + 21/2*a*c^4 - 17/2*a*c^3 + 5/2*a*c^2 - 1/2*b*c^5 + 3/2*b*c^4 - 3/2*b*c^3 + 1/2*b*c^2 - 1/2*c^6 + 2*c^5 - 3*c^4 + 2*c^3 - 1/2*c^2)*$.1^2)/ ** WARNING: Output too long, hence truncated. '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:31:46 2005 Input: R:=PolynomialRing(RationalField(),3); print Factorization((a^2*c + a*c^2 - a*c)); print Factorization((-2*a^2*b*c + 1/2*a^2 + 1/2*a*b + 1/2*a*c - 1/2*a)); print Factorization((a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)); print Factorization((a^2*c - a*c + 1/4*c)); Output: Magma V2.11-10 Tue Dec 6 2005 08:31:46 on modular [Seed = 2655538967] ------------------------------------- [ , , ] [ , ] [ , , ] [ , ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:27:45 2005 Input: R:=PolynomialRing(RationalField(),3); print Factorization((-a^2*c + a*c^2)); print Factorization((a^3 - 2*a^2*b*c + a^2*b - a^2 + 1/2*a*c)); print Factorization((a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)); print Factorization((a^2*c - a*c + 1/4*c)); Output: Magma V2.11-10 Tue Dec 6 2005 08:27:44 on modular [Seed = 2605533440] ------------------------------------- [ , , ] [ , ] [ , , ] [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:26:25 2005 Input: K:=FunctionField(RationalField(),3); S:=PolynomialRing(K,1); p1:= (2*a*c - c)*t; p2:= c*t + (-a*b + 1/2*a)/(a - 1/2); py:= (-a^2*c + a*c^2)*t^2 + (a^3 - 2*a^2*b*c + a^2*b - a^2 + 1/2*a*c)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c); q := (a^2*c + a*c^2 - a*c)*t^2 + (-2*a^2*b*c + 1/2*a^2 + 1/2*a*b + 1/2*a*c - 1/2*a)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c); print Factorization(p1); print Factorization(p2); print Factorization(py); print Factorization(q); Output: Magma V2.11-10 Tue Dec 6 2005 08:26:24 on modular [Seed = 2169983818] ------------------------------------- [ ] [ ] [ ] [ ] Total time: 0.210 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:24:56 2005 Input: K:=FunctionField(RationalField(),3); S:=PolynomialRing(K,1); print Factorization((-a^2*c + a*c^2)); Output: Magma V2.11-10 Tue Dec 6 2005 08:24:56 on modular [Seed = 2287885035] ------------------------------------- >> print Factorization((-a^2*c + a*c^2)); ^ Runtime error in 'Factorization': Bad argument types Argument types given: FldFunRatMElt Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:24:03 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); t0 := (a+b-1)/(c*(1-2*a)); print Evaluate(p1,[t+t0]); print Evaluate(p2,[t+t0]); print Evaluate(py,[t+t0]); print Evaluate(q,[t+t0]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:24:02 on modular [Seed = 4045873736] ------------------------------------- (2*a*c - c)*t c*t + (-a*b + 1/2*a)/(a - 1/2) (-a^2*c + a*c^2)*t^2 + (a^3 - 2*a^2*b*c + a^2*b - a^2 + 1/2*a*c)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) (a^2*c + a*c^2 - a*c)*t^2 + (-2*a^2*b*c + 1/2*a^2 + 1/2*a*b + 1/2*a*c - 1/2*a)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) 0 0 0 0 Total time: 1.730 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:20:28 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); t0 := (a+b-1)/(c*(1-2*a)); print Evaluate(p1,[t+t0]); print Evaluate(p2,[t+t0]); print Evaluate(py,[t+t0]); print Evaluate(q,[t+t0]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:20:26 on modular [Seed = 3590325639] ------------------------------------- (2*a*c - c)*t c*t + (-a*b + 1/2*a)/(a - 1/2) (-a^2*c + a*c^2)*t^2 + (a^3 - 2*a^2*b*c + a^2*b - a^2 + 1/2*a*c)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) (a^2*c + a*c^2 - a*c)*t^2 + (-2*a^2*b*c + 1/2*a^2 + 1/2*a*b + 1/2*a*c - 1/2*a)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) 0 0 0 0 Total time: 1.610 seconds, Total memory usage: 4.37MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:19:52 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); t0 := (a+b-1)/(c*(1-2*a)); print Evaluate(p1,[t+t0]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:19:50 on modular [Seed = 3658225255] ------------------------------------- (2*a*c - c)*t 0 0 0 0 Total time: 1.610 seconds, Total memory usage: 4.37MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:18:46 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(px,[(a+b-1)/(c*(1-2*a))]); print Evaluate(py,[(a+b-1)/(c*(1-2*a))]); print Evaluate(pz,[(a+b-1)/(c*(1-2*a))]); print Evaluate(q,[(a+b-1)/(c*(1-2*a))]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:18:44 on modular [Seed = 3239513406] ------------------------------------- 0 (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) 0 (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) 0 0 0 0 Total time: 1.600 seconds, Total memory usage: 4.37MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:18:18 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(px,[(a+b-1)/(c*(1-2*a))]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:18:16 on modular [Seed = 3457679901] ------------------------------------- 0 0 0 0 0 Total time: 1.629 seconds, Total memory usage: 4.37MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:17:34 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(px,[(a+b-1)/(c*(1-2*a))]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:17:33 on modular [Seed = 820444182] ------------------------------------- >> print Evaluate(px,[(a+b-1)/(c*(1-2*a))]); ^ Runtime error in 'Evaluate': Bad argument types Argument types given: FldFunRatMElt, SeqEnum[FldFunRatMElt] 0 0 0 0 Total time: 1.409 seconds, Total memory usage: 4.46MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:17:18 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(px,(a+b-1)/(c*(1-2*a))); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:17:16 on modular [Seed = 1055453737] ------------------------------------- >> print Evaluate(px,(a+b-1)/(c*(1-2*a))); ^ Runtime error in 'Evaluate': Bad argument types Argument types given: FldFunRatMElt, FldFunRatMElt 0 0 0 0 Total time: 1.419 seconds, Total memory usage: 4.46MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:16:01 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:16:00 on modular [Seed = 569896314] ------------------------------------- 0 0 0 0 Total time: 1.409 seconds, Total memory usage: 4.46MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:15:39 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:15:37 on modular [Seed = 788063132] ------------------------------------- 0 Total time: 1.340 seconds, Total memory usage: 4.33MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:14:35 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:14:33 on modular [Seed = 348314086] ------------------------------------- >> p1 := t * c*(2*a-1) + (a+b-1); ^ User error: Identifier 't' has not been declared or assigned >> p2 := t * c + (1-b); ^ User error: Identifier 't' has not been declared or assigned >> px := -a * p1; ^ User error: Identifier 'p1' has not been declared or assigned >> pz := p1 * p2; ^ User error: Identifier 'p1' has not been declared or assigned >> py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); ^ User error: Identifier 't' has not been declared or assigned >> q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); ^ User error: Identifier 't' has not been declared or assigned >> print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); ^ User error: Identifier 'pz' has not been declared or assigned Total time: 1.459 seconds, Total memory usage: 4.60MB '66.69.1' ************** MAGMA ***************** Host 66.69.190.149 (66.69.190.149) Time: Tue Dec 6 08:03:45 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Tue Dec 6 2005 08:03:45 on modular [Seed = 2692116437] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.210 seconds, Total memory usage: 3.24MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:09:58 2005 Input: G:=GroupOfLieType("A2", 9) Output: Magma V2.11-10 Tue Dec 6 2005 07:09:57 on modular [Seed = 2091532460] ------------------------------------- Total time: 0.240 seconds, Total memory usage: 4.95MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:06:40 2005 Input: ?GroupOfLieType ?2 Output: Magma V2.11-10 Tue Dec 6 2005 07:06:39 on modular [Seed = 904071356] ------------------------------------- 12 matches: 1 I /magma/group/fiurg/related/GroupOfLieType 2 I /magma/group/groups-lie-type/construction/GroupOfLieType 3 I /magma/group/groups-lie-type/construction/GroupOfLieType 4 I /magma/group/groups-lie-type/construction/GroupOfLieType 5 I /magma/group/groups-lie-type/construction/GroupOfLieType 6 I /magma/group/groups-lie-type/construction/GroupOfLieType 7 I /magma/group/groups-lie-type/construction/GroupOfLieType 8 I /magma/group/groups-lie-type/construction/GroupOfLieType 9 I /magma/group/groups-lie-type/construction/GroupOfLieType 10 I /magma/group/root-data/related-structures/GroupOfLieType 11 I /magma/lie-theory/coxeter-description/related/GroupOfLieType 12 I /magma/lie-theory/finite-coxeter-groups/related/GroupOfLieType To view an entry, type ? followed by the number next to it =============================================================================== PATH: /magma/group/groups-lie-type/construction/GroupOfLieType KIND: Intrinsic =============================================================================== GroupOfLieType(C, k) : Mtrx, Rng -> GrpLie GroupOfLieType(D, k) : Mtrx, Rng -> GrpLie Isogeny: BoolElt Default: "Ad" Normalising: BoolElt Default: true Construct the group of Lie type with Cartan matrix C or Dynkin digraph D, over the ring k. The optional parameter Isogeny can take the values described in Section H84E17. =============================================================================== Total time: 0.210 seconds, Total memory usage: 3.24MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:06:25 2005 Input: ?GroupOfLieType Output: Magma V2.11-10 Tue Dec 6 2005 07:06:25 on modular [Seed = 687740335] ------------------------------------- 12 matches: 1 I /magma/group/fiurg/related/GroupOfLieType 2 I /magma/group/groups-lie-type/construction/GroupOfLieType 3 I /magma/group/groups-lie-type/construction/GroupOfLieType 4 I /magma/group/groups-lie-type/construction/GroupOfLieType 5 I /magma/group/groups-lie-type/construction/GroupOfLieType 6 I /magma/group/groups-lie-type/construction/GroupOfLieType 7 I /magma/group/groups-lie-type/construction/GroupOfLieType 8 I /magma/group/groups-lie-type/construction/GroupOfLieType 9 I /magma/group/groups-lie-type/construction/GroupOfLieType 10 I /magma/group/root-data/related-structures/GroupOfLieType 11 I /magma/lie-theory/coxeter-description/related/GroupOfLieType 12 I /magma/lie-theory/finite-coxeter-groups/related/GroupOfLieType To view an entry, type ? followed by the number next to it Total time: 0.200 seconds, Total memory usage: 3.24MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:06:16 2005 Input: GroupOfLieType? Output: Magma V2.11-10 Tue Dec 6 2005 07:06:16 on modular [Seed = 804852433] ------------------------------------- >> GroupOfLieType?; ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:05:57 2005 Input: ? Output: Magma V2.11-10 Tue Dec 6 2005 07:05:56 on modular [Seed = 586424256] ------------------------------------- =============================================================================== PATH: /language/statement KIND: Overview =============================================================================== A statement is a complete command to Magma. Every statement must end with a semicolon ( ; ). If you press the key before typing ; then Magma may give you a special prompt symbol such as print> or if> to remind you that the statement is not yet finished. EXAMPLE: > print 5+8; > DivBy3 := func< n | IsZero(n mod 3) >; =============================================================================== Total time: 0.200 seconds, Total memory usage: 3.24MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:05:52 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Tue Dec 6 2005 07:05:48 on modular [Seed = 653270012] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.230 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:29:16 2005 Input: G :=DirichletGroup(784,CyclotomicField(168)); G; X :=Elements(G); X; Output: Magma V2.11-10 Tue Dec 6 2005 04:29:16 on modular [Seed = 1757311683] ------------------------------------- Group of Dirichlet characters of modulus 784 over Cyclotomic Field of order 168 and degree 48 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.2^2*$.3, $.1*$.2^2*$.3, $.2^3*$.3, $.1*$.2^3*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.2^2*$.3^2, $.1*$.2^2*$.3^2, $.2^3*$.3^2, $.1*$.2^3*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.2^2*$.3^3, $.1*$.2^2*$.3^3, $.2^3*$.3^3, $.1*$.2^3*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.2^2*$.3^4, $.1*$.2^2*$.3^4, $.2^3*$.3^4, $.1*$.2^3*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5, $.2^2*$.3^5, $.1*$.2^2*$.3^5, $.2^3*$.3^5, $.1*$.2^3*$.3^5, $.3^6, $.1*$.3^6, $.2*$.3^6, $.1*$.2*$.3^6, $.2^2*$.3^6, $.1*$.2^2*$.3^6, $.2^3*$.3^6, $.1*$.2^3*$.3^6, $.3^7, $.1*$.3^7, $.2*$.3^7, $.1*$.2*$.3^7, $.2^2*$.3^7, $.1*$.2^2*$.3^7, $.2^3*$.3^7, $.1*$.2^3*$.3^7, $.3^8, $.1*$.3^8, $.2*$.3^8, $.1*$.2*$.3^8, $.2^2*$.3^8, $.1*$.2^2*$.3^8, $.2^3*$.3^8, $.1*$.2^3*$.3^8, $.3^9, $.1*$.3^9, $.2*$.3^9, $.1*$.2*$.3^9, $.2^2*$.3^9, $.1*$.2^2*$.3^9, $.2^3*$.3^9, $.1*$.2^3*$.3^9, $.3^10, $.1*$.3^10, $.2*$.3^10, $.1*$.2*$.3^10, $.2^2*$.3^10, $.1*$.2^2*$.3^10, $.2^3*$.3^10, $.1*$.2^3*$.3^10, $.3^11, $.1*$.3^11, $.2*$.3^11, $.1*$.2*$.3^11, $.2^2*$.3^11, $.1*$.2^2*$.3^11, $.2^3*$.3^11, $.1*$.2^3*$.3^11, $.3^12, $.1*$.3^12, $.2*$.3^12, $.1*$.2*$.3^12, $.2^2*$.3^12, $.1*$.2^2*$.3^12, $.2^3*$.3^12, $.1*$.2^3*$.3^12, $.3^13, $.1*$.3^13, $.2*$.3^13, $.1*$.2*$.3^13, $.2^2*$.3^13, $.1*$.2^2*$.3^13, $.2^3*$.3^13, $.1*$.2^3*$.3^13, $.3^14, $.1*$.3^14, $.2*$.3^14, $.1*$.2*$.3^14, $.2^2*$.3^14, $.1*$.2^2*$.3^14, $.2^3*$.3^14, $.1*$.2^3*$.3^14, $.3^15, $.1*$.3^15, $.2*$.3^15, $.1*$.2*$.3^15, $.2^2*$.3^15, $.1*$.2^2*$.3^15, $.2^3*$.3^15, $.1*$.2^3*$.3^15, $.3^16, $.1*$.3^16, $.2*$.3^16, $.1*$.2*$.3^16, $.2^2*$.3^16, $.1*$.2^2*$.3^16, $.2^3*$.3^16, $.1*$.2^3*$.3^16, $.3^17, $.1*$.3^17, $.2*$.3^17, $.1*$.2*$.3^17, $.2^2*$.3^17, $.1*$.2^2*$.3^17, $.2^3*$.3^17, $.1*$.2^3*$.3^17, $.3^18, $.1*$.3^18, $.2*$.3^18, $.1*$.2*$.3^18, $.2^2*$.3^18, $.1*$.2^2*$.3^18, $.2^3*$.3^18, $.1*$.2^3*$.3^18, $.3^19, $.1*$.3^19, $.2*$.3^19, $.1*$.2*$.3^19, $.2^2*$.3^19, $.1*$.2^2*$.3^19, $.2^3*$.3^19, $.1*$.2^3*$.3^19, $.3^20, $.1*$.3^20, $.2*$.3^20, $.1*$.2*$.3^20, $.2^2*$.3^20, $.1*$.2^2*$.3^20, $.2^3*$.3^20, $.1*$.2^3*$.3^20, $.3^21, $.1*$.3^21, $.2*$.3^21, $.1*$.2*$.3^21, $.2^2*$.3^21, $.1*$.2^2*$.3^21, $.2^3*$.3^21, $.1*$.2^3*$.3^21, $.3^22, $.1*$.3^22, $.2*$.3^22, $.1*$.2*$.3^22, $.2^2*$.3^22, $.1*$.2^2*$.3^22, $.2^3*$.3^22, $.1*$.2^3*$.3^22, $.3^23, $.1*$.3^23, $.2*$.3^23, $.1*$.2*$.3^23, $.2^2*$.3^23, $.1*$.2^2*$.3^23, $.2^3*$.3^23, $.1*$.2^3*$.3^23, $.3^24, $.1*$.3^24, $.2*$.3^24, $.1*$.2*$.3^24, $.2^2*$.3^24, $.1*$.2^2*$.3^24, $.2^3*$.3^24, $.1*$.2^3*$.3^24, $.3^25, $.1*$.3^25, $.2*$.3^25, $.1*$.2*$.3^25, $.2^2*$.3^25, $.1*$.2^2*$.3^25, $.2^3*$.3^25, $.1*$.2^3*$.3^25, $.3^26, $.1*$.3^26, $.2*$.3^26, $.1*$.2*$.3^26, $.2^2*$.3^26, $.1*$.2^2*$.3^26, $.2^3*$.3^26, $.1*$.2^3*$.3^26, $.3^27, $.1*$.3^27, $.2*$.3^27, $.1*$.2*$.3^27, $.2^2*$.3^27, $.1*$.2^2*$.3^27, $.2^3*$.3^27, $.1*$.2^3*$.3^27, $.3^28, $.1*$.3^28, $.2*$.3^28, $.1*$.2*$.3^28, $.2^2*$.3^28, $.1*$.2^2*$.3^28, $.2^3*$.3^28, $.1*$.2^3*$.3^28, $.3^29, $.1*$.3^29, $.2*$.3^29, $.1*$.2*$.3^29, $.2^2*$.3^29, $.1*$.2^2*$.3^29, $.2^3*$.3^29, $.1*$.2^3*$.3^29, $.3^30, $.1*$.3^30, $.2*$.3^30, $.1*$.2*$.3^30, $.2^2*$.3^30, $.1*$.2^2*$.3^30, $.2^3*$.3^30, $.1*$.2^3*$.3^30, $.3^31, $.1*$.3^31, $.2*$.3^31, $.1*$.2*$.3^31, $.2^2*$.3^31, $.1*$.2^2*$.3^31, $.2^3*$.3^31, $.1*$.2^3*$.3^31, $.3^32, $.1*$.3^32, $.2*$.3^32, $.1*$.2*$.3^32, $.2^2*$.3^32, $.1*$.2^2*$.3^32, $.2^3*$.3^32, $.1*$.2^3*$.3^32, $.3^33, $.1*$.3^33, $.2*$.3^33, $.1*$.2*$.3^33, $.2^2*$.3^33, $.1*$.2^2*$.3^33, $.2^3*$.3^33, $.1*$.2^3*$.3^33, $.3^34, $.1*$.3^34, $.2*$.3^34, $.1*$.2*$.3^34, $.2^2*$.3^34, $.1*$.2^2*$.3^34, $.2^3*$.3^34, $.1*$.2^3*$.3^34, $.3^35, $.1*$.3^35, $.2*$.3^35, $.1*$.2*$.3^35, $.2^2*$.3^35, $.1*$.2^2*$.3^35, $.2^3*$.3^35, $.1*$.2^3*$.3^35, $.3^36, $.1*$.3^36, $.2*$.3^36, $.1*$.2*$.3^36, $.2^2*$.3^36, $.1*$.2^2*$.3^36, $.2^3*$.3^36, $.1*$.2^3*$.3^36, $.3^37, $.1*$.3^37, $.2*$.3^37, $.1*$.2*$.3^37, $.2^2*$.3^37, $.1*$.2^2*$.3^37, $.2^3*$.3^37, $.1*$.2^3*$.3^37, $.3^38, $.1*$.3^38, $.2*$.3^38, $.1*$.2*$.3^38, $.2^2*$.3^38, $.1*$.2^2*$.3^38, $.2^3*$.3^38, $.1*$.2^3*$.3^38, $.3^39, $.1*$.3^39, $.2*$.3^39, $.1*$.2*$.3^39, $.2^2*$.3^39, $.1*$.2^2*$.3^39, $.2^3*$.3^39, $.1*$.2^3*$.3^39, $.3^40, $.1*$.3^40, $.2*$.3^40, $.1*$.2*$.3^40, $.2^2*$.3^40, $.1*$.2^2*$.3^40, $.2^3*$.3^40, $.1*$.2^3*$.3^40, $.3^41, $.1*$.3^41, $.2*$.3^41, $.1*$.2*$.3^41, $.2^2*$.3^41, $.1*$.2^2*$.3^41, $.2^3*$.3^41, $.1*$.2^3*$.3^41 ] Total time: 0.210 seconds, Total memory usage: 3.53MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:21:54 2005 Input: G :=DirichletGroup(784,CyclotomicField(168)); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Tue Dec 6 2005 04:21:50 on modular [Seed = 3161106386] ------------------------------------- Group of Dirichlet characters of modulus 784 over Cyclotomic Field of order 168 and degree 48 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.2^2*$.3, $.1*$.2^2*$.3, $.2^3*$.3, $.1*$.2^3*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.2^2*$.3^2, $.1*$.2^2*$.3^2, $.2^3*$.3^2, $.1*$.2^3*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.2^2*$.3^3, $.1*$.2^2*$.3^3, $.2^3*$.3^3, $.1*$.2^3*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.2^2*$.3^4, $.1*$.2^2*$.3^4, $.2^3*$.3^4, $.1*$.2^3*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5, $.2^2*$.3^5, $.1*$.2^2*$.3^5, $.2^3*$.3^5, $.1*$.2^3*$.3^5, $.3^6, $.1*$.3^6, $.2*$.3^6, $.1*$.2*$.3^6, $.2^2*$.3^6, $.1*$.2^2*$.3^6, $.2^3*$.3^6, $.1*$.2^3*$.3^6, $.3^7, $.1*$.3^7, $.2*$.3^7, $.1*$.2*$.3^7, $.2^2*$.3^7, $.1*$.2^2*$.3^7, $.2^3*$.3^7, $.1*$.2^3*$.3^7, $.3^8, $.1*$.3^8, $.2*$.3^8, $.1*$.2*$.3^8, $.2^2*$.3^8, $.1*$.2^2*$.3^8, $.2^3*$.3^8, $.1*$.2^3*$.3^8, $.3^9, $.1*$.3^9, $.2*$.3^9, $.1*$.2*$.3^9, $.2^2*$.3^9, $.1*$.2^2*$.3^9, $.2^3*$.3^9, $.1*$.2^3*$.3^9, $.3^10, $.1*$.3^10, $.2*$.3^10, $.1*$.2*$.3^10, $.2^2*$.3^10, $.1*$.2^2*$.3^10, $.2^3*$.3^10, $.1*$.2^3*$.3^10, $.3^11, $.1*$.3^11, $.2*$.3^11, $.1*$.2*$.3^11, $.2^2*$.3^11, $.1*$.2^2*$.3^11, $.2^3*$.3^11, $.1*$.2^3*$.3^11, $.3^12, $.1*$.3^12, $.2*$.3^12, $.1*$.2*$.3^12, $.2^2*$.3^12, $.1*$.2^2*$.3^12, $.2^3*$.3^12, $.1*$.2^3*$.3^12, $.3^13, $.1*$.3^13, $.2*$.3^13, $.1*$.2*$.3^13, $.2^2*$.3^13, $.1*$.2^2*$.3^13, $.2^3*$.3^13, $.1*$.2^3*$.3^13, $.3^14, $.1*$.3^14, $.2*$.3^14, $.1*$.2*$.3^14, $.2^2*$.3^14, $.1*$.2^2*$.3^14, $.2^3*$.3^14, $.1*$.2^3*$.3^14, $.3^15, $.1*$.3^15, $.2*$.3^15, $.1*$.2*$.3^15, $.2^2*$.3^15, $.1*$.2^2*$.3^15, $.2^3*$.3^15, $.1*$.2^3*$.3^15, $.3^16, $.1*$.3^16, $.2*$.3^16, $.1*$.2*$.3^16, $.2^2*$.3^16, $.1*$.2^2*$.3^16, $.2^3*$.3^16, $.1*$.2^3*$.3^16, $.3^17, $.1*$.3^17, $.2*$.3^17, $.1*$.2*$.3^17, $.2^2*$.3^17, $.1*$.2^2*$.3^17, $.2^3*$.3^17, $.1*$.2^3*$.3^17, $.3^18, $.1*$.3^18, $.2*$.3^18, $.1*$.2*$.3^18, $.2^2*$.3^18, $.1*$.2^2*$.3^18, $.2^3*$.3^18, $.1*$.2^3*$.3^18, $.3^19, $.1*$.3^19, $.2*$.3^19, $.1*$.2*$.3^19, $.2^2*$.3^19, $.1*$.2^2*$.3^19, $.2^3*$.3^19, $.1*$.2^3*$.3^19, $.3^20, $.1*$.3^20, $.2*$.3^20, $.1*$.2*$.3^20, $.2^2*$.3^20, $.1*$.2^2*$.3^20, $.2^3*$.3^20, $.1*$.2^3*$.3^20, $.3^21, $.1*$.3^21, $.2*$.3^21, $.1*$.2*$.3^21, $.2^2*$.3^21, $.1*$.2^2*$.3^21, $.2^3*$.3^21, $.1*$.2^3*$.3^21, $.3^22, $.1*$.3^22, $.2*$.3^22, $.1*$.2*$.3^22, $.2^2*$.3^22, $.1*$.2^2*$.3^22, $.2^3*$.3^22, $.1*$.2^3*$.3^22, $.3^23, $.1*$.3^23, $.2*$.3^23, $.1*$.2*$.3^23, $.2^2*$.3^23, $.1*$.2^2*$.3^23, $.2^3*$.3^23, $.1*$.2^3*$.3^23, $.3^24, $.1*$.3^24, $.2*$.3^24, $.1*$.2*$.3^24, $.2^2*$.3^24, $.1*$.2^2*$.3^24, $.2^3*$.3^24, $.1*$.2^3*$.3^24, $.3^25, $.1*$.3^25, $.2*$.3^25, $.1*$.2*$.3^25, $.2^2*$.3^25, $.1*$.2^2*$.3^25, $.2^3*$.3^25, $.1*$.2^3*$.3^25, $.3^26, $.1*$.3^26, $.2*$.3^26, $.1*$.2*$.3^26, $.2^2*$.3^26, $.1*$.2^2*$.3^26, $.2^3*$.3^26, $.1*$.2^3*$.3^26, $.3^27, $.1*$.3^27, $.2*$.3^27, $.1*$.2*$.3^27, $.2^2*$.3^27, $.1*$.2^2*$.3^27, $.2^3*$.3^27, $.1*$.2^3*$.3^27, $.3^28, $.1*$.3^28, $.2*$.3^28, $.1*$.2*$.3^28, $.2^2*$.3^28, $.1*$.2^2*$.3^28, $.2^3*$.3^28, $.1*$.2^3*$.3^28, $.3^29, $.1*$.3^29, $.2*$.3^29, $.1*$.2*$.3^29, $.2^2*$.3^29, $.1*$.2^2*$.3^29, $.2^3*$.3^29, $.1*$.2^3*$.3^29, $.3^30, $.1*$.3^30, $.2*$.3^30, $.1*$.2*$.3^30, $.2^2*$.3^30, $.1*$.2^2*$.3^30, $.2^3*$.3^30, $.1*$.2^3*$.3^30, $.3^31, $.1*$.3^31, $.2*$.3^31, $.1*$.2*$.3^31, $.2^2*$.3^31, $.1*$.2^2*$.3^31, $.2^3*$.3^31, $.1*$.2^3*$.3^31, $.3^32, $.1*$.3^32, $.2*$.3^32, $.1*$.2*$.3^32, $.2^2*$.3^32, $.1*$.2^2*$.3^32, $.2^3*$.3^32, $.1*$.2^3*$.3^32, $.3^33, $.1*$.3^33, $.2*$.3^33, $.1*$.2*$.3^33, $.2^2*$.3^33, $.1*$.2^2*$.3^33, $.2^3*$.3^33, $.1*$.2^3*$.3^33, $.3^34, $.1*$.3^34, $.2*$.3^34, $.1*$.2*$.3^34, $.2^2*$.3^34, $.1*$.2^2*$.3^34, $.2^3*$.3^34, $.1*$.2^3*$.3^34, $.3^35, $.1*$.3^35, $.2*$.3^35, $.1*$.2*$.3^35, $.2^2*$.3^35, $.1*$.2^2*$.3^35, $.2^3*$.3^35, $.1*$.2^3*$.3^35, $.3^36, $.1*$.3^36, $.2*$.3^36, $.1*$.2*$.3^36, $.2^2*$.3^36, $.1*$.2^2*$.3^36, $.2^3*$.3^36, $.1*$.2^3*$.3^36, $.3^37, $.1*$.3^37, $.2*$.3^37, $.1*$.2*$.3^37, $.2^2*$.3^37, $.1*$.2^2*$.3^37, $.2^3*$.3^37, $.1*$.2^3*$.3^37, $.3^38, $.1*$.3^38, $.2*$.3^38, $.1*$.2*$.3^38, $.2^2*$.3^38, $.1*$.2^2*$.3^38, $.2^3*$.3^38, $.1*$.2^3*$.3^38, $.3^39, $.1*$.3^39, $.2*$.3^39, $.1*$.2*$.3^39, $.2^2*$.3^39, $.1*$.2^2*$.3^39, $.2^3*$.3^39, $.1*$.2^3*$.3^39, $.3^40, $.1*$.3^40, $.2*$.3^40, $.1*$.2*$.3^40, $.2^2*$.3^40, $.1*$.2^2*$.3^40, $.2^3*$.3^40, $.1*$.2^3*$.3^40, $.3^41, $.1*$.3^41, $.2*$.3^41, $.1*$.2*$.3^41, $.2^2*$.3^41, $.1*$.2^2*$.3^41, $.2^3*$.3^41, $.1*$.2^3*$.3^41 ] 4 2 Current total memory usage: 95.4MB, failed memory request: 0.0MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Integer Ring Total time: 3.520 seconds, Total memory usage: 95.37MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:16:34 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[12]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:16:33 on modular [Seed = 2758979861] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 56 6 [ Modular symbols space of level 56, weight 3, character $.1*$.2*$.3^2, and dimension 1 over Cyclotomic Field of order 12 and degree 4, Modular symbols space of level 56, weight 3, character $.1*$.2*$.3^2, and dimension 1 over Cyclotomic Field of order 12 and degree 4, Modular symbols space of level 56, weight 3, character $.1*$.2*$.3^2, and dimension 6 over Cyclotomic Field of order 12 and degree 4, Modular symbols space of level 56, weight 3, character $.1*$.2*$.3^2, and dimension 6 over Cyclotomic Field of order 12 and degree 4 ] q + 2*zeta_12^2*q^2 - zeta_12^2*q^3 + (4*zeta_12^2 - 4)*q^4 + (3*zeta_12^2 + 3)*q^5 + (-2*zeta_12^2 + 2)*q^6 + (8*zeta_12^2 - 3)*q^7 - 8*q^8 + (-8*zeta_12^2 + 8)*q^9 + (12*zeta_12^2 - 6)*q^10 - 17*zeta_12^2*q^11 + O(q^12) Power series ring in q over Cyclotomic Field of order 12 and degree 4 Total time: 0.560 seconds, Total memory usage: 4.94MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:16:25 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[11]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:16:24 on modular [Seed = 2672396855] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 56 6 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.32MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:16:16 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[10]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:16:15 on modular [Seed = 2588708613] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 28 6 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.700 seconds, Total memory usage: 4.94MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:15:26 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[9]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:15:25 on modular [Seed = 2504228984] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 7 3 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.33MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:15:15 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:15:14 on modular [Seed = 2420540786] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 56 6 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.32MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:14:55 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:14:54 on modular [Seed = 2338168389] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 56 6 [ Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 14 over Cyclotomic Field of order 12 and degree 4 ] q + (1/2348959680*(4579*zeta_12^2 + 4579)*a^13 + 15397/104398208*zeta_12^2*a^12 + 1/2348959680*(1467542*zeta_12^2 - 733771)*a^11 + 1/26099552*(249503*zeta_12^2 - 249503)*a^10 + 1/2348959680*(34198157*zeta_12^2 - 68396314)*a^9 - 21938035/104398208*a^8 + 1/391493280*(-107476613*zeta_12^2 - 107476613)*a^7 - 50610471/26099552*zeta_12^2*a^6 + 1/2348959680*(-10540068074*zeta_12^2 + 5270034037)*a^5 + 1/104398208*(-738593311*zeta_12^2 + 738593311)*a^4 + 1/2348959680*(-16484360123*zeta_12^2 + 32968720246)*a^3 + 11367885/1631222*a^2 + 1/2348959680*(11603758463*zeta_12^2 + 11603758463)*a + 9801249/104398208*zeta_12^2)*q^2 + a*q^3 + (1/1565973120*(52502*zeta_12^2 - 26251)*a^13 + 1/104398208*(19851*zeta_12^2 - 19851)*a^12 + 1/130497760*(141337*zeta_12^2 - 282674)*a^11 - 622539/52199104*a^10 + 1/521991040*(-12229781*zeta_12^2 - 12229781)*a^9 - 25556357/104398208*zeta_12^2*a^8 + 1/130497760*(-52658222*zeta_12^2 + 26329111)*a^7 + 1/6524888*(-12557077*zeta_12^2 + 12557077)*a^6 + 1/521991040*(-262718751*zeta_12^2 + 525437502)*a^5 + 421293737/104398208*a^4 + 1/32624440*(-34379021*zeta_12^2 - 34379021)*a^3 - 321222443/52199104*zeta_12^2*a^2 + 1/1565973120*(-9362800046*zeta_12^2 + 4681400023)*a + 1/104398208*(-438492713*zeta_12^2 + 438492713))*q^4 + (1/195746640*(-9343*zeta_12^2 + 9343)*a^13 + 1276409/391493280*a^11 + 31072933/391493280*zeta_12^2*a^9 + 1/65248880*(58403837*zeta_12^2 - 58403837)*a^7 - 60668884/12234165*a^5 - 5142634777/391493280*zeta_12^2*a^3 + 1/391493280*(-5843345917*zeta_12^2 + 5843345917)*a)*q^5 + (15397/104398208*zeta_12^2*a^13 + 1/78298656*(24802*zeta_12^2 - 12401)*a^12 + 1/26099552*(249503*zeta_12^2 - 249503)*a^11 + 1/78298656*(783845*zeta_12^2 - 1567690)*a^10 - 21938035/104398208*a^9 + 1/156597312*(-33018499*zeta_12^2 - 33018499)*a^8 - 50610471/26099552*zeta_12^2*a^7 + 1/39149328*(-142014746*zeta_12^2 + 71007373)*a^6 + 1/104398208*(-738593311*zeta_12^2 + 738593311)*a^5 + 1/39149328*(-227569759*zeta_12^2 + 455139518)*a^4 + 11367885/1631222*a^3 + 1/78298656*(324656903*zeta_12^2 + 324656903)*a^2 + 9801249/104398208*zeta_12^2*a + 1/52199104*(-4807950*zeta_12^2 + 2403975))*q^6 + (1/78298656*(8393*zeta_12^2 + 40739)*a^12 + 1/78298656*(3157175*zeta_12^2 - 609496)*a^10 + 1/26099552*(17440828*zeta_12^2 - 22769715)*a^8 + 1/39149328*(-95698738*zeta_12^2 - 210647131)*a^6 + 1/78298656*(-2129129966*zeta_12^2 + 1052988271)*a^4 + 1/26099552*(45576087*zeta_12^2 + 706336949)*a^2 + 1/26099552*(238877441*zeta_12^2 - 24851810))*q^7 + (1/2348959680*(-291967*zeta_12^2 + 583934)*a^13 - 183/3262444*a^12 + 1/2348959680*(20993293*zeta_12^2 + 20993293)*a^11 - 210395/52199104*zeta_12^2*a^10 + 1/1174479840*(548226986*zeta_12^2 - 274113493)*a^9 + 1/52199104*(-5382883*zeta_12^2 + 5382883)*a^8 + 1/130497760*(370662333*zeta_12^2 - 741324666)*a^7 + 30698417/26099552*a^6 + 1/2348959680*(-38895998071*zeta_12^2 - 38895998071)*a^5 + 158397319/26099552*zeta_12^2*a^4 + 1/2348959680*(-188548838578*zeta_12^2 + 94274419289)*a^3 + 1/52199104*(670342267*zeta_12^2 - 670342267)*a^2 + 1/146809980*(-2731818449*zeta_12^2 + 5463636898)*a - 237835863/52199104)*q^8 + (a^2 + 9*zeta_12^2)*q^9 + (-47433/260995520*a^13 + 1/104398208*(-41981*zeta_12^2 - 41981)*a^12 - 1435511/130497760*zeta_12^2*a^11 + 1/156597312*(-8089606*zeta_12^2 + 4044803)*a^10 + 1/65248880*(-13715531*zeta_12^2 + 13715531)*a^9 + 1/313194624*(-174679391*zeta_12^2 + 349358782)*a^8 + 45371317/32624440*a^7 + 1/3262444*(16183267*zeta_12^2 + 16183267)*a^6 + 196747729/260995520*zeta_12^2*a^5 + 1/313194624*(10380831650*zeta_12^2 - 5190415825)*a^4 + 1/130497760*(-1423689617*zeta_12^2 + 1423689617)*a^3 + 1/156597312*(2019684241*zeta_12^2 - 4039368482)*a^2 - 505558583/130497760*a + 1/104398208*(-504246155*zeta_12^2 - 504246155))*q^10 + (1/293619960*(-28826*zeta_12^2 + 14413)*a^13 + 1/73404990*(-139903*zeta_12^2 + 279806)*a^11 + 1/293619960*(-3530371*zeta_12^2 - 3530371)*a^9 + 1/8156110*(-18231238*zeta_12^2 + 9115619)*a^7 + 1/293619960*(-3824890421*zeta_12^2 + 7649780842)*a^5 + 1/36702495*(1660220933*zeta_12^2 + 1660220933)*a^3 + 1/293619960*(12225510218*zeta_12^2 - 6112755109)*a)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 12 and degree 4 with modulus a^14 + 79*zeta_12^2*a^12 + (2333*zeta_12^2 - 2333)*a^10 - 32667*a^8 - 220483*zeta_12^2*a^6 + (-618077*zeta_12^2 + 618077)*a^4 + 407087*a^2 + 23625*zeta_12^2 Total time: 0.680 seconds, Total memory usage: 4.94MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:14:43 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:14:43 on modular [Seed = 2253427525] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 28 6 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.220 seconds, Total memory usage: 4.33MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:59:20 2005 Input: LogIntegral(100); Output: Magma V2.11-10 Tue Dec 6 2005 03:59:20 on modular [Seed = 820388248] ------------------------------------- 30.12614158407962992590172 Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:58:40 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:58:39 on modular [Seed = 738015956] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 7 6 [ Modular symbols space of level 56, weight 3, character $.3, and dimension 4 over Cyclotomic Field of order 12 and degree 4 ] q + a*q^3 + (-1/8*zeta_12^2*a^3 + 1/8*(6*zeta_12^2 - 3)*a^2 + 1/8*(17*zeta_12^2 - 17)*a + 1/8*(-39*zeta_12^2 + 78))*q^5 + (-1/8*a^3 + 1/8*(3*zeta_12^2 - 5)*a^2 + 25/8*zeta_12^2*a + 1/8*(10*zeta_12^2 + 47))*q^7 + (a^2 - 9*zeta_12^2)*q^9 + (-zeta_12^2*a^2 + (-2*zeta_12^2 + 1)*a + (13*zeta_12^2 - 13))*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 12 and degree 4 with modulus a^4 - 28*zeta_12^2*a^2 + (-8*zeta_12^2 + 4)*a + 119*zeta_12^2 - 119 Total time: 0.700 seconds, Total memory usage: 4.94MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:57:20 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:57:19 on modular [Seed = 653276937] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Cyclotomic Field of order 12 and degree 4, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Cyclotomic Field of order 12 and degree 4 ] q + (55/26448*a^3 - 13/3306*a^2 + 7639/13224*a - 1251/1102)*q^2 + (-1/6612*a^3 + 31/3306*a^2 - 199/3306*a + 1979/551)*q^3 + (1/456*a^3 - 5/456*a^2 + 199/228*a - 443/76)*q^4 + (-1/13224*a^3 + 31/6612*a^2 - 3505/6612*a + 1979/1102)*q^5 + (7/1102*a^3 - 83/4408*a^2 + 2235/1102*a - 9035/2204)*q^6 + (-1/456*a^3 + 5/456*a^2 - 199/228*a + 215/76)*q^7 + (-47/13224*a^3 - 49/1653*a^2 - 6047/6612*a - 2257/551)*q^8 + (-1/1653*a^3 + 62/1653*a^2 - 398/1653*a + 1855/551)*q^9 + (-7/2204*a^3 + 317/4408*a^2 - 421/551*a + 11405/2204)*q^10 + (1/1102*a^3 - 31/551*a^2 + 199/551*a - 3058/551)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 12 and degree 4 with modulus a^4 - 8*a^3 + 356*a^2 - 2256*a + 6948 Total time: 0.640 seconds, Total memory usage: 4.85MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:57:07 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:57:06 on modular [Seed = 569847811] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 8 2 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:56:36 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:56:35 on modular [Seed = 365102102] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 4 2 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.770 seconds, Total memory usage: 4.75MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:55:51 2005 Input: G :=DirichletGroup(14); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:55:51 on modular [Seed = 249044825] ------------------------------------- Group of Dirichlet characters of modulus 14 over Rational Field [ 1, $.1 ] 7 2 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.270 seconds, Total memory usage: 4.50MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:55:42 2005 Input: G :=DirichletGroup(14); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:55:42 on modular [Seed = 165615687] ------------------------------------- Group of Dirichlet characters of modulus 14 over Rational Field [ 1, $.1 ] >> Y :=X[4]; Conductor(Y); Order(Y); ^ Runtime error in '[]': Sequence element 4 not defined >> Y :=X[4]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[4]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 3, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Set of sequences over Group of Dirichlet characters of modulus 14 over Rational Field Total time: 0.190 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:55:31 2005 Input: G :=DirichletGroup(28); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:55:31 on modular [Seed = 80876871] ------------------------------------- Group of Dirichlet characters of modulus 28 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 28 2 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 3.91MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:54:13 2005 Input: G :=DirichletGroup(28); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:54:13 on modular [Seed = 2141810058] ------------------------------------- Group of Dirichlet characters of modulus 28 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 7 2 [ Modular symbols space of level 28, weight 3, character $.2, and dimension 2 over Rational Field ] q + a*q^3 - a*q^5 + (-a + 5)*q^7 - 15*q^9 - 6*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 + 24 Total time: 0.330 seconds, Total memory usage: 4.69MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:53:28 2005 Input: G :=DirichletGroup(28); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:53:28 on modular [Seed = 2007066047] ------------------------------------- Group of Dirichlet characters of modulus 28 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 4 2 [ Modular symbols space of level 28, weight 3, character $.1, and dimension 6 over Rational Field ] q + (1/64*a^5 - 1/32*a^4 + 7/16*a^3 - 5/8*a^2 + 9/4*a - 3/2)*q^2 + a*q^3 + (1/64*a^5 - 1/32*a^4 + 5/16*a^3 - 9/8*a^2 + 1/4*a - 13/2)*q^4 + (1/8*a^4 + 3*a^2 + 10)*q^5 + (-1/32*a^5 - 1/16*a^4 - 5/8*a^3 - 7/4*a^2 - 3/2*a - 7)*q^6 + (-1/8*a^3 - 2*a)*q^7 + (-1/64*a^5 + 1/32*a^4 - 9/16*a^3 + 9/8*a^2 - 25/4*a + 9/2)*q^8 + (a^2 + 9)*q^9 + (1/8*a^4 + 1/4*a^3 + 3*a^2 + 5*a + 6)*q^10 + (-1/8*a^5 - 3*a^3 - 12*a)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^6 + 32*a^4 + 256*a^2 + 448 Total time: 0.320 seconds, Total memory usage: 4.72MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:50:42 2005 Input: LogIntegral(200); Output: Magma V2.11-10 Tue Dec 6 2005 03:50:42 on modular [Seed = 1874427085] ------------------------------------- 50.192171165963783809779121 Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:48:12 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:48:11 on modular [Seed = 1739682924] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] q + (31/48768*a^3 - 7/12192*a^2 + 2425/6096*a - 275/508)*q^2 + (-1/32512*a^3 + 33/8128*a^2 - 111/4064*a + 3599/1016)*q^3 + (1/1524*a^3 - 5/1524*a^2 + 74/127*a - 1868/381)*q^4 + (-1/97536*a^3 + 11/8128*a^2 - 4175/12192*a + 3599/3048)*q^5 + (47/24384*a^3 - 9/2032*a^2 + 4201/3048*a - 1513/762)*q^6 + (-1/1524*a^3 + 5/1524*a^2 - 74/127*a + 725/381)*q^7 + (-7/6096*a^3 - 23/1524*a^2 - 523/762*a - 646/127)*q^8 + (-1/8128*a^3 + 33/2032*a^2 - 111/1016*a + 805/254)*q^9 + (-7/8128*a^3 + 185/6096*a^2 - 1315/3048*a + 3443/762)*q^10 + (3/16256*a^3 - 99/4064*a^2 + 333/2032*a - 2669/508)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 - 8*a^3 + 776*a^2 - 5056*a + 24448 Total time: 0.410 seconds, Total memory usage: 4.64MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:47:44 2005 Input: #[n: n in [100 .. 200] | IsSquare(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:47:44 on modular [Seed = 1655992594] ------------------------------------- 5 Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:47:35 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[2],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:47:34 on modular [Seed = 1569410619] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] q + (-14817/2874195968*a^7 - 26645/718548992*a^6 - 10231/89818624*a^5 - 9201/44909312*a^4 - 329307/25662464*a^3 - 524939/6415616*a^2 + 14383/168832*a + 634373/801952)*q^2 + (-1251/302546944*a^7 - 1723/75636736*a^6 - 1363/4727296*a^5 - 103/147728*a^4 - 72009/2701312*a^3 - 44577/675328*a^2 - 92401/337664*a - 57833/84416)*q^3 + (-331/179637248*a^7 + 1423/25662464*a^6 + 29587/89818624*a^5 + 138223/44909312*a^4 - 23141/3207808*a^3 + 87083/400976*a^2 + 13159/21104*a + 506731/100244)*q^4 + (-1251/302546944*a^7 - 1723/75636736*a^6 - 1363/4727296*a^5 - 103/147728*a^4 - 72009/2701312*a^3 - 44577/675328*a^2 - 430065/337664*a - 57833/84416)*q^5 + (473/44909312*a^7 + 11735/89818624*a^6 + 2511/6415616*a^5 + 138053/22454656*a^4 + 73027/1603904*a^3 + 12557/25061*a^2 + 4731/10552*a + 282483/50122)*q^6 + (195/44909312*a^7 + 3823/44909312*a^6 + 4421/11227328*a^5 + 33199/11227328*a^4 + 1065/50122*a^3 + 60859/200488*a^2 + 1209/1319*a + 83369/25061)*q^7 + (4975/179637248*a^7 + 4203/25662464*a^6 + 147831/89818624*a^5 + 266099/44909312*a^4 + 363411/3207808*a^3 + 242029/400976*a^2 + 66663/21104*a + 1007587/100244)*q^8 + (779/75636736*a^7 + 803/18909184*a^6 + 61/168832*a^5 - 459/73864*a^4 + 22977/675328*a^3 - 9895/168832*a^2 + 29737/84416*a - 182703/21104)*q^9 + (241/37818368*a^7 - 1587/9454592*a^6 - 855/1181824*a^5 - 3845/590912*a^4 + 891/337664*a^3 - 40149/84416*a^2 - 69053/42208*a - 23973/10552)*q^10 + (-7247/151273472*a^7 - 7687/37818368*a^6 - 4311/2363648*a^5 - 409/73864*a^4 - 231309/1350656*a^3 - 191429/337664*a^2 - 298965/168832*a - 406957/42208)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^8 + 8*a^7 + 80*a^6 + 256*a^5 + 4944*a^4 + 24192*a^3 + 172928*a^2 + 250880*a + 1533952 Total time: 0.410 seconds, Total memory usage: 4.64MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:47:16 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[4],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:47:15 on modular [Seed = 1485720331] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] >> qEigenform(D[4],12);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Set of sequences over Power Structure of ModSym Total time: 0.400 seconds, Total memory usage: 4.45MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:45:59 2005 Input: [n: n in [200 .. 625] | IsSquare(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:45:58 on modular [Seed = 1384406740] ------------------------------------- [ 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625 ] Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:45:54 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[4],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:45:54 on modular [Seed = 1302020559] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 56 2 [ Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 8 over Rational Field ] q + (-61/779607751680*a^7 + 19/13319658240*a^6 - 33617/151590396160*a^5 + 367/221994304*a^4 - 5068603/28423199280*a^3 - 163153/138746440*a^2 + 1166340923/42634798920*a - 445123/424734)*q^2 + (-54449/44567576471040*a^7 - 321481/742792941184*a^5 + 34597151/58030698530*a^3 - 382734991/1421159964*a)*q^3 + (118831/267405458826240*a^7 - 19/6659829120*a^6 + 2324557/2785473529440*a^5 - 367/110997152*a^4 + 295787251/928491176480*a^3 + 163153/69373220*a^2 - 86314273/3045342780*a - 191978/212367)*q^4 + (313/1856982352960*a^7 + 2368551/3713964705920*a^5 + 56897591/464245588240*a^3 - 877243639/2368599940*a)*q^5 + (-61961/89135152942080*a^7 + 185/1997948736*a^6 - 993989/1856982352960*a^5 + 174781/3329914560*a^4 + 219879617/928491176480*a^3 - 2885719/83247864*a^2 + 359027981/7105799820*a + 66335023/6371010)*q^6 + (-41/815489280*a^6 - 39/11326240*a^4 + 352067/8494680*a^2 - 40889569/3185505)*q^7 + (-8159/11141894117760*a^7 - 19/2219943040*a^6 - 13654757/11141894117760*a^5 - 1101/110997152*a^4 - 390638659/1392736764720*a^3 + 489459/69373220*a^2 + 4673211/2368599940*a + 303545/70789)*q^8 + (19/1664957280*a^6 + 367/27749288*a^4 - 163153/17343305*a^2 - 718657/212367)*q^9 + (-15397/17827030588416*a^7 + 361/7991794944*a^6 - 4356529/3713964705920*a^5 + 34865/665982912*a^4 + 21216887/185698235296*a^3 + 61459/13874644*a^2 + 1495379449/3552899910*a - 1936955/1274202)*q^10 + (15397/26740545882624*a^7 + 4356529/5570947058880*a^5 - 21216887/278547352944*a^3 + 281070506/5329349865*a)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^8 + 1520*a^6 + 55296*a^4 - 331287040*a^2 + 140439061504 Total time: 0.350 seconds, Total memory usage: 4.54MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:45:44 2005 Input: [n: n in [200 .. 500] | IsSquare(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:45:44 on modular [Seed = 1218331824] ------------------------------------- [ 225, 256, 289, 324, 361, 400, 441, 484 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:45:37 2005 Input: [n: n in [300 .. 500] | IsSquare(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:45:37 on modular [Seed = 1133858738] ------------------------------------- [ 324, 361, 400, 441, 484 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:45:25 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:45:24 on modular [Seed = 3211375450] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 56 2 [ Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 8 over Rational Field ] q + (1/16*a + 3/2)*q^2 + (3/16*a + 1/2)*q^4 + 7*q^7 + (5/16*a - 9/2)*q^8 - 9*q^9 - a*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 + 448 Total time: 0.340 seconds, Total memory usage: 4.54MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:45:05 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:45:04 on modular [Seed = 3094262889] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 28 2 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.34MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:44:37 2005 Input: [n: n in [100 .. 200] | IsSquare(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:44:36 on modular [Seed = 3043211648] ------------------------------------- [ 100, 121, 144, 169, 196 ] Total time: 0.180 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:43:19 2005 Input: [n: n in [100 .. 200] | IsPrime(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:43:18 on modular [Seed = 2927141577] ------------------------------------- [ 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:43:00 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:42:59 on modular [Seed = 2877138937] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 56, weight 3, character $.3, and dimension 4 over Rational Field ] q + a*q^3 + (1/2*a^3 + 7*a)*q^5 + (-1/2*a^3 + 1/2*a^2 - 5*a + 5)*q^7 + (a^2 + 9)*q^9 + (-2*a^2 - 14)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 + 16*a^2 + 32 Total time: 0.430 seconds, Total memory usage: 4.64MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:42:29 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 03:42:29 on modular [Seed = 2758977808] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 56, weight 3, character $.3, and dimension 4 over Rational Field ] Total time: 0.420 seconds, Total memory usage: 4.45MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:41:40 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; #qEigenform(D[2],12),11;Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:41:39 on modular [Seed = 2655556380] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 56, weight 3, character $.3, and dimension 4 over Rational Field ] >> #qEigenform(D[2],12),11;Parent($1); ^ Runtime error in '[]': Sequence element 2 not defined Set of sequences over Power Structure of ModSym Total time: 0.420 seconds, Total memory usage: 4.45MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:39:14 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[2],12),11;Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:39:14 on modular [Seed = 2605553384] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] q + (-571500746715/1775060251923248128*a^7 - 388629337553/887530125961624064*a^6 - 1028834904749/11678027973179264*a^5 + 4918368039129/55470632872601504*a^4 - 4651621445267367/443765062980812032*a^3 + 6509171845645319/221882531490406016*a^2 - 25815516211983173/110941265745203008*a + 7150300887080843/2411766646634848)*q^2 + (-202925409/2919506993294816*a^7 + 833622367/729876748323704*a^6 - 11090313389/729876748323704*a^5 + 122109745219/364938374161852*a^4 - 3634150197939/729876748323704*a^3 + 3921605990478/91234593540463*a^2 - 39299581757551/182469187080926*a + 5869895733196/3966721458281)*q^3 + (299769622133/1775060251923248128*a^7 + 6555626450207/887530125961624064*a^6 + 1138059007615/11678027973179264*a^5 + 122284993648093/55470632872601504*a^4 + 6459555605921081/443765062980812032*a^3 + 66760393194614839/221882531490406016*a^2 + 95645252448681499/110941265745203008*a + 31634693493840411/2411766646634848)*q^4 + (-202925409/5839013986589632*a^7 + 833622367/1459753496647408*a^6 - 11090313389/1459753496647408*a^5 + 122109745219/729876748323704*a^4 - 3634150197939/1459753496647408*a^3 + 1960802995239/91234593540463*a^2 - 221768768838477/364938374161852*a + 2934947866598/3966721458281)*q^5 + (711085480935/887530125961624064*a^7 + 235869057485/443765062980812032*a^6 + 1200042577193/5839013986589632*a^5 + 911209802185/3466914554537594*a^4 + 6181260606368675/221882531490406016*a^3 + 9166844150804637/110941265745203008*a^2 + 52756368685049465/55470632872601504*a + 4498614222256633/1205883323317424)*q^6 + (17241/35534698624*a^7 + 183763/35534698624*a^6 + 35797/233780912*a^5 + 1454255/1110459332*a^4 + 190238637/8883674656*a^3 + 1587947027/8883674656*a^2 + 2428447959/2220918664*a + 13762616057/2220918664)*q^7 + (1226158137473/1775060251923248128*a^7 - 986896349357/887530125961624064*a^6 + 2866470193731/11678027973179264*a^5 - 27191192590141/55470632872601504*a^4 + 16767816630761589/443765062980812032*a^3 - 4166678493727589/221882531490406016*a^2 + 262678257101054031/110941265745203008*a + 8568098621402639/2411766646634848)*q^8 + (247894637/364938374161852*a^7 + 470257646/91234593540463*a^6 + 34730559821/182469187080926*a^5 - 7463559867/182469187080926*a^4 + 2215406259756/91234593540463*a^3 - 12060094738132/91234593540463*a^2 + 74854681732280/91234593540463*a - 57822477939285/3966721458281)*q^9 + (-15609054847/23356055946358528*a^7 - 241973420793/11678027973179264*a^6 - 784427920321/2919506993294816*a^5 - 8645457727271/1459753496647408*a^4 - 243983987349699/5839013986589632*a^3 - 2142335224733605/2919506993294816*a^2 - 3401476783420169/1459753496647408*a - 680021364237737/31733771666248)*q^10 + (-3247072067/2919506993294816*a^7 + 2525600507/182469187080926*a^6 - 113545926879/364938374161852*a^5 + 375687815222/91234593540463*a^4 - 32070491902875/729876748323704*a^3 + 48544544489462/91234593540463*a^2 - 284832833856747/182469187080926*a + 57382437550564/3966721458281)*q^11 + O(q^12) 11 Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^8 + 8*a^7 + 404*a^6 + 2032*a^5 + 70164*a^4 + 254976*a^3 + 5537696*a^2 + 8220416*a + 144702016 Total time: 0.410 seconds, Total memory usage: 4.64MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:28:53 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12),11;Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:28:52 on modular [Seed = 2220002305] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] q + (7/480*a^3 - 1/24*a^2 + 61/60*a - 13/5)*q^2 + (-1/480*a^3 + 1/24*a^2 - 13/60*a + 19/5)*q^3 + (1/60*a^3 - 1/12*a^2 + 26/15*a - 42/5)*q^4 + (-1/480*a^3 + 1/24*a^2 - 73/60*a + 19/5)*q^5 + (11/240*a^3 - 1/6*a^2 + 113/30*a - 48/5)*q^6 + (-1/60*a^3 + 1/12*a^2 - 26/15*a + 27/5)*q^7 + (-1/48*a^3 - 1/12*a^2 - 7/6*a - 2)*q^8 + (-1/120*a^3 + 1/6*a^2 - 13/15*a + 21/5)*q^9 + (-7/240*a^3 + 1/3*a^2 - 61/30*a + 36/5)*q^10 + (1/80*a^3 - 1/4*a^2 + 13/10*a - 34/5)*q^11 + O(q^12) 11 Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 - 8*a^3 + 104*a^2 - 576*a + 1152 Total time: 0.400 seconds, Total memory usage: 4.64MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:28:23 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:28:22 on modular [Seed = 2169997670] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] >> qEigenform(D[1],12),11);Parent($1); ^ User error: bad syntax Total time: 0.380 seconds, Total memory usage: 4.45MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:26:47 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 03:26:47 on modular [Seed = 4214092797] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] Total time: 0.390 seconds, Total memory usage: 4.45MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:26:37 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 03:26:36 on modular [Seed = 4112776585] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [] Total time: 0.210 seconds, Total memory usage: 4.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:26:24 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 03:26:23 on modular [Seed = 3996706432] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 4 2 [] Total time: 0.490 seconds, Total memory usage: 4.45MB '131.156' ************** MAGMA ***************** Host 131.156.146.48 (131.156.146.48) Time: Tue Dec 6 02:05:37 2005 Input: P:=PolynomialRing(RationalField(),10); I:=ideal; GroebnerBasis(EliminationIdeal(I,{x,a,b,c,d,e,f})); GroebnerBasis(EliminationIdeal(I,{y,a,b,c,d,e,f})); Output: Magma V2.11-10 Tue Dec 6 2005 02:05:37 on modular [Seed = 499869213] ------------------------------------- [ x^2 - x*a - x*f + a*f - b*e, x*a*c - x*c*f + x*d*e - x*e^2 - a*c*f + b*c*e + c*f^2 - d*e*f, x*a*f + x*b*d - x*b*e - x*f^2 - a^2*f - a*b*d + a*b*e + a*f^2 + b^2*c - b*e*f, x*b*c - x*e*f + a*e*f - b*c*f + b*d*e - b*e^2, a^2*c*f + a*b*c*d - a*b*c*e - 2*a*c*f^2 + a*d*e*f - a*e^2*f - b^2*c^2 - b*c*d*f + 3*b*c*e*f + b*d^2*e - 2*b*d*e^2 + b*e^3 + c*f^3 - d*e*f^2 ] [ y^2 - y*d - y*e - c*f + d*e, y*a*c - y*c*f + y*d*e - y*e^2 - a*c*d + b*c^2 + c*d*f - c*e*f - d^2*e + d*e^2, y*a*f + y*b*d - y*b*e - y*f^2 - a*e*f + b*c*f - b*d*e + b*e^2, y*b*c - y*e*f + a*c*f - b*c*e - c*f^2 + d*e*f, a^2*c*f + a*b*c*d - a*b*c*e - 2*a*c*f^2 + a*d*e*f - a*e^2*f - b^2*c^2 - b*c*d*f + 3*b*c*e*f + b*d^2*e - 2*b*d*e^2 + b*e^3 + c*f^3 - d*e*f^2 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '131.156' ************** MAGMA ***************** Host 131.156.146.48 (131.156.146.48) Time: Tue Dec 6 02:02:24 2005 Input: P:=PolynomialRing(RationalField(),10); I:=ideal; EliminationIdeal(I,{x,a,b,c,d,e,f}); EliminationIdeal(I,{y,a,b,c,d,e,f}); Output: Magma V2.11-10 Tue Dec 6 2005 02:02:23 on modular [Seed = 381706792] ------------------------------------- Ideal of Polynomial ring of rank 10 over Rational Field Lexicographical Order Variables: xi, yi, x, y, a, b, c, d, e, f Basis: [ -x^2*c + 2*x*c*f - x*d*e + x*e^2 - c*f^2 + d*e*f, -x^2*d + x^2*e + x*a*d - x*a*e - x*b*c + x*d*f - a*d*f + b*c*f, -x^3 + 2*x^2*a - x*a^2 + x*b*d - a*b*d + b^2*c, -x^2 + x*a + x*f - a*f + b*e ] Ideal of Polynomial ring of rank 10 over Rational Field Lexicographical Order Variables: xi, yi, x, y, a, b, c, d, e, f Basis: [ -y^2*b - y*a*f + 2*y*b*e + y*f^2 + a*e*f - b*e^2, -y^2*e - y*a*c + y*c*f + 2*y*e^2 + a*c*d - b*c^2 - c*d*f + 2*c*e*f + d^2*e - 2*d*e^2, y*b*c - y*e*f + a*c*f - b*c*e - c*f^2 + d*e*f, -y^2 + y*d + y*e + c*f - d*e ] Total time: 0.200 seconds, Total memory usage: 3.34MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 22:09:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 22:09:31 on modular [Seed = 4197234989] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 Total time: 1.889 seconds, Total memory usage: 39.58MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:59:46 2005 Input: 67+56=x Output: Magma V2.11-10 Mon Dec 5 2005 21:59:46 on modular [Seed = 1251866580] ------------------------------------- >> 67+56=x; ^ User error: Identifier 'x' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:59:37 2005 Input: 67+56 Output: Magma V2.11-10 Mon Dec 5 2005 21:59:37 on modular [Seed = 1301999848] ------------------------------------- 123 Total time: 0.190 seconds, Total memory usage: 3.24MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:59:31 2005 Input: x=67+56 Output: Magma V2.11-10 Mon Dec 5 2005 21:59:30 on modular [Seed = 1351085566] ------------------------------------- >> x=67+56; ^ User error: Identifier 'x' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:59:18 2005 Input: x=67x+56 Output: Magma V2.11-10 Mon Dec 5 2005 21:59:17 on modular [Seed = 1535960802] ------------------------------------- >> x=67x+56; ^ User error: Invalid hexadecimal integer Total time: 0.190 seconds, Total memory usage: 3.24MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:59:00 2005 Input: x=67x+56 Output: Magma V2.11-10 Mon Dec 5 2005 21:59:00 on modular [Seed = 1990314814] ------------------------------------- >> x=67x+56; ^ User error: Invalid hexadecimal integer Total time: 0.190 seconds, Total memory usage: 3.24MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:58:53 2005 Input: x=67y+56 Output: Magma V2.11-10 Mon Dec 5 2005 21:58:53 on modular [Seed = 2041496616] ------------------------------------- >> x=67y+56; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:54:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 21:54:23 on modular [Seed = 64273760] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x, 9*x + 324, 2) ] 1.13829905958751855639529004360387879793177840048426356187981 11.991636334011282544968217138124976622401 8.862903321989681361582647123238902219762 Errors: /bin/sh: line 1: 32575 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:52:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:51:49 on modular [Seed = 80985085] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x, 9*x + 324, 2) ] 1.13829905958751855639529004360387879793177840048426356187981 11.991636334011282544968217138124976622401 8.862903321989681361582647123238902219762 {@ (1, 0, 0), (x^2, 324, 2), (x^2, -324, 2), (x, 324, 1), (x, -324, 1) @} Total time: 17.890 seconds, Total memory usage: 39.53MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:50:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:49:55 on modular [Seed = 314944893] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x, 9*x + 324, 2) ] 1.13829905958751855639529004360387879793177840048426356187981 11.991636334011282544968217138124976622401 8.862903321989681361582647123238902219762 Total time: 18.010 seconds, Total memory usage: 39.53MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:49:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:49:20 on modular [Seed = 365079065] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 {@ @} Total time: 1.940 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:49:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:49:10 on modular [Seed = 416261890] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 {@ <29826, 1, 4, 1>, <82452, 1, 4, 1>, <31836, 1, 4, 1>, <83364, 1, 4, 1>, <51712, 1, 4, 1>, <22684, 1, 4, 1>, <17708, 1, 4, 1>, <3618, 1, 4, 1> @} Total time: 2.850 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:49:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:49:01 on modular [Seed = 603233282] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 {@ <346, 1, 4, 1>, <15020, 1, 4, 1>, <12736, 1, 4, 1>, <0, 1, 4, 1>, <21425, 1, 4, 1> @} Total time: 2.419 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:48:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:48:48 on modular [Seed = 653367562] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 {@ <0, 1, 4, 1> @} Total time: 1.940 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:48:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:48:37 on modular [Seed = 704550416] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 {@ <536, 1, 4, 1>, <1258, 1, 4, 1>, <766, 1, 4, 1> @} Total time: 2.020 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:44:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:44:45 on modular [Seed = 937463574] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 Total time: 1.520 seconds, Total memory usage: 39.39MB '172.130' ************** MAGMA ***************** Host 172.130.251.86 (172.130.251.86) Time: Mon Dec 5 21:38:13 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Mon Dec 5 2005 21:38:13 on modular [Seed = 3306323821] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:37:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:37:25 on modular [Seed = 3424351892] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3*x + 36, 3*x + 72, 2) ] 2.6951401700620685956297684449255413869414734353341592770870 10.1142760525465394052441122341781093637312 7.398086937098868439722320140675534613557 Total time: 2.629 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:37:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:37:11 on modular [Seed = 3506858939] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3*x + 36, 3*x + 72, 2) ] 2.6951401700620685956297684449255413869414734353341592770870 10.1142760525465394052441122341781093637312 7.398086937098868439722320140675534613557 {@ (1, 0, 0), (x^2, 108, 2), (x^2, -108, 2), (x, 108, 1), (x, -108, 1) @} Total time: 2.750 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:36:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:36:30 on modular [Seed = 3675021768] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3*x + 36, 3*x + 72, 2) ] 2.6951401700620685956297684449255413869414734353341592770870 10.1142760525465394052441122341781093637312 7.398086937098868439722320140675534613557 Total time: 2.629 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:34:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:34:14 on modular [Seed = 3708446278] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 2, -20, 1) ] 0.887004017177288147926102307775262838447 6.501416595843100382592300278524636022103 5.2008623597626490569318296668304832042724 {@ <130319, 1, 4, 1> @} Total time: 2.089 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:34:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:34:06 on modular [Seed = 3861994821] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 2, -20, 1) ] 0.887004017177288147926102307775262838447 6.501416595843100382592300278524636022103 5.2008623597626490569318296668304832042724 {@ <83519, 1, 4, 1>, <71129, 1, 4, 1>, <32700, 1, 4, 1>, <41477, 1, 4, 1>, <79888, 1, 4, 1>, <6059, 1, 4, 1>, <21532, 1, 4, 1> @} Total time: 2.790 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:34:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:33:58 on modular [Seed = 3946598464] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 2, -20, 1) ] 0.887004017177288147926102307775262838447 6.501416595843100382592300278524636022103 5.2008623597626490569318296668304832042724 {@ <28559, 1, 4, 1>, <18034, 1, 4, 1>, <26, 1, 4, 1>, <0, 1, 4, 1>, <6858, 1, 4, 1>, <5292, 1, 4, 1>, <5644, 1, 4, 1> @} Total time: 2.450 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:32:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:32:20 on modular [Seed = 3963312101] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 2, -20, 1) ] 0.887004017177288147926102307775262838447 6.501416595843100382592300278524636022103 5.2008623597626490569318296668304832042724 {@ <14639, 1, 4, 1>, <1835, 1, 4, 1> @} Total time: 1.919 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:29:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:29:51 on modular [Seed = 4264118876] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 81, -9*x + 324, 2) ] 3.8147298171104413244492568924472542117389814640423666477508 10.0853919220257459932592798346033758733630 9.1332133940617909495679892002151349774645 {@ @} Total time: 2.200 seconds, Total memory usage: 39.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:29:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:29:35 on modular [Seed = 2203334449] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 81, -9*x + 324, 2) ] 3.8147298171104413244492568924472542117389814640423666477508 10.0853919220257459932592798346033758733630 9.1332133940617909495679892002151349774645 {@ @} Total time: 2.209 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:25:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:25:45 on modular [Seed = 2672306699] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 21/4*x - 63/4, 147/8*x + 117/8, 2) ] 4.5968876495584691807324090445792654211205124632896604980007 5.781716759018522400865294183991207198153 5.471172431834758644917171743806715961974 {@ @} Total time: 1.770 seconds, Total memory usage: 39.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:23:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:23:48 on modular [Seed = 2692158889] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 5, 1) ] 1.10576830681247164656097111182671309904513044221613666621520 3.0583747280739988265523538592098468041222 3.273947854498539262126681269961664552701 {@ <1, 1, 4, 1>, <40452, 1, 4, 1> @} Total time: 2.049 seconds, Total memory usage: 39.27MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:23:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:23:17 on modular [Seed = 2775715981] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1> @} Total time: 1.919 seconds, Total memory usage: 39.27MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:23:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:23:08 on modular [Seed = 2927172511] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1>, <71940, 1, 4, 1>, <3394, 1, 4, 1>, <0, 1, 4, 1>, <51125, 1, 4, 1>, <27105, 1, 4, 1>, <16153, 1, 4, 1>, <64966, 1, 4, 1> @} Total time: 2.830 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:22:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:22:56 on modular [Seed = 2976253068] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1> @} Total time: 1.830 seconds, Total memory usage: 39.28MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:22:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:22:48 on modular [Seed = 3127708799] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1>, <6605, 1, 4, 1>, <12049, 1, 4, 1>, <10607, 1, 4, 1>, <13847, 1, 4, 1> @} Total time: 2.200 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:22:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,1); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:22:40 on modular [Seed = 3211266418] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 >> Chabauty(P,1); ^ Runtime error in 'Chabauty': p must be a prime number Total time: 1.429 seconds, Total memory usage: 39.31MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:21:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:21:44 on modular [Seed = 1285290697] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1>, <2238, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 1.970 seconds, Total memory usage: 39.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:21:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:21:31 on modular [Seed = 1367795655] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] >> P:=J!B[1]; Q:=J![x,54]; ^ Runtime error in '!': Points specified by second polynomial are not on the curve 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1>, <2238, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 1.990 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:08:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:08:51 on modular [Seed = 1774115047] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x - 27/2, -135/8*x + 27/4, 2) ] 2.367463309945994887046414980850866680181 8.937165595523057578067849122405558415787 6.473890696352274693832677312064632522802 {@ <0, 1, 4, 1> @} (x^2 - 10*x - 72, 44*x + 198, 2) (x^2 + 6*x + 36, -12*x + 18, 2) Total time: 3.009 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:08:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:08:15 on modular [Seed = 1923466176] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x - 27/2, -135/8*x + 27/4, 2) ] 2.367463309945994887046414980850866680181 8.937165595523057578067849122405558415787 6.473890696352274693832677312064632522802 {@ <0, 1, 4, 1> @} (x^2 - 10*x - 72, 44*x + 198, 2) (x^2 + 6*x + 36, -12*x + 18, 2) Total time: 3.049 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:07:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:07:20 on modular [Seed = 1973601296] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x - 27/2, -135/8*x + 27/4, 2) ] 2.367463309945994887046414980850866680181 8.937165595523057578067849122405558415787 6.473890696352274693832677312064632522802 {@ <0, 1, 4, 1> @} Total time: 3.020 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:07:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:07:12 on modular [Seed = 2125057811] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x - 27/2, -135/8*x + 27/4, 2) ] 2.367463309945994887046414980850866680181 8.937165595523057578067849122405558415787 6.473890696352274693832677312064632522802 {@ <0, 1, 4, 1> @} Total time: 3.060 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:06:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:06:14 on modular [Seed = 14135595] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <707278, 1, 4, 1> @} Total time: 2.220 seconds, Total memory usage: 39.60MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:06:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:06:03 on modular [Seed = 97694771] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <279838, 1, 4, 1>, <227084, 1, 4, 1>, <170618, 1, 4, 1>, <70671, 1, 4, 1>, <202525, 1, 4, 1>, <0, 1, 4, 1>, <172723, 1, 4, 1>, <131279, 1, 4, 1>, <278961, 1, 4, 1>, <88671, 1, 4, 1>, <119888, 1, 4, 1> @} Total time: 3.350 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:05:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:05:53 on modular [Seed = 249151282] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <130318, 1, 4, 1>, <25679, 1, 4, 1> @} Total time: 2.250 seconds, Total memory usage: 39.72MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:05:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:05:45 on modular [Seed = 331653591] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <83518, 1, 4, 1>, <68269, 1, 4, 1> @} Total time: 2.089 seconds, Total memory usage: 39.61MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:05:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:05:32 on modular [Seed = 348366533] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <28558, 1, 4, 1>, <16269, 1, 4, 1> @} Total time: 2.029 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:05:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:05:19 on modular [Seed = 432974843] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <14638, 1, 4, 1>, <10148, 1, 4, 1>, <8477, 1, 4, 1>, <7929, 1, 4, 1>, <1577, 1, 4, 1>, <12853, 1, 4, 1>, <3367, 1, 4, 1>, <0, 1, 4, 1>, <1495, 1, 4, 1>, <6129, 1, 4, 1> @} Total time: 3.020 seconds, Total memory usage: 39.62MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:03:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:03:03 on modular [Seed = 586518600] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 Total time: 1.610 seconds, Total memory usage: 39.61MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:47:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:47:06 on modular [Seed = 3457769578] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 2*x + 12, -4*x - 6, 2) ] 0.333877813949881714276569288505833032214 7.0893735547043793827827401012245389724008 5.00907431146146177197235032950126491659705 {@ <86443, 1, 4, 1>, <0, 1, 4, 1>, <130318, 1, 4, 1>, <45920, 1, 4, 1>, <6, 1, 4, 1> @} Total time: 5.870 seconds, Total memory usage: 39.53MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:45:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:45:35 on modular [Seed = 3556997845] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 2*x + 12, -4*x - 6, 2) ] 0.333877813949881714276569288505833032214 7.0893735547043793827827401012245389724008 5.00907431146146177197235032950126491659705 {@ <86443, 1, 4, 1>, <0, 1, 4, 1>, <130318, 1, 4, 1>, <45920, 1, 4, 1>, <6, 1, 4, 1> @} Total time: 5.660 seconds, Total memory usage: 39.53MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:44:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:44:56 on modular [Seed = 3675026941] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 {@ @} Total time: 1.659 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:44:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:44:48 on modular [Seed = 3725160688] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 {@ <45766, 1, 4, 1>, <58628, 1, 4, 1>, <8022, 1, 4, 1>, <65546, 1, 4, 1>, <10803, 1, 4, 1>, <22990, 1, 4, 1>, <52424, 1, 4, 1>, <5012, 1, 4, 1> @} Total time: 2.640 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:44:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:44:38 on modular [Seed = 3845278685] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 {@ <14058, 1, 4, 1>, <0, 1, 4, 1>, <24455, 1, 4, 1>, <12913, 1, 4, 1>, <14425, 1, 4, 1> @} Total time: 2.160 seconds, Total memory usage: 39.42MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:44:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:44:31 on modular [Seed = 3896461097] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 {@ <0, 1, 5, 1> @} Total time: 1.600 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:44:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:44:22 on modular [Seed = 4013441063] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 {@ <794, 1, 4, 1>, <1049, 1, 4, 1> @} Total time: 1.710 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:42:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,41); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:42:33 on modular [Seed = 4062534981] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 Total time: 1.229 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:42:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,41); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:42:25 on modular [Seed = 4180563526] ------------------------------------- 1 [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 Total time: 1.199 seconds, Total memory usage: 38.67MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:35:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,41); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:35:48 on modular [Seed = 2354783558] ------------------------------------- 1 [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <0, 1, 5, 1>, <115856199, 1, 5, 1>, <112103779, 1, 5, 1> @} Total time: 4.559 seconds, Total memory usage: 38.67MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:35:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,37); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:35:34 on modular [Seed = 2470722123] ------------------------------------- 1 [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <1165288, 1, 4, 1>, <1589581, 1, 4, 1>, <148749, 1, 4, 1>, <430481, 1, 4, 1>, <1363809, 1, 4, 1>, <0, 1, 4, 1>, <1874159, 1, 4, 1>, <187917, 1, 4, 1>, <564161, 1, 4, 1>, <1289506, 1, 4, 1>, <129135, 1, 4, 1>, <1403053, 1, 4, 1>, <1171182, 1, 4, 1>, <450502, 1, 4, 1>, <1721883, 1, 4, 1>, <369145, 1, 4, 1>, <1802688, 1, 4, 1> @} Total time: 5.910 seconds, Total memory usage: 38.67MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:35:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,31); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:35:20 on modular [Seed = 2520858480] ------------------------------------- 1 [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <0, 1, 4, 1>, <923519, 1, 4, 1>, <804103, 1, 4, 1>, <275093, 1, 4, 1>, <65871, 1, 4, 1>, <275737, 1, 4, 1>, <907036, 1, 4, 1>, <222533, 1, 4, 1>, <849946, 1, 4, 1>, <438850, 1, 4, 1>, <665474, 1, 4, 1> @} Total time: 4.589 seconds, Total memory usage: 38.67MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:35:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:35:04 on modular [Seed = 2638885995] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <279839, 1, 4, 1>, <183996, 1, 4, 1>, <156556, 1, 4, 1>, <0, 1, 4, 1>, <131380, 1, 4, 1>, <183336, 1, 4, 1>, <234426, 1, 4, 1>, <70852, 1, 4, 1>, <175397, 1, 4, 1>, <255331, 1, 4, 1> @} Total time: 4.879 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:34:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:34:49 on modular [Seed = 2692155687] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <481069, 1, 4, 1>, <707279, 1, 4, 1>, <0, 1, 4, 1>, <480287, 1, 4, 1>, <176530, 1, 4, 1>, <148044, 1, 4, 1>, <95583, 1, 4, 1> @} Total time: 4.389 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:34:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:34:35 on modular [Seed = 2809138738] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <6554, 1, 4, 1>, <130319, 1, 4, 1>, <0, 1, 4, 1>, <2044, 1, 4, 1>, <25657, 1, 4, 1> @} Total time: 4.250 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:34:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:34:23 on modular [Seed = 2860319499] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <279, 1, 4, 1>, <83519, 1, 4, 1>, <47775, 1, 4, 1>, <0, 1, 4, 1>, <38178, 1, 4, 1>, <31851, 1, 4, 1>, <64072, 1, 4, 1>, <3372, 1, 4, 1> @} Total time: 4.450 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:34:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:34:10 on modular [Seed = 2976258077] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <19662, 1, 4, 1>, <0, 1, 4, 1>, <21569, 1, 4, 1>, <28559, 1, 4, 1>, <14650, 1, 4, 1> @} Total time: 4.080 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:34:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:33:56 on modular [Seed = 3026394382] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <14639, 1, 4, 1>, <6873, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 3.629 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:33:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:33:41 on modular [Seed = 3144422128] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 Total time: 3.259 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:32:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:32:15 on modular [Seed = 3194558367] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 Total time: 3.250 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:30:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:30:38 on modular [Seed = 1150545604] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <2, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 3.540 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:30:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:30:26 on modular [Seed = 1268572615] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <37142, 1, 4, 1>, <7232, 1, 4, 1>, <16290, 1, 4, 1>, <2, 1, 4, 1>, <0, 1, 4, 1>, <63333, 1, 4, 1>, <12209, 1, 4, 1> @} Total time: 4.200 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:30:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:30:16 on modular [Seed = 1318708440] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <21536, 1, 4, 1>, <2, 1, 4, 1>, <2144, 1, 4, 1>, <21325, 1, 4, 1>, <10144, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 3.790 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:29:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); 2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:29:52 on modular [Seed = 1434648571] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <21536, 1, 4, 1>, <2, 1, 4, 1>, <2144, 1, 4, 1>, <21325, 1, 4, 1>, <10144, 1, 4, 1>, <0, 1, 4, 1> @} >> 2*Q;; ^ User error: Identifier 'Q' has not been declared or assigned Total time: 3.790 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:29:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); 2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:29:41 on modular [Seed = 1485828835] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <2, 1, 4, 1>, <7806, 1, 4, 1>, <14167, 1, 4, 1>, <12775, 1, 4, 1>, <8028, 1, 4, 1>, <12820, 1, 4, 1>, <10090, 1, 4, 1>, <8185, 1, 4, 1>, <0, 1, 4, 1>, <9523, 1, 4, 1> @} >> 2*Q;; ^ User error: Identifier 'Q' has not been declared or assigned Total time: 4.280 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:29:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); 2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:29:25 on modular [Seed = 1602811156] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <2, 1, 4, 1>, <623, 1, 4, 1>, <0, 1, 4, 1>, <416, 1, 4, 1>, <0, 1, 0, 6> @} >> 2*Q;; ^ User error: Identifier 'Q' has not been declared or assigned Total time: 5.889 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:26:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,11); //Order(Q);Order(P); 2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:26:15 on modular [Seed = 1706217196] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 (x^2, 2, 2) Total time: 2.930 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:25:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:25:36 on modular [Seed = 1757397958] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ (1, 0, 0), (x^2, 2, 2), (x^2, -2, 2), (x, 2, 1), (x, -2, 1), (x - 2, 6, 1), (x - 2, -6, 1), (x^2 + x + 2, x, 2), (x^2 + x + 2, -x, 2), (x^2 - 2*x - 4, 4*x + 6, 2), (x^2 - 2*x - 4, -4*x - 6, 2), (x^2 - 2*x, 4*x - 2, 2), (x^2 - 2*x, -4*x + 2, 2), (x^2 - 2*x, 2*x + 2, 2), (x^2 - 2*x, -2*x - 2, 2), (x^2 + 3/4*x - 3/4, 9/8*x + 11/8, 2), (x^2 + 3/4*x - 3/4, -9/8*x - 11/8, 2) @} {@ <2, 1, 4, 1>, <7806, 1, 4, 1>, <14167, 1, 4, 1>, <12775, 1, 4, 1>, <8028, 1, 4, 1>, <12820, 1, 4, 1>, <10090, 1, 4, 1>, <8185, 1, 4, 1>, <0, 1, 4, 1>, <9523, 1, 4, 1> @} Total time: 4.299 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:24:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:24:42 on modular [Seed = 1874379787] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <2, 1, 4, 1>, <7806, 1, 4, 1>, <14167, 1, 4, 1>, <12775, 1, 4, 1>, <8028, 1, 4, 1>, <12820, 1, 4, 1>, <10090, 1, 4, 1>, <8185, 1, 4, 1>, <0, 1, 4, 1>, <9523, 1, 4, 1> @} Total time: 4.240 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:21:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:21:41 on modular [Seed = 2007028716] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 {@ @} Total time: 1.679 seconds, Total memory usage: 39.25MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:20:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=5000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:20:54 on modular [Seed = 2058214962] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 5.450 seconds, Total memory usage: 40.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:37 on modular [Seed = 30844182] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.459 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:34 on modular [Seed = 80977948] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (462313490571*2^3 + O(2^53))*$.1^2 +... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.449 seconds, Total memory usage: 39.66MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:30 on modular [Seed = 199012119] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreePart( F: $.1^4 + O(2^50)*$.1^3 + (4949974972277*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/loclib.m", line 120, column 5: >> assert R eq 0; ^ Runtime error in assert: Assertion failed >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.820 seconds, Total memory usage: 40.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:27 on modular [Seed = 249145903] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (1407999997813*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.280 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:24 on modular [Seed = 365080887] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (2845492574069*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.129 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:20 on modular [Seed = 416267326] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (588297271157*2^3 + O(2^53))*$.1^2 +... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.159 seconds, Total memory usage: 39.16MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:18:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:18:11 on modular [Seed = 533248754] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1> @} Total time: 1.530 seconds, Total memory usage: 38.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:18:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:18:04 on modular [Seed = 586516432] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <63266, 1, 4, 1>, <54395, 1, 4, 1>, <77867, 1, 4, 1>, <46987, 1, 4, 1>, <77545, 1, 4, 1>, <54827, 1, 4, 1>, <16840, 1, 4, 1> @} Total time: 2.410 seconds, Total memory usage: 38.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:17:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:17:54 on modular [Seed = 704548053] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <13181, 1, 4, 1>, <16533, 1, 4, 1>, <20252, 1, 4, 1>, <24913, 1, 4, 1>, <18857, 1, 4, 1> @} Total time: 1.940 seconds, Total memory usage: 38.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:17:48 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:17:45 on modular [Seed = 754682283] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <875, 1, 4, 1>, <9660, 1, 4, 1>, <2248, 1, 4, 1>, <6456, 1, 4, 1>, <5430, 1, 4, 1>, <13630, 1, 4, 1>, <2351, 1, 4, 1>, <636, 1, 4, 1> @} Total time: 2.359 seconds, Total memory usage: 38.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:17:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:17:38 on modular [Seed = 870618815] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 486 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 486 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 486, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (12817354456845*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <875, 1, 4, 1>, <9660, 1, 4, 1>, <2248, 1, 4, 1>, <6456, 1, 4, 1>, <5430, 1, 4, 1>, <13630, 1, 4, 1>, <2351, 1, 4, 1>, <636, 1, 4, 1> @} Total time: 2.810 seconds, Total memory usage: 39.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:17:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:17:24 on modular [Seed = 920753068] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <2369, 1, 4, 1> @} Total time: 1.860 seconds, Total memory usage: 39.31MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:15:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:15:18 on modular [Seed = 3222758873] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <63266, 1, 4, 1>, <54395, 1, 4, 1>, <77867, 1, 4, 1>, <46987, 1, 4, 1>, <77545, 1, 4, 1>, <54827, 1, 4, 1>, <16840, 1, 4, 1> @} Total time: 2.870 seconds, Total memory usage: 39.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:15:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:15:12 on modular [Seed = 3306315994] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 486 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 486 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 486, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (34471543027*2^3 + O(2^53))*$.1^2 + ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <63266, 1, 4, 1>, <54395, 1, 4, 1>, <77867, 1, 4, 1>, <46987, 1, 4, 1>, <77545, 1, 4, 1>, <54827, 1, 4, 1>, <16840, 1, 4, 1> @} Total time: 2.680 seconds, Total memory usage: 39.11MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:15:02 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:15:00 on modular [Seed = 3390926817] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <63266, 1, 4, 1>, <54395, 1, 4, 1>, <77867, 1, 4, 1>, <46987, 1, 4, 1>, <77545, 1, 4, 1>, <54827, 1, 4, 1>, <16840, 1, 4, 1> @} Total time: 2.410 seconds, Total memory usage: 38.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:06:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:06:23 on modular [Seed = 3741874415] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] 3.10929166505257268510955302447913689669881188175885053195997 5.0871519665718709241496748503892215814574 4.546976191088164899027528915195813871219 {@ <2090, 1, 4, 1> @} Total time: 1.740 seconds, Total memory usage: 39.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:06:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:06:19 on modular [Seed = 3828564944] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 162 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 162 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 162, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (3086848360367*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] 3.10929166505257268510955302447913689669881188175885053195997 5.0871519665718709241496748503892215814574 4.546976191088164899027528915195813871219 {@ <2090, 1, 4, 1> @} Total time: 2.020 seconds, Total memory usage: 40.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:06:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:06:10 on modular [Seed = 3913174740] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 162 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 162 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 162, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (6289834180527*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] 3.10929166505257268510955302447913689669881188175885053195997 5.0871519665718709241496748503892215814574 4.546976191088164899027528915195813871219 {@ <2090, 1, 4, 1> @} Total time: 1.700 seconds, Total memory usage: 39.49MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:57:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 19:57:17 on modular [Seed = 2992967776] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] Errors: /bin/sh: line 1: 31885 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:56:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 19:56:36 on modular [Seed = 3094288963] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] Errors: /bin/sh: line 1: 31879 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:54:04 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:54:02 on modular [Seed = 3177848744] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3*x + 9, -3*x + 9, 2) ] 2.26560762534307442838198741030006503529825419756590246025749 6.02129893666715068547855842318402917446829 4.817286263160274487012870992172046628945 {@ <0, 1, 4, 1> @} Total time: 1.929 seconds, Total memory usage: 39.68MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:53:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:53:13 on modular [Seed = 1117118852] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2, -2*x - 1, 2) ] 1.5626377849846081444970869663211590195034831524679039584881 4.1571752748544702272869038549736433015765 2.620061685824055104222380518326995219648 {@ <0, 1, 5, 1> @} Total time: 2.779 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:51:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:51:37 on modular [Seed = 1251863020] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2, -2*x - 1, 2) ] 1.5626377849846081444970869663211590195034831524679039584881 4.1571752748544702272869038549736433015765 2.620061685824055104222380518326995219648 Total time: 2.629 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:50:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:50:40 on modular [Seed = 1335421968] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ <1, 1, 4, 1> @} Total time: 1.790 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:50:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:50:31 on modular [Seed = 1417934652] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ <1, 1, 4, 1>, <27726, 1, 4, 1>, <4450, 1, 4, 1>, <45574, 1, 4, 1>, <54682, 1, 4, 1>, <3462, 1, 4, 1>, <39256, 1, 4, 1>, <67968, 1, 4, 1> @} Total time: 2.529 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:50:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:50:21 on modular [Seed = 1502542376] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ <1, 1, 4, 1>, <419, 1, 4, 1>, <4227, 1, 4, 1>, <7898, 1, 4, 1>, <19914, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.140 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:50:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:50:11 on modular [Seed = 1586101538] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ <1, 1, 4, 1>, <13410, 1, 4, 1> @} Total time: 1.679 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:50:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:50:02 on modular [Seed = 1672790118] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ <1, 1, 4, 1>, <1133, 1, 4, 1>, <2194, 1, 4, 1> @} Total time: 1.790 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:49:52 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:49:51 on modular [Seed = 1757399928] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ (1, 0, 0), (x - 1, 2, 1), (x - 1, -2, 1) @} Total time: 1.350 seconds, Total memory usage: 39.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:49:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:49:16 on modular [Seed = 1840959070] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ (1, 0, 0), (x - 1, 2, 1), (x - 1, -2, 1) @} Total time: 1.290 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:49:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=12); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:49:04 on modular [Seed = 1923469645] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 Total time: 0.950 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:47:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:47:22 on modular [Seed = 2007034834] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 Total time: 1.350 seconds, Total memory usage: 39.47MB '168.168' ************** MAGMA ***************** Host 168.168.67.112 (168.168.67.112) Time: Mon Dec 5 18:57:32 2005 Input: 1024^1024 Output: Magma V2.11-10 Mon Dec 5 2005 18:57:32 on modular [Seed = 754695423] ------------------------------------- 3524971412108382657134814839800281546439142134396647106039138260573107027685474\ 9365048330296473663862456968155395298373973259049475943113619888338673116133666\ 8147068707652719076562056460186083699855587212676703217390319386338332818891926\ 2015842653180692314423926972687639995196119198034802329170347230576378241039458\ 9758934585631111078120435303032688818751446435291371357171755632775362932694795\ 0763134366874696380043276893902467353218558306108568659249137608267637760032658\ 5171655733421064227734347575779978049902155982241243427508708431729345512957040\ 6707590002071704673135527533543217355987568107697577946785796412456048360072965\ 6168710248662446500810590681830381345185142229871868373945980198595129936003792\ 3619019757683890508073335998909468700899941624772202006199255993140187235737970\ 8488585003666965930609730430774107407494018065365845077094320534700692354400169\ 8241315783891536569167546822524255627428950268220861122361857689319404333240786\ 9238646364237802929158238455090401228426527712466745281698565933749758099159251\ 0201479766500877427834566619156314388107585743546289067551052434075678195345373\ 3639195713232101136226155117651343296272079557936053768928759383576728708813056\ 7930552129335997542780192199753489147409086811346735778435978338309108571710080\ 7228425031226776985197364359404683041506613943646666199454899363685801848776729\ 6858378032282161138338547424434092214804502325631304177096253207949716727377373\ 8598397552004773997816512490691685793196090240739784153665765037875801240915720\ 5939513085324282439290108909069036515430690359963152986587749930516880670326145\ 0369876070529616967815564185509662018228218579780200625368240156976209572227380\ 6553883218709740985950266919658902596119944875899737379297319172333554977239487\ 8874050854532785922475822836403793986623193174020931432381418437022760412682276\ 3829893548396254532412898071082609051342346791309548675704473545497601746910070\ 7852845274502799494385322948054451236883137876111968161671932763730814231510512\ 0528704683515182038320225078665313911731749364255621284434304945437214609406008\ 6405209720295099554355680948888157014704194108891565239711821728144232741409554\ 2807059432838166704828677197285770343552580354470783456777402720661414341998241\ 0109261930698311010857874866840743851472857645330929169548403751084494725893729\ 3554504737710599868010583420219027353676279009748723681378389963973798981614548\ 2597091073285820278128297393764284797338183867298069339903942934261300159514896\ 8082010016061022316242842367672741265405434553107296623559604413326352140529618\ 1711754506578842550993346187227316979201855824371823913976733011681606825166392\ 1470656698146596173137480894913174236475299307832636771411700140421093025153813\ 2442219335072672096865184691303027156962439777053707286583949764055151291816402\ 5464624527191347971790992102335775962779256460318241722748740845621134400433973\ 9519106547362071710425068604089658092870084259391917328384453147095220560087448\ 2302488523867074532907781264990865351844684807012208039108287564534854500486391\ 5388760636114766656202302948114683518353740720605302159079093112818161319422197\ 76 Total time: 0.190 seconds, Total memory usage: 3.24MB '168.168' ************** MAGMA ***************** Host 168.168.67.112 (168.168.67.112) Time: Mon Dec 5 18:57:21 2005 Input: 1024^1024^3 Output: Magma V2.11-10 Mon Dec 5 2005 18:57:20 on modular [Seed = 837198318] ------------------------------------- >> 1024^1024^3; ^ Runtime error in '^': Argument 2 (1073741824) is too large Total time: 0.190 seconds, Total memory usage: 3.24MB '168.168' ************** MAGMA ***************** Host 168.168.67.112 (168.168.67.112) Time: Mon Dec 5 18:56:56 2005 Input: 1024^9999 Output: Magma V2.11-10 Mon Dec 5 2005 18:56:56 on modular [Seed = 920757953] ------------------------------------- 9755879814593598710390944964160484286918253311075993083706340565069783520158763\ 1672606184700434536172184022895073813958009648404371642349247153040568823172609\ 3641565417350692699838908923155515553300808450316512571981016141330897477008661\ 2330908863206560463571340681293649220239094087378460205364049336725544698214590\ 9201484794690929739558378875956299966990925505720468315641797669143507857597160\ 4419853088273531816198589663932375536698956361797485279322445111455574275652932\ 7671029471425412924795787458748107524029918332158777143869477511729305162470255\ 2752303235936113628941152557352618706704888742055285829812888619353928394329586\ 3173179418709840035072036793204211427803584460006365175429561534101598040502501\ 7666603138064354062789905857966821056102473678761936386847047945271833319810447\ 7503950902186280963889085279623407172102513494899737825067814628282473651444032\ 2617545242496240500599602944609772575382316787435729974159614833623173066138322\ 0426985882367921437870420793230646877395269289154985002134710803579719476788410\ 2390060757338801904323415527970984630974829583492192943919286764438313175595233\ 3589430583448210275822044964144575769737220498571840505091794591971217290739243\ 6361877429217111702392889302251437611689388835919607588139768372048891791803022\ 8567765182201952551733390530747143987660297906559154257877865062766762051167774\ 6218775486739477296309358858615706192412506699581996109556224771136667212252436\ 0595410463388564544103778274164481157691821750247233900899592022905652709285383\ 4396490556886108721966456683024660217345266716745494327567429310992723897738458\ 7420266731157517138219781390453083323040983694346296626645359123811737827073045\ 2965269691176506806197174484735472789380793089194963999896938690529181761552481\ 3553870957258158700775280925160187788847596690939008281285519678409331944453085\ 9524471592551701272678752369339423192696880168707971275238849839315883327824522\ 2848931451445631364000438812267316806399748372353252229452317938743177037522765\ 3902394829054410908526989350803290551371114681520070289492348907695530745428526\ 3433537300827781190285175638771225486909034277877856114964271790557611172453444\ 2337486854381810963875592801401857505431669481789531299605978090741878188580804\ 7861013896015177359548022880835156346555736112072369979402060248160258183873280\ 7660676151327331128934498085301740830153797624302042783834813392958364166482870\ 2355090032520222559118221720886731305233792674873914094694210558257774640949217\ 1005109619949865042205863666480204948848602888865081893827824708235614601510675\ 8566829641344293578928104280653132324148128089203156113717150264041971124319613\ 6546357823187799405734305020694311704335168288744516592345277039485292030705080\ 6811825945806690751338881338734337320513242710423198925065438880617888956468820\ 6471478277838844293698821689581076693635750039949178450150802104619853715837070\ 1586162852020712629792275528912455039922421096052125864890480703215602647938607\ 9027509722726286975570669277130672553458073741321811754995512076165145407202008\ 3867274205270989683391527949614430395118383849453516793582995378301081103355587\ 6928207694434324068707221352372289781380971812077323527130828179739967269669070\ 6522053137296112837902191527435056474251142988305961119743114866393904889184427\ 8342012877885178104325116182820362751478648629375524387027790094238132767349333\ 4519656833404372401522241535503140243589764917186866224826413049821302458417321\ 7821477493782178721829032732776199406170133317152890102693633290861876140118655\ 9355983191275655956997001655212602128405873694151458006056712598958039497168494\ 6658741395315892722850258900956369809473625976793687664604779466252011645871059\ 8095682866046477965882265335662203237014873771357275732956272353781004397786800\ 0810574329385824384096724354485662493612626269037382423370093370967667864211762\ 6049065707780279534346847287823990826949318327931840395106460839379594473633943\ 4677379235657513558172548334069915372821380740296861178196294813751718738715600\ 4504888979791871100594222389407010720445002933865127223838445686818502957004745\ 6383136826722719007724340410107115393464862894669681577899293542961356005773585\ 6815666702570787582170385320692280682551186044799860931670051266577555634329397\ 4304708916491617529404093753536156129714372692380954845212855087568783998058997\ 1370680546782224059176070796587362817124396484730244457905255960284504798400698\ 4080609476712080607362580417220691879429220441442266487422860780678075569162144\ 0395535305737410849473113400859665412659301268489449735957262950057276274444139\ 9715480469952804604670669670015321790172054780675899876744444377829802599070467\ 1064119616864741746350122118761982858542111617385494944950291053490911884642177\ 1691937675034697108152327192407608377806002903574521244546530742722044212184301\ 6262459995112633158327103418416850574886303640492788332071094228627720597938560\ 0032753876902156908902475779158579799379117023156181680839740632717464319897952\ 5379695969250769880595471337813850335602928996806859257087590257107957734501685\ 0962960906546662581116919824031723279788894879158597575499875678106898978175478\ 5766894757984290459812062655287844718254055634222288739363605797790703057990324\ 9596682055296712033030929222341025236053486705094952004145316716438828394505191\ 5235607664378013016883775541561739675174323643534006098368664710519751530439237\ 7048509194756028215511060889428234474295596888031891413474583717885370305663458\ 1853401127091971965515995145061612299066529346869711839639645149370142955634874\ 5004045181799775205150051475516117753389513191498100173630726856938378865853676\ 4653595903933457117812751068676213418398771905666052852293530644521641274077353\ 3884842830990949242485413662828498195437558597834719660691918742959398020273298\ 2626518195667803644949463585467579859017324034691173822935803975147104589698490\ 5604253853530672859980608336188220100895440257117718037123901139014832332863025\ 1491095656871109897060601187574972016954849165134221189362896865829423830900094\ 2233978542037763152818939098071707641837543785793138596275906713129836292287407\ 5306771966136131077505386830385440741212602424090200448402341053834244298244196\ 3671555408547812253820848531388602271607398199384921156818705190617400122453090\ 9213227337578485407230286188044791319347144837099190972984881570076163973498916\ 7631786640815184786609877119914573753850525407237787941787166448811813952824999\ 4295545375085670498217686028636606783733396411513726690582254711049530484193086\ 9258073632469358233994560827410878750327287226854996690279485512390699164577114\ 5005567095258321802221615726902860527221798691148898344350736460767403701564180\ 8427314680423537831033964209658598138530623793994488003503210273593445806770619\ 6754982007518921329196770873160147655385705787892302772412995901407406640698813\ 9341415790680367034429953938744300957466706326608203626570060100472780395081576\ 0119674713763323330280099292037091730639112633097056053042437559615204355909061\ 2988578305253079568644263440244758517683628736387602121543573441638262073385937\ 6550283939078454934018186511424979373131557167273892308761948289919597694037631\ 5877815615976304227812569261862438544594436477607471394833145751247207497675142\ 4811831256087533423953258861820471211399853122476248908206880319115614209354680\ 0240177453569435856826123343246307979375813766690509536728620202081621295850196\ 2541831254555403648551120767979131957721208379151425723561221580473306962755239\ 1459835186191312075949538737984430846735621889057551327511101591187475199332105\ 1187499212912761662408998884017200816336949969155081624297784448308604605191553\ 3472808262417796544166850026204128391196220202719930643637789339378469529149558\ 1605219061019524877571582354465329114070718653290483584351415962646854946307779\ 3065053738228575496042254868240718850623875449564681564213374080821846976022464\ 9982722678299481767628137385922461866230814763117556962126647067976198770744325\ 4695448118047046692489546894678463214823790515767903723315328375863070655447732\ 0240825949343108098909984225700645349435643941861448323671407906021299467029023\ 2607380257854523040134103033193493283046851814946819869093392583510652398987043\ 9917346219655219746314612432521895307623299737663879887912246819301519720755646\ 6625515948213101115341109752861381445428976350612905893610505228571021997306405\ 5218395689792171512891812739908407582233848193441645444071272521436140387746948\ 7153591485020721052798955362757118097891309613171420906105465975779677817045504\ 8790987679684799764294431704906634615381339687973494328062115889902031431395291\ 2066514095238127038419701502459544498349735620840750057308594902817870522410032\ 8987255242955700393244733592487945592223392359091367529021872466372563739673064\ 7991224110456887055837846872631996921011190707675817915610209677682983337857576\ 5726602953112867513203300343617661780488437359017767020182510988491547494999142\ 3984006586450558900604672596455660800724786401761341465272010575392392778174724\ 3337807473203381274705102463957319943135378613551360235922597605510303395809603\ 7113622229776717641076896027194191158455095791143991929554442043771608833072449\ 5769407758825331577698496435330283195719296007423923278800226914831913409194387\ 1260228884995639709039228687228275264274913993356598245732698265076363134244085\ 2388719036324269055162805039514224014020036457310230294036869260026779672563150\ 8528926067211752016736146948601316085822480564833673847445524144546378177109801\ 2088354294635094315482598960185652388280845189254744772548220621673244373003291\ 4577524358929795506239059545033501080996217877502783939164045245006623034013493\ 3022641814718255481553675765646398875315843653703665831763627102743588870073054\ 3362226394196694473075748700714622109407799819372104467040018536220111845966571\ 07 ** WARNING: Output too long, hence truncated. '168.168' ************** MAGMA ***************** Host 168.168.67.112 (168.168.67.112) Time: Mon Dec 5 18:56:33 2005 Input: 1024^1024 Output: Magma V2.11-10 Mon Dec 5 2005 18:56:32 on modular [Seed = 1022078131] ------------------------------------- 3524971412108382657134814839800281546439142134396647106039138260573107027685474\ 9365048330296473663862456968155395298373973259049475943113619888338673116133666\ 8147068707652719076562056460186083699855587212676703217390319386338332818891926\ 2015842653180692314423926972687639995196119198034802329170347230576378241039458\ 9758934585631111078120435303032688818751446435291371357171755632775362932694795\ 0763134366874696380043276893902467353218558306108568659249137608267637760032658\ 5171655733421064227734347575779978049902155982241243427508708431729345512957040\ 6707590002071704673135527533543217355987568107697577946785796412456048360072965\ 6168710248662446500810590681830381345185142229871868373945980198595129936003792\ 3619019757683890508073335998909468700899941624772202006199255993140187235737970\ 8488585003666965930609730430774107407494018065365845077094320534700692354400169\ 8241315783891536569167546822524255627428950268220861122361857689319404333240786\ 9238646364237802929158238455090401228426527712466745281698565933749758099159251\ 0201479766500877427834566619156314388107585743546289067551052434075678195345373\ 3639195713232101136226155117651343296272079557936053768928759383576728708813056\ 7930552129335997542780192199753489147409086811346735778435978338309108571710080\ 7228425031226776985197364359404683041506613943646666199454899363685801848776729\ 6858378032282161138338547424434092214804502325631304177096253207949716727377373\ 8598397552004773997816512490691685793196090240739784153665765037875801240915720\ 5939513085324282439290108909069036515430690359963152986587749930516880670326145\ 0369876070529616967815564185509662018228218579780200625368240156976209572227380\ 6553883218709740985950266919658902596119944875899737379297319172333554977239487\ 8874050854532785922475822836403793986623193174020931432381418437022760412682276\ 3829893548396254532412898071082609051342346791309548675704473545497601746910070\ 7852845274502799494385322948054451236883137876111968161671932763730814231510512\ 0528704683515182038320225078665313911731749364255621284434304945437214609406008\ 6405209720295099554355680948888157014704194108891565239711821728144232741409554\ 2807059432838166704828677197285770343552580354470783456777402720661414341998241\ 0109261930698311010857874866840743851472857645330929169548403751084494725893729\ 3554504737710599868010583420219027353676279009748723681378389963973798981614548\ 2597091073285820278128297393764284797338183867298069339903942934261300159514896\ 8082010016061022316242842367672741265405434553107296623559604413326352140529618\ 1711754506578842550993346187227316979201855824371823913976733011681606825166392\ 1470656698146596173137480894913174236475299307832636771411700140421093025153813\ 2442219335072672096865184691303027156962439777053707286583949764055151291816402\ 5464624527191347971790992102335775962779256460318241722748740845621134400433973\ 9519106547362071710425068604089658092870084259391917328384453147095220560087448\ 2302488523867074532907781264990865351844684807012208039108287564534854500486391\ 5388760636114766656202302948114683518353740720605302159079093112818161319422197\ 76 Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:33:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:33:13 on modular [Seed = 3523567571] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1> @} Total time: 2.529 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:33:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:33:01 on modular [Seed = 3708440776] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1>, <117425, 1, 4, 1> @} Total time: 2.560 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:32:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:32:52 on modular [Seed = 3624883652] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1>, <25746, 1, 4, 1>, <41974, 1, 4, 1>, <66925, 1, 4, 1>, <1795, 1, 4, 1>, <83509, 1, 4, 1>, <5869, 1, 4, 1>, <44342, 1, 4, 1> @} Total time: 3.279 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:32:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:32:43 on modular [Seed = 3811864256] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1>, <6861, 1, 4, 1>, <16555, 1, 4, 1>, <25976, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.830 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:32:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:32:35 on modular [Seed = 3996738501] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1>, <5637, 1, 4, 1>, <7726, 1, 4, 1>, <4854, 1, 4, 1>, <13605, 1, 4, 1> @} Total time: 2.759 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:32:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:32:26 on modular [Seed = 3913180378] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1>, <176, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.430 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:26:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:25:58 on modular [Seed = 4062543180] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 Total time: 1.970 seconds, Total memory usage: 39.21MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:25:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=100); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:25:45 on modular [Seed = 4247417532] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 Total time: 1.960 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:25:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:25:32 on modular [Seed = 4163860396] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 Total time: 1.980 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:25:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:25:20 on modular [Seed = 2203323611] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 Total time: 1.960 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:23:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:23:52 on modular [Seed = 2388197822] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ <4, 1, 4, 1> @} Total time: 2.970 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:23:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:23:43 on modular [Seed = 2304639310] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ <4, 1, 4, 1>, <77093, 1, 4, 1>, <56288, 1, 4, 1>, <69699, 1, 4, 1>, <76710, 1, 4, 1>, <80434, 1, 4, 1>, <61220, 1, 4, 1>, <44724, 1, 4, 1> @} Total time: 3.830 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:23:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:23:36 on modular [Seed = 2487426132] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ <12745, 1, 4, 1>, <4, 1, 4, 1> @} Total time: 2.960 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:23:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:23:23 on modular [Seed = 2672301900] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ <4, 1, 4, 1>, <4975, 1, 4, 1>, <3147, 1, 4, 1>, <283, 1, 4, 1>, <12220, 1, 4, 1>, <13979, 1, 4, 1>, <7190, 1, 4, 1>, <10940, 1, 4, 1>, <8901, 1, 4, 1> @} Total time: 3.839 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:23:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:23:14 on modular [Seed = 2588742208] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ <4, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.910 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:22:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:22:07 on modular [Seed = 2742299396] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 Total time: 2.480 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:21:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1300); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:21:48 on modular [Seed = 2927175194] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ (1, 0, 0), (x^2 - 8*x + 16, 40*x - 144, 2), (x^2 - 8*x + 16, -40*x + 144, 2), (x - 4, 16, 1), (x - 4, -16, 1), (x^2 + 3*x + 12, 11*x + 12, 2), (x^2 + 3*x + 12, -11*x - 12, 2), (x^2 + 16, 8*x + 16, 2), (x^2 + 16, -8*x - 16, 2), (x^2 - 56/25*x + 448/25, 196/125*x + 2832/125, 2), (x^2 - 56/25*x + 448/25, -196/125*x - 2832/125, 2) @} Total time: 2.669 seconds, Total memory usage: 39.44MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:20:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); 2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:20:00 on modular [Seed = 2826903728] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ (1, 0, 0), (x - 4, 16, 1), (x - 4, -16, 1), (x^2 + 3*x + 12, 11*x + 12, 2), (x^2 + 3*x + 12, -11*x - 12, 2), (x^2 + 16, 8*x + 16, 2), (x^2 + 16, -8*x - 16, 2), (x^2 - 56/25*x + 448/25, 196/125*x + 2832/125, 2), (x^2 - 56/25*x + 448/25, -196/125*x - 2832/125, 2) @} (x - 4, 16, 1) Total time: 2.390 seconds, Total memory usage: 39.06MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:19:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:19:30 on modular [Seed = 3009690505] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ (1, 0, 0), (x - 4, 16, 1), (x - 4, -16, 1), (x^2 + 3*x + 12, 11*x + 12, 2), (x^2 + 3*x + 12, -11*x - 12, 2), (x^2 + 16, 8*x + 16, 2), (x^2 + 16, -8*x - 16, 2), (x^2 - 56/25*x + 448/25, 196/125*x + 2832/125, 2), (x^2 - 56/25*x + 448/25, -196/125*x - 2832/125, 2) @} Total time: 2.370 seconds, Total memory usage: 39.06MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:15:52 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:15:49 on modular [Seed = 3127718823] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 Total time: 2.370 seconds, Total memory usage: 39.06MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:03:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:03:33 on modular [Seed = 1622664221] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 {@ @} Total time: 1.139 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:03:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:03:27 on modular [Seed = 1807544172] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 {@ <1721, 1, 4, 1>, <738, 1, 4, 1> @} Total time: 1.300 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:03:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:03:19 on modular [Seed = 1990324857] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 0.820 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:02:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:02:34 on modular [Seed = 1906767171] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 0.830 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:02:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=3000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:02:24 on modular [Seed = 2091644996] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 2.229 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:51:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:51:30 on modular [Seed = 955236210] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 1.360 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:51:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:51:27 on modular [Seed = 3256173972] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (14185465184247*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 1.159 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:51:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:51:19 on modular [Seed = 3474475153] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 0.820 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:49:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:49:49 on modular [Seed = 3357495292] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 {@ @} Total time: 1.129 seconds, Total memory usage: 38.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:49:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:49:39 on modular [Seed = 3573699311] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 {@ <1311, 1, 4, 1>, <769, 1, 4, 1>, <1594, 1, 4, 1> @} Total time: 1.270 seconds, Total memory usage: 38.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:47:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:47:20 on modular [Seed = 3725155999] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 Total time: 1.360 seconds, Total memory usage: 39.61MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:47:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:47:16 on modular [Seed = 3675020182] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (137394841321471*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 Total time: 2.069 seconds, Total memory usage: 40.90MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:47:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:47:13 on modular [Seed = 3828573356] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (278930146590721*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 Total time: 1.169 seconds, Total memory usage: 39.57MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:46:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:46:56 on modular [Seed = 3778437560] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 Total time: 0.790 seconds, Total memory usage: 38.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:38:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:38:12 on modular [Seed = 2220033769] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 48, -4*x - 24, 2) ] 1.802849529668960157376835690894363395849 9.531058522619703794561201943405304565704 7.589874985400055724681799478004752901208 {@ @} Total time: 1.320 seconds, Total memory usage: 38.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:38:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:38:03 on modular [Seed = 2169897971] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 48, -4*x - 24, 2) ] 1.802849529668960157376835690894363395849 9.531058522619703794561201943405304565704 7.589874985400055724681799478004752901208 {@ <6462, 1, 4, 1>, <18387, 1, 4, 1>, <25670, 1, 4, 1>, <0, 1, 4, 1>, <25738, 1, 4, 1>, <22629, 1, 4, 1> @} Total time: 1.790 seconds, Total memory usage: 38.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:37:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:37:55 on modular [Seed = 2287930098] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 48, -4*x - 24, 2) ] 1.802849529668960157376835690894363395849 9.531058522619703794561201943405304565704 7.589874985400055724681799478004752901208 {@ <4765, 1, 4, 1>, <13493, 1, 4, 1>, <8772, 1, 4, 1>, <7875, 1, 4, 1>, <5842, 1, 4, 1>, <2126, 1, 4, 1>, <3119, 1, 4, 1>, <12152, 1, 4, 1>, <4354, 1, 4, 1> @} Total time: 2.250 seconds, Total memory usage: 38.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:37:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:37:47 on modular [Seed = 2504136599] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 48, -4*x - 24, 2) ] 1.802849529668960157376835690894363395849 9.531058522619703794561201943405304565704 7.589874985400055724681799478004752901208 {@ <727, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 1.389 seconds, Total memory usage: 38.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:33:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:33:56 on modular [Seed = 2672303346] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 48, -4*x - 24, 2) ] 1.802849529668960157376835690894363395849 9.531058522619703794561201943405304565704 7.589874985400055724681799478004752901208 {@ <0, 1, 4, 1> @} Total time: 1.419 seconds, Total memory usage: 38.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:52 on modular [Seed = 2555321269] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ @} Total time: 1.330 seconds, Total memory usage: 38.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:44 on modular [Seed = 2775721616] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ <48522, 1, 4, 1>, <54152, 1, 4, 1> @} Total time: 1.540 seconds, Total memory usage: 38.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:36 on modular [Seed = 2927177107] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ <20536, 1, 4, 1>, <0, 1, 4, 1>, <28317, 1, 4, 1>, <21380, 1, 4, 1>, <23888, 1, 4, 1> @} Total time: 1.800 seconds, Total memory usage: 38.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:27 on modular [Seed = 2877052777] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ <9360, 1, 4, 1>, <10998, 1, 4, 1>, <1096, 1, 4, 1>, <10363, 1, 4, 1>, <9477, 1, 4, 1>, <3855, 1, 4, 1>, <11310, 1, 4, 1>, <12235, 1, 4, 1>, <9156, 1, 4, 1> @} Total time: 2.250 seconds, Total memory usage: 38.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:10 on modular [Seed = 3026405479] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ <1706, 1, 4, 1>, <1622, 1, 4, 1>, <1354, 1, 4, 1>, <557, 1, 4, 1> @} Total time: 2.319 seconds, Total memory usage: 39.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:04 on modular [Seed = 2976271724] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 64 ov... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 64 over Rational ..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 64, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (559154624462847*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ <1706, 1, 4, 1>, <1622, 1, 4, 1>, <1354, 1, 4, 1>, <557, 1, 4, 1> @} Total time: 2.240 seconds, Total memory usage: 40.02MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:29:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:29:29 on modular [Seed = 3144435659] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 Total time: 1.429 seconds, Total memory usage: 39.25MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:29 on modular [Seed = 1150531696] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 1.449 seconds, Total memory usage: 6.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:13 on modular [Seed = 1100397701] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 7.790638689351186835930798024215308990709 6.587156720882539675766293909746836340427 Total time: 1.080 seconds, Total memory usage: 5.96MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:10 on modular [Seed = 1251850121] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 3456 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 3456 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 3456, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (110447995125787*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 7.790638689351186835930798024215308990709 6.587156720882539675766293909746836340427 Total time: 1.540 seconds, Total memory usage: 6.92MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:08 on modular [Seed = 1468050070] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 3456 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 3456 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 3456, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (479125739405285*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 7.790638689351186835930798024215308990709 6.587156720882539675766293909746836340427 Total time: 0.750 seconds, Total memory usage: 5.79MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:04 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:03 on modular [Seed = 1351069082] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 3456 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 3456 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 3456, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (363636024410139*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 7.790638689351186835930798024215308990709 6.587156720882539675766293909746836340427 Total time: 0.670 seconds, Total memory usage: 5.69MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:00 on modular [Seed = 1569366164] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 3456 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 3456 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 3456, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (532686333542373*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 7.790638689351186835930798024215308990709 6.587156720882539675766293909746836340427 Total time: 1.229 seconds, Total memory usage: 6.36MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:21:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:21:37 on modular [Seed = 1656069161] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 18, 1296, 1) ] 1.386675839719476251203965940701820868253 11.892918185334237139088150509239494492736 9.325001442362978234527468537544353265163 Total time: 9.410 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:49:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:49:24 on modular [Seed = 920764651] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 {@ @} Total time: 1.740 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:49:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:49:15 on modular [Seed = 870630870] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 {@ <34002, 1, 4, 1>, <41826, 1, 4, 1>, <7056, 1, 4, 1>, <37191, 1, 4, 1>, <3354, 1, 4, 1>, <0, 1, 4, 1>, <68570, 1, 4, 1>, <39624, 1, 4, 1> @} Total time: 2.660 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:49:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:49:08 on modular [Seed = 1022085841] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 {@ <12435, 1, 4, 1>, <5943, 1, 4, 1>, <27236, 1, 4, 1>, <12180, 1, 4, 1>, <21368, 1, 4, 1>, <8039, 1, 4, 1> @} Total time: 2.209 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:48:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:48:53 on modular [Seed = 971952069] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 {@ <0, 1, 5, 1> @} Total time: 1.600 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:48:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:48:43 on modular [Seed = 3239458624] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 {@ <866, 1, 4, 1>, <1342, 1, 4, 1>, <920, 1, 4, 1>, <648, 1, 4, 1> @} Total time: 1.870 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:45:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:45:21 on modular [Seed = 3474470916] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 Total time: 1.350 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:42:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:42:17 on modular [Seed = 3573686998] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1> @} Total time: 2.069 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:42:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:42:07 on modular [Seed = 3490129861] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1>, <219276, 1, 4, 1>, <12216, 1, 4, 1>, <184709, 1, 4, 1>, <77609, 1, 4, 1>, <50827, 1, 4, 1>, <88495, 1, 4, 1>, <199560, 1, 4, 1>, <1347, 1, 4, 1>, <216043, 1, 4, 1>, <181837, 1, 4, 1> @} Total time: 3.319 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:41:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:41:54 on modular [Seed = 3741853895] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1>, <104373, 1, 4, 1> @} Total time: 2.009 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:41:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:41:43 on modular [Seed = 3845264072] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1>, <0, 1, 4, 1>, <31710, 1, 4, 1>, <70973, 1, 4, 1>, <40187, 1, 4, 1>, <15766, 1, 4, 1>, <76595, 1, 4, 1> @} Total time: 2.819 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:41:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:41:34 on modular [Seed = 3761706964] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1>, <9054, 1, 4, 1> @} Total time: 1.940 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:40:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); Order(Q); Output: Magma V2.11-10 Mon Dec 5 2005 16:40:57 on modular [Seed = 4079223584] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <0, 1, 0, 1> @} 2 Total time: 1.810 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:39:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:39:56 on modular [Seed = 2520868352] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <0, 1, 0, 1> @} Total time: 1.780 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:39:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:39:24 on modular [Seed = 2638900523] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1>, <43, 1, 4, 1>, <63414557, 1, 10, 1>, <6923584, 1, 10, 1> @} Total time: 2.129 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:39:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:38:59 on modular [Seed = 2893765861] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x - 2, 0, 1), (x - 6, 88, 1), (x - 6, -88, 1), (x^2 - 8*x + 12, 22*x - 44, 2), (x^2 - 8*x + 12, -22*x + 44, 2) @} (x^2 - 8*x + 12, -22*x + 44, 2) Total time: 1.500 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:38:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:38:45 on modular [Seed = 2877053940] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x - 2, 0, 1), (x - 6, 88, 1), (x - 6, -88, 1), (x^2 - 8*x + 12, 22*x - 44, 2), (x^2 - 8*x + 12, -22*x + 44, 2) @} Total time: 1.500 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:36:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:36:22 on modular [Seed = 3009692255] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x - 2, 0, 1), (x - 6, 88, 1), (x - 6, -88, 1), (x^2 - 8*x + 12, 22*x - 44, 2), (x^2 - 8*x + 12, -22*x + 44, 2) @} Total time: 1.490 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:35:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:35:41 on modular [Seed = 3194572585] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 Total time: 1.490 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:32:44 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:32:43 on modular [Seed = 1167243140] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 {@ @} Total time: 1.770 seconds, Total memory usage: 39.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:32:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:32:29 on modular [Seed = 1083683479] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 {@ @} Total time: 1.740 seconds, Total memory usage: 39.19MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:32:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:32:20 on modular [Seed = 1335410065] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 {@ <12692, 1, 4, 1>, <13291, 1, 4, 1> @} Total time: 1.840 seconds, Total memory usage: 39.21MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:32:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:32:12 on modular [Seed = 1251850412] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 {@ <13950, 1, 4, 1>, <4406, 1, 4, 1>, <11263, 1, 4, 1>, <3142, 1, 4, 1>, <12460, 1, 4, 1> @} Total time: 2.149 seconds, Total memory usage: 39.17MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:32:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:32:03 on modular [Seed = 1367778232] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 {@ <157, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 1.810 seconds, Total memory usage: 39.18MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:27:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:27:29 on modular [Seed = 1602792612] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 Total time: 1.340 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:24:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1804); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:24:56 on modular [Seed = 1706202743] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ @} Total time: 2.029 seconds, Total memory usage: 39.09MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:23:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:23:41 on modular [Seed = 1622641799] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ @} Total time: 1.500 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:23:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:23:34 on modular [Seed = 1874363267] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ @} Total time: 1.500 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:23:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:23:20 on modular [Seed = 1790805656] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ @} Total time: 1.500 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:22:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:22:25 on modular [Seed = 1940168664] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ @} Total time: 1.500 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:22:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:22:16 on modular [Seed = 2125042380] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ <47711, 1, 4, 1>, <33808, 1, 4, 1>, <21178, 1, 4, 1>, <46068, 1, 4, 1>, <0, 1, 4, 1>, <32331, 1, 4, 1>, <14664, 1, 4, 1>, <48079, 1, 4, 1> @} Total time: 2.500 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:22:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:22:07 on modular [Seed = 97708352] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ <132, 1, 4, 1>, <23152, 1, 4, 1>, <21272, 1, 4, 1>, <307, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.020 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:22:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:21:58 on modular [Seed = 14150732] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ <5956, 1, 4, 1>, <13803, 1, 4, 1>, <10084, 1, 4, 1>, <728, 1, 4, 1>, <5209, 1, 4, 1>, <8231, 1, 4, 1>, <8488, 1, 4, 1>, <10891, 1, 4, 1>, <13875, 1, 4, 1>, <10460, 1, 4, 1> @} Total time: 2.450 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:21:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:21:49 on modular [Seed = 265872216] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ <1090, 1, 4, 1> @} Total time: 1.409 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:20:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:20:16 on modular [Seed = 381811260] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 10, 316, 1) ] 3.42486350460119354543892077010836727047869921855726085241610 5.586784795732860417404869965061177273340 4.468454167317242596001666175548799401169 {@ <10, 1, 4, 1>, <381, 1, 4, 1> @} Total time: 1.459 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:20:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:20:00 on modular [Seed = 499839271] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 10, 316, 1) ] 3.42486350460119354543892077010836727047869921855726085241610 5.586784795732860417404869965061177273340 4.468454167317242596001666175548799401169 {@ <10, 1, 5, 1> @} Total time: 1.409 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:17:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:17:28 on modular [Seed = 636668023] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 10, 316, 1) ] 3.42486350460119354543892077010836727047869921855726085241610 5.586784795732860417404869965061177273340 4.468454167317242596001666175548799401169 Total time: 1.120 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:15:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:15:34 on modular [Seed = 754696928] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x + 28, 38*x - 104, 2) ] 3.5469721771777763679248622062281245897778981929410244076538 4.547022304274841971998782326210084579449 3.736045974871836135071502684267115598090 {@ @} Total time: 1.449 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:15:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:15:23 on modular [Seed = 737986042] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x + 28, 38*x - 104, 2) ] 3.5469721771777763679248622062281245897778981929410244076538 4.547022304274841971998782326210084579449 3.736045974871836135071502684267115598090 {@ <848, 1, 4, 1>, <677, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 1.570 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:12:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:12:27 on modular [Seed = 920770615] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x + 28, 38*x - 104, 2) ] 3.5469721771777763679248622062281245897778981929410244076538 4.547022304274841971998782326210084579449 3.736045974871836135071502684267115598090 Total time: 1.100 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:09:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:09:42 on modular [Seed = 1022088080] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1> @} Total time: 1.740 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:09:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,23); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:09:32 on modular [Seed = 1005376653] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <149367, 1, 4, 1>, <249855, 1, 4, 1>, <165386, 1, 4, 1>, <259234, 1, 4, 1>, <128789, 1, 4, 1>, <137229, 1, 4, 1>, <155916, 1, 4, 1>, <63763, 1, 4, 1>, <67400, 1, 4, 1>, <127439, 1, 4, 1> @} Total time: 3.029 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:09:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:09:21 on modular [Seed = 3339724925] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <62911, 1, 4, 1> @} Total time: 1.710 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:09:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:09:10 on modular [Seed = 3424330625] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <8607, 1, 4, 1>, <0, 1, 4, 1>, <6846, 1, 4, 1>, <10605, 1, 4, 1>, <13759, 1, 4, 1>, <61626, 1, 4, 1> @} Total time: 2.529 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:09:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:09:01 on modular [Seed = 3374196383] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <20355, 1, 4, 1>, <0, 1, 4, 1>, <7240, 1, 4, 1>, <24204, 1, 4, 1> @} Total time: 2.060 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:08:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:08:51 on modular [Seed = 3658298754] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <10898, 1, 4, 1>, <5044, 1, 4, 1>, <3222, 1, 4, 1> @} Total time: 1.929 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:08:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:08:44 on modular [Seed = 3878684800] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <1811, 1, 4, 1>, <153, 1, 4, 1>, <930, 1, 4, 1> @} Total time: 1.669 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:07:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:07:42 on modular [Seed = 3996713419] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <62911, 1, 4, 1> @} Total time: 1.700 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:07:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:07:33 on modular [Seed = 3946579141] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <8607, 1, 4, 1>, <0, 1, 4, 1>, <6846, 1, 4, 1>, <10605, 1, 4, 1>, <13759, 1, 4, 1>, <61626, 1, 4, 1> @} Total time: 2.500 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:07:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:07:22 on modular [Seed = 4029095873] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <20355, 1, 4, 1>, <0, 1, 4, 1>, <7240, 1, 4, 1>, <24204, 1, 4, 1> @} Total time: 2.089 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:07:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:07:15 on modular [Seed = 4247392965] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <10898, 1, 4, 1>, <5044, 1, 4, 1>, <3222, 1, 4, 1> @} Total time: 1.940 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:07:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:07:05 on modular [Seed = 4197257784] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <1811, 1, 4, 1>, <153, 1, 4, 1>, <930, 1, 4, 1> @} Total time: 1.659 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:04:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,31); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:04:01 on modular [Seed = 2203347961] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 Total time: 1.169 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:03:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,31); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:03:35 on modular [Seed = 2287952020] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] Total time: 1.030 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:51:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,31); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:51:22 on modular [Seed = 653380951] ------------------------------------- [ (x - 9, -81, 1) ] {@ <9, 1, 4, 1>, <178093, 1, 4, 1>, <402744, 1, 4, 1>, <210384, 1, 4, 1>, <381074, 1, 4, 1>, <456991, 1, 4, 1>, <597056, 1, 4, 1>, <517834, 1, 4, 1>, <218317, 1, 4, 1>, <486504, 1, 4, 1>, <48365, 1, 4, 1>, <210439, 1, 4, 1>, <310588, 1, 4, 1>, <784202, 1, 4, 1>, <530206, 1, 4, 1> @} Total time: 11.279 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:51:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:51:05 on modular [Seed = 603244577] ------------------------------------- [ (x - 9, -81, 1) ] {@ <9, 1, 4, 1>, <683665, 1, 4, 1> @} Total time: 9.890 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:50:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,23); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:50:43 on modular [Seed = 687850288] ------------------------------------- [ (x - 9, -81, 1) ] {@ <9, 1, 4, 1>, <143999, 1, 4, 1>, <42845, 1, 4, 1>, <106066, 1, 4, 1>, <19302, 1, 4, 1>, <53815, 1, 4, 1>, <5678, 1, 4, 1>, <30257, 1, 4, 1>, <15766, 1, 4, 1>, <31409, 1, 4, 1>, <110347, 1, 4, 1>, <263785, 1, 4, 1> @} Total time: 11.130 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:50:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:50:23 on modular [Seed = 904060417] ------------------------------------- [ (x - 9, -81, 1) ] {@ <9, 1, 4, 1> @} Total time: 9.689 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:50:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:49:57 on modular [Seed = 853923580] ------------------------------------- [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1> @} Total time: 11.599 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:49:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:49:39 on modular [Seed = 1072225255] ------------------------------------- [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1>, <9591, 1, 4, 1>, <21818, 1, 4, 1>, <21730, 1, 4, 1>, <23727, 1, 4, 1>, <14172, 1, 4, 1>, <3128, 1, 4, 1> @} Total time: 12.009 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:49:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:49:12 on modular [Seed = 3407621943] ------------------------------------- [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1>, <9591, 1, 4, 1>, <21818, 1, 4, 1>, <21730, 1, 4, 1>, <23727, 1, 4, 1>, <14172, 1, 4, 1>, <3128, 1, 4, 1> @} Total time: 12.160 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:49:04 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:48:48 on modular [Seed = 3691722840] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 52488... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 52488 over Ration..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 52488, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 14, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (467387893081695*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1>, <9591, 1, 4, 1>, <21818, 1, 4, 1>, <21730, 1, 4, 1>, <23727, 1, 4, 1>, <14172, 1, 4, 1>, <3128, 1, 4, 1> @} Total time: 12.449 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:48:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:48:22 on modular [Seed = 3607114613] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1>, <8071, 1, 4, 1>, <10913, 1, 4, 1>, <2668, 1, 4, 1>, <0, 1, 4, 1>, <409, 1, 4, 1>, <1657, 1, 4, 1>, <13764, 1, 4, 1>, <4218, 1, 4, 1>, <1539, 1, 4, 1> @} Total time: 13.169 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:48:04 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:47:52 on modular [Seed = 3913158053] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1>, <1615, 1, 4, 1>, <2148, 1, 4, 1> @} Total time: 12.080 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:41:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:41:44 on modular [Seed = 2270192315] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 Total time: 11.609 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:41:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:41:00 on modular [Seed = 2588769394] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 Total time: 11.609 seconds, Total memory usage: 39.42MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:40:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:40:10 on modular [Seed = 2638904067] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 {@ @} Total time: 1.960 seconds, Total memory usage: 39.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:40:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:40:00 on modular [Seed = 2420602372] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 {@ <11307, 1, 4, 1> @} Total time: 2.100 seconds, Total memory usage: 39.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:39:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:39:52 on modular [Seed = 2877050119] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 {@ <16183, 1, 4, 1>, <1782, 1, 4, 1>, <6337, 1, 4, 1>, <2738, 1, 4, 1>, <15695, 1, 4, 1>, <18082, 1, 4, 1> @} Total time: 2.430 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:39:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:39:42 on modular [Seed = 2927183892] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 {@ <0, 1, 4, 1> @} Total time: 1.909 seconds, Total memory usage: 39.42MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:39:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:39:34 on modular [Seed = 2708883219] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 {@ <2150, 1, 4, 1>, <535, 1, 4, 1>, <125, 1, 4, 1> @} Total time: 1.960 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:38:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1275); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:38:40 on modular [Seed = 2959562452] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 Total time: 1.649 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:37:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:37:33 on modular [Seed = 1235136368] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 Total time: 1.500 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:37:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:37:24 on modular [Seed = 1285270140] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 Total time: 1.560 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:37:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:37:14 on modular [Seed = 1200660617] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 Total time: 1.490 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:35:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:35:13 on modular [Seed = 1502527212] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 8.803102712951222835961923441572148585428 5.471172431834758644917171743806715961974 Total time: 1.470 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:24:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:24:28 on modular [Seed = 1384493210] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 43/9*x + 73/9, -251/27*x + 554/27, 2) ] 5.16057625047072430311345959953099031630616226408707241792455 5.713556585783996031183901371149946289857 3.273947854498539262126681269961664552701 {@ @} Total time: 1.429 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:24:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:24:20 on modular [Seed = 1434630037] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 43/9*x + 73/9, -251/27*x + 554/27, 2) ] 5.16057625047072430311345959953099031630616226408707241792455 5.713556585783996031183901371149946289857 3.273947854498539262126681269961664552701 {@ <0, 1, 0, 1>, <1935, 1, 4, 1>, <0, 1, 4, 1>, <268, 1, 4, 1> @} Total time: 1.679 seconds, Total memory usage: 39.49MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:23:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1111); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:23:29 on modular [Seed = 1757383325] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 43/9*x + 73/9, -251/27*x + 554/27, 2) ] 5.16057625047072430311345959953099031630616226408707241792455 5.713556585783996031183901371149946289857 3.273947854498539262126681269961664552701 Total time: 1.229 seconds, Total memory usage: 39.49MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:20:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:20:53 on modular [Seed = 148892185] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 43/9*x + 73/9, -251/27*x + 554/27, 2) ] 5.16057625047072430311345959953099031630616226408707241792455 5.713556585783996031183901371149946289857 3.273947854498539262126681269961664552701 Total time: 1.240 seconds, Total memory usage: 39.56MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:17:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:17:07 on modular [Seed = 14148471] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + x + 9, 5*x + 8, 2) ] 3.47426304826961249980683698951257553773742512428567231010826 4.830829590104535747512612652516993975442 2.541539662053132801196517778679980749598 {@ @} Total time: 1.520 seconds, Total memory usage: 39.56MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:16:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:16:56 on modular [Seed = 416283412] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + x + 9, 5*x + 8, 2) ] 3.47426304826961249980683698951257553773742512428567231010826 4.830829590104535747512612652516993975442 2.541539662053132801196517778679980749598 {@ <626, 1, 4, 1>, <2399, 1, 4, 1>, <95, 1, 4, 1> @} Total time: 1.639 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:13:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:13:18 on modular [Seed = 314953336] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + x + 9, 5*x + 8, 2) ] 3.47426304826961249980683698951257553773742512428567231010826 4.830829590104535747512612652516993975442 2.541539662053132801196517778679980749598 Total time: 1.149 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:12:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:12:09 on modular [Seed = 603258638] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + x + 9, 5*x + 8, 2) ] Total time: 1.060 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:11:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:11:53 on modular [Seed = 653392411] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] Total time: 1.169 seconds, Total memory usage: 39.18MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Mon Dec 5 14:51:38 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); B:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-B)); G:=GroebnerBasis(I); print Factorization(G[10]); Output: Magma V2.11-10 Mon Dec 5 2005 14:51:37 on modular [Seed = 3323011002] ------------------------------------- [ <$.8^2 + (a - 2*b*c + b + c - 1)/(a*b + b^2 - b)*$.8*$.9 + (-2*b*c + c)/(a*b + b^2 - b)*$.8 + (a*c + c^2 - c)/(a*b + b^2 - b)*$.9^2 + (2*c^2 - c)/(a*b + b^2 - b)*$.9 + (-a*c + c^2)/(a*b + b^2 - b), 1>, <$.8^2 + (a - 2*b*c + b + c - 1)/(a*b + b^2 - b)*$.8*$.9 + (2*b*c - c)/(a*b + b^2 - b)*$.8 + (a*c + c^2 - c)/(a*b + b^2 - b)*$.9^2 + (-2*c^2 + c)/(a*b + b^2 - b)*$.9 + (-a*c + c^2)/(a*b + b^2 - b), 1> ] Total time: 0.980 seconds, Total memory usage: 4.27MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Mon Dec 5 14:35:21 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); B:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Mon Dec 5 2005 14:35:19 on modular [Seed = 3896458666] ------------------------------------- [ $.1 + (-a*b + b*c - b - c + 1)/(a*c + b*c - c)*$.7 + (b^2 - b*c - 2*b + 1)/(a*c + b*c - c)*$.8 + (-2*b + 1)/(a + b - 1)*$.9, $.2 + (a*b + a*c - a)/(a*c + b*c - c)*$.7 + (-a - b^2 + b*c + b)/(a*c + b*c - c)*$.8 + (-a + b)/(a + b - 1)*$.9, $.3 + b/c*$.8, $.4 + (a^2 - a*c - a - b + 1)/(a*c + b*c - c)*$.7 + (-a*b + a - b*c)/(a*c + b*c - c)*$.8 + (a - b)/(a + b - 1)*$.9, $.5 + (-a^2 + a*c)/(a*c + b*c - c)*$.7 + (a*b - a*c - 2*a - b + c + 1)/(a*c + b*c - c)*$.8 + (-2*a + 1)/(a + b - 1)*$.9, $.6 + a/c*$.7 + 1/c*$.8, $.7^2 + (b + c - 1)/(a*b - 1/2*a)*$.7*$.9 + (a*b^2 - a*b + 1/2*b^2 - b + 1/2)/(a^2*b - 1/2*a^2)*$.8^2 + (a*b - a - b*c + c)/(a^2*b - 1/2*a^2)*$.8*$.9 + (a*b*c - a*c + 1/2*c^2)/(a^2*b - 1/2*a^2)*$.9^2 + (-a*b*c + a*c - 1/2*c^2)/(a^2*b - 1/2*a^2), $.7*$.8 + (-c + 1/2)/(b - 1/2)*$.7*$.9 + (1/2*a*b - 1/2*b^2 + b - 1/2)/(a*b - 1/2*a)*$.8^2 + (1/2*a + b*c - c)/(a*b - 1/2*a)*$.8*$.9 + (1/2*a*c - 1/2*c^2)/(a*b - 1/2*a)*$.9^2 + (-1/2*a*c + 1/2*c^2)/(a*b - 1/2*a), $.7*$.9^2 + (-a*b^2*c + a*b*c - 1/4*a*c + b^2*c^2 - b*c^2 + 1/4*c^2)/(a*b^2*c + a*b*c^2 - a*b*c - 1/4*a*b - 1/4*a*c + 1/4*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)*$.7 + (-1/2*a^2*b^3 + 1/4*a^2*b^2 - a*b^4 + 3/2*a*b^3 - 1/2*a*b^2 - 1/2*b^5 + 5/4*b^4 - b^3 + 1/4*b^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^3 + (-1/2*a^2*b^2*c - 3/4*a^2*b^2 + 1/2*a^2*b + a*b^3*c - 3/2*a*b^3 - a*b^2*c + 5/2*a*b^2 + 1/2*a*b*c - a*b + 3/2*b^4*c - 3/4*b^4 - 3*b^3*c + 2*b^3 + 2*b^2*c - 7/4*b^2 - 1/2*b*c + 1/2*b)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^2*$.9 + (-1/2*a^2*b^2*c - 1/4*a^2*b*c - 1/4*a^2*b + 1/4*a^2 - 3/2*a*b^3*c - 1/2*a*b^2*c^2 + 15/4*a*b^2*c - 1/4*a*b^2 + 3/4*a*b*c^2 - 2*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a - 3/2*b^3*c^2 + 3/2*b^3*c + 9/4*b^2*c^2 - 5/2*b^2*c - b*c^2 + b*c)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8*$.9^2 + (1/2*a^2*b^2*c - 1/4*a^2*b*c + 3/2*a*b^3*c - 1/2*a*b^2*c^2 - 11/4*a*b^2*c + 1/4*a*b*c^2 + 3/2*a*b*c - 1/4*a*c + 1/2*b^3*c^2 - 1/4*b^2*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8 + (-1/2*a^2*b*c^2 - 1/4*a^2*b*c + 1/4*a^2*c + 3/2*a*b^2*c^2 - 1/4*a*b^2*c + 1/2*a*b*c^3 - 5/4*a*b*c^2 + 1/2*a*b*c + 1/4*a*c^2 - 1/4*a*c + 1/2*b^2*c^3 - 3/4*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9^3 + (1/2*a^2*b*c^2 + 1/4*a^2*b*c - 1/4*a^2*c - 3/2*a*b^2*c^2 + 1/4*a*b^2*c - 1/2*a*b*c^3 + 5/4*a*b*c^2 - 1/2*a*b*c - 1/4*a*c^2 + 1/4*a*c - 1/2*b^2*c^3 + 3/4*b^2*c^2 + 1/2*b*c^3 - 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9, $.8^4 + (2*a - 4*b*c + 2*b + 2*c - 2)/(a*b + b^2 - b)*$.8^3*$.9 + (2*a^2*b*c + a^2 + 2*a*b^2*c + 2*a*b*c^2 - 8*a*b*c + 2*a*b + 2*a*c - 2*a + 6*b^2*c^2 - 6*b^2*c + b^2 - 6*b*c^2 + 8*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2*$.9^2 + (-2*a^2*b*c - 2*a*b^2*c + 2*a*b*c^2 + 2*a*b*c - 2*b^2*c^2 + 2*b*c^2 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2 + (2*a^2*c - 4*a*b*c^2 + 2*a*b*c + 4*a*c^2 - 4*a*c - 4*b*c^3 + 6*b*c^2 - 2*b*c + 2*c^3 - 4*c^2 + 2*c)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9^3 + (-2*a^2*c + 4*a*b*c^2 - 2*a*b*c + 2*a*c + 4*b*c^3 - 2*b*c^2 - 2*c^3)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9 + (a^2*c^2 + 2*a*c^3 - 2*a*c^2 + c^4 - 2*c^3 + c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^4 + (-2*a^2*c^2 + 2*a*c^2 - 2*c^4 + 2*c^3 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^2 + (a^2*c^2 - 2*a*c^3 + c^4)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2) ] Total time: 0.980 seconds, Total memory usage: 4.30MB '80.240.' ************** MAGMA ***************** Host 80.240.228.94 (80.240.228.94) Time: Mon Dec 5 11:14:42 2005 Input: 5! Output: Magma V2.11-10 Mon Dec 5 2005 11:14:41 on modular [Seed = 2588786397] ------------------------------------- >> 5!; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 06:18:27 2005 Input: G :=DirichletGroup(1680); G; X :=Elements(G); X; Y :=X[30]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Mon Dec 5 2005 06:18:21 on modular [Seed = 4230716976] ------------------------------------- Group of Dirichlet characters of modulus 1680 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4, $.5, $.1*$.5, $.2*$.5, $.1*$.2*$.5, $.3*$.5, $.1*$.3*$.5, $.2*$.3*$.5, $.1*$.2*$.3*$.5, $.4*$.5, $.1*$.4*$.5, $.2*$.4*$.5, $.1*$.2*$.4*$.5, $.3*$.4*$.5, $.1*$.3*$.4*$.5, $.2*$.3*$.4*$.5, $.1*$.2*$.3*$.4*$.5 ] 420 2 Current total memory usage: 91.9MB, failed memory request: 81.0MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned Total time: 5.740 seconds, Total memory usage: 91.93MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 05:44:54 2005 Input: G :=DirichletGroup(1680); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Mon Dec 5 2005 05:44:48 on modular [Seed = 1602750030] ------------------------------------- Group of Dirichlet characters of modulus 1680 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4, $.5, $.1*$.5, $.2*$.5, $.1*$.2*$.5, $.3*$.5, $.1*$.3*$.5, $.2*$.3*$.5, $.1*$.2*$.3*$.5, $.4*$.5, $.1*$.4*$.5, $.2*$.4*$.5, $.1*$.2*$.4*$.5, $.3*$.4*$.5, $.1*$.3*$.4*$.5, $.2*$.3*$.4*$.5, $.1*$.2*$.3*$.4*$.5 ] 4 2 Current total memory usage: 92.0MB, failed memory request: 80.9MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned Total time: 5.679 seconds, Total memory usage: 92.02MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:29:30 2005 Input: G :=DirichletGroup(840); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 04:29:09 on modular [Seed = 670067827] ------------------------------------- Group of Dirichlet characters of modulus 840 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4, $.5, $.1*$.5, $.2*$.5, $.1*$.2*$.5, $.3*$.5, $.1*$.3*$.5, $.2*$.3*$.5, $.1*$.2*$.3*$.5, $.4*$.5, $.1*$.4*$.5, $.2*$.4*$.5, $.1*$.2*$.4*$.5, $.3*$.4*$.5, $.1*$.3*$.4*$.5, $.2*$.3*$.4*$.5, $.1*$.2*$.3*$.4*$.5 ] 4 2 Errors: /bin/sh: line 1: 29206 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:26:49 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:26:46 on modular [Seed = 553084103] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 210, weight 3, character $.1, and dimension 16 over Rational Field ] Total time: 2.799 seconds, Total memory usage: 9.98MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:26:30 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:26:30 on modular [Seed = 988643021] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 105 2 [] Total time: 0.220 seconds, Total memory usage: 4.59MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:26:18 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:26:16 on modular [Seed = 937456574] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 35 2 [ Modular symbols space of level 210, weight 3, character $.2*$.3, and dimension 16 over Rational Field ] Total time: 2.299 seconds, Total memory usage: 9.98MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:26:05 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:26:05 on modular [Seed = 820475052] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 21 2 [] Total time: 0.220 seconds, Total memory usage: 4.60MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:21:55 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],20);Parent($1); Output: Magma V2.11-10 Mon Dec 5 2005 04:21:52 on modular [Seed = 3357510986] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 210, weight 3, character $.3, and dimension 8 over Rational Field ] q + (-415353/140631020042240*a^7 - 301369/10817770772480*a^6 - 100645827/70315510021120*a^5 - 1687471593/70315510021120*a^4 - 6165419051/7031551002112*a^3 - 7544009613/1004507286016*a^2 - 2395504268061/17578877505280*a + 952079037763/2511268215040)*q^2 + (61/270466846720*a^7 + 1051/19319060480*a^6 + 102729/135233423360*a^5 + 1445303/67616711680*a^4 + 8801395/13523342336*a^3 + 101272359/6761671168*a^2 + 6674090427/33808355840*a + 6266199807/2414882560)*q^3 + 2*q^4 + (105129/67123253550080*a^7 + 38499/368809085440*a^6 + 65880289/33561626775040*a^5 + 985815917/16780813387520*a^4 + 19244054491/16780813387520*a^3 + 248612660747/8390406693760*a^2 + 339284794093/1198629527680*a + 2679931711057/599314763840)*q^5 + (337287/154526132773376*a^7 + 357451/11886625597952*a^6 + 102999019/77263066386688*a^5 + 2547290899/77263066386688*a^4 + 26135542317/38631533193344*a^3 + 82937289347/5518790456192*a^2 + 718594671095/2759395228096*a + 5318504874287/2759395228096)*q^6 + (73866797867/35134297756416954368*a^7 + 42074836351/241306990085281280*a^6 + 340062766719647/87835744391042385920*a^5 + 386359492218157/4391787219552119296*a^4 + 65227685421317269/43917872195521192960*a^3 + 9493755569266281/171554188263754660*a^2 + 1783706007473175811/3136990871108656640*a + 5859779321633194923/784247717777164160)*q^7 + (-415353/70315510021120*a^7 - 301369/5408885386240*a^6 - 100645827/35157755010560*a^5 - 1687471593/35157755010560*a^4 - 6165419051/3515775501056*a^3 - 7544009613/502253643008*a^2 - 2395504268061/8789438752640*a + 952079037763/1255634107520)*q^8 - 3*q^9 + (7209/3315131348480*a^7 + 656157/13260525393920*a^6 + 7380803/3315131348480*a^5 + 36675163/947180385280*a^4 + 454198953/414391418560*a^3 + 40618382071/3315131348480*a^2 + 302621545967/828782837120*a + 300249322681/236795096320)*q^10 + (-2473932893/320568410186286080*a^7 - 68024963/3522729782266880*a^6 - 796914927311/160284205093143040*a^5 - 9589847773573/160284205093143040*a^4 - 233568053516403/80142102546571520*a^3 - 2084080830459627/80142102546571520*a^2 - 3440859888022621/8014210254657152*a - 15675074276252681/5724435896183680)*q\ ^11 + (61/135233423360*a^7 + 1051/9659530240*a^6 + 102729/67616711680*a^5 + 1445303/33808355840*a^4 + 8801395/6761671168*a^3 + 101272359/3380835584*a^2 + 6674090427/16904177920*a + 6266199807/1207441280)*q^12 + (2473932893/641136820372572160*a^7 + 68024963/7045459564533760*a^6 + 796914927311/320568410186286080*a^5 + 9589847773573/320568410186286080*a^4 + 233568053516403/160284205093143040*a^3 + 2084080830459627/160284205093143040*a^2 + 11455070142679773/16028420509314304*a + 15675074276252681/11448871792367360)*q^13 + (2703342821/182610695199672320*a^7 + 1159611501/3260905271422720*a^6 + 875460355197/91305347599836160*a^5 + 4664601406499/22826336899959040*a^4 + 266625016194911/45652673799918080*a^3 + 104881037503867/1426646056247440*a^2 + 987000415519317/652181054284544*a + 7102726963661553/815226317855680)*q^1\ 4 + (121/44910161920*a^7 - 53/6045598720*a^6 + 231887/157185566720*a^5 - 284741/19648195840*a^4 + 3017419/11227540480*a^3 - 77204177/19648195840*a^2 + 574080053/39296391680*a - 1458574873/350860640)*q^15 + 4*q^16 + (336932919/66744486619333120*a^7 + 37905946599/71878677897743360*a^6 + 61112850011/8343060827416640*a^5 + 128501335337131/467211406335331840*a^4 + 547642963535037/116802851583832960*a^3 + 35806771762695601/233605703167665920*a^2 + 28268746826087251/29200712895958240*a + 405372914402092591/16686121654833280)*q^17 + (1246059/140631020042240*a^7 + 904107/10817770772480*a^6 + 301937481/70315510021120*a^5 + 5062414779/70315510021120*a^4 + 18496257153/7031551002112*a^3 + 22632028839/1004507286016*a^2 + 7186512804183/17578877505280*a - 2856237113289/2511268215040)*q^18 + (113207027013/17567148878208477184*a^7 - 89451024127/211143616324621120*a^6 - 22617155986209/6273981742217313280*a^5 - 202900838349135/2195893609776059648*a^4 - 80258710845285461/21958936097760596480*a^3 - 187840457305308673/2744867012220074560*a^2 - 7223781603714594173/10979468048880298240*a - 5315748266184103957/392123858888582080)*q^19 + O(q^20) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^8 + 16*a^7 + 760*a^6 + 13600*a^5 + 432984*a^4 + 6066368*a^3 + 133021408*a^2 + 750049664*a + 14968881424 Total time: 2.970 seconds, Total memory usage: 9.99MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:21:07 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:21:04 on modular [Seed = 3323036532] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 210, weight 3, character $.3, and dimension 8 over Rational Field ] Total time: 2.950 seconds, Total memory usage: 9.99MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:20:54 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:20:51 on modular [Seed = 3708459164] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 15 2 [ Modular symbols space of level 210, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] Total time: 2.299 seconds, Total memory usage: 9.99MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:20:43 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:20:43 on modular [Seed = 3624902017] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 5 2 [] Total time: 0.220 seconds, Total memory usage: 4.63MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:20:31 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:20:28 on modular [Seed = 3607139001] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 210, weight 3, character $.1, and dimension 16 over Rational Field ] Total time: 2.810 seconds, Total memory usage: 9.98MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:19:55 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:19:54 on modular [Seed = 3523581914] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 105, weight 3, character $.1, and dimension 16 over Rational Field ] Total time: 0.740 seconds, Total memory usage: 4.81MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:19:45 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:19:45 on modular [Seed = 3913198826] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 5 2 [] Total time: 0.210 seconds, Total memory usage: 4.50MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:19:32 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:19:32 on modular [Seed = 3828589042] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 15 2 [ Modular symbols space of level 105, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] Total time: 0.600 seconds, Total memory usage: 4.81MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:19:22 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:19:21 on modular [Seed = 3811878600] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 105, weight 3, character $.3, and dimension 12 over Rational Field ] Total time: 0.740 seconds, Total memory usage: 4.62MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:19:01 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:19:01 on modular [Seed = 4264147913] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 21 2 [] Total time: 0.220 seconds, Total memory usage: 4.38MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:18:45 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:18:44 on modular [Seed = 4112692019] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 35 2 [ Modular symbols space of level 105, weight 3, character $.2*$.3, and dimension 16 over Rational Field ] Total time: 0.590 seconds, Total memory usage: 4.62MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:18:34 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:18:33 on modular [Seed = 4029134379] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 105 2 [] Total time: 0.210 seconds, Total memory usage: 3.91MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:18:20 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:18:20 on modular [Seed = 2404960609] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] >> Y :=X[16]; Conductor(Y); Order(Y); ^ Runtime error in '[]': Sequence element 16 not defined >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 3, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:54:20 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 03:54:00 on modular [Seed = 1367741003] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 420 2 Errors: /bin/sh: line 1: 29027 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:53:50 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:53:49 on modular [Seed = 1824206659] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 1 1 [] Total time: 0.230 seconds, Total memory usage: 4.82MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:53:32 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:53:21 on modular [Seed = 1807493208] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 4 2 [ Modular symbols space of level 420, weight 3, character $.1, and dimension 48 over Rational Field ] Total time: 11.220 seconds, Total memory usage: 25.76MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:53:00 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:52:51 on modular [Seed = 1622203569] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 3 2 [ Modular symbols space of level 420, weight 3, character $.2, and dimension 16 over Rational Field ] Total time: 8.779 seconds, Total memory usage: 25.76MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:52:39 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:52:39 on modular [Seed = 2142225306] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 12 2 [] Total time: 0.220 seconds, Total memory usage: 4.47MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:52:14 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:52:13 on modular [Seed = 2058012405] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 5 2 [] Total time: 0.220 seconds, Total memory usage: 4.77MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:51:57 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:51:47 on modular [Seed = 1906429408] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 20 2 [ Modular symbols space of level 420, weight 3, character $.1*$.3, and dimension 72 over Rational Field ] Total time: 9.310 seconds, Total memory usage: 25.76MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:51:35 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:51:27 on modular [Seed = 266318770] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 15 2 [ Modular symbols space of level 420, weight 3, character $.2*$.3, and dimension 24 over Rational Field ] Total time: 7.230 seconds, Total memory usage: 25.77MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:51:15 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:51:15 on modular [Seed = 182105787] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 60 2 [] Total time: 0.210 seconds, Total memory usage: 4.43MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:50:35 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[9]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:50:26 on modular [Seed = 30522721] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 7 2 [ Modular symbols space of level 420, weight 3, character $.4, and dimension 12 over Rational Field ] Total time: 9.199 seconds, Total memory usage: 25.75MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:50:12 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[10]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:50:12 on modular [Seed = 483172458] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 28 2 [] Total time: 0.220 seconds, Total memory usage: 4.59MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:49:59 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[11]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:49:59 on modular [Seed = 466331422] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 21 2 [] Total time: 0.220 seconds, Total memory usage: 4.67MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:49:32 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[12]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:49:21 on modular [Seed = 382118453] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 84 2 [ Modular symbols space of level 420, weight 3, character $.1*$.2*$.4, and dimension 128 over Rational Field ] Total time: 11.460 seconds, Total memory usage: 33.97MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:48:19 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[13]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:48:12 on modular [Seed = 754749906] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 35 2 [ Modular symbols space of level 420, weight 3, character $.3*$.4, and dimension 16 over Rational Field ] Total time: 7.750 seconds, Total memory usage: 25.76MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:48:13 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[13]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:48:05 on modular [Seed = 704222430] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 35 2 [ Modular symbols space of level 420, weight 3, character $.3*$.4, and dimension 16 over Rational Field ] Total time: 7.710 seconds, Total memory usage: 25.76MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:47:55 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[14]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:47:54 on modular [Seed = 620009900] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 140 2 [] Total time: 0.220 seconds, Total memory usage: 4.54MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:47:37 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[15]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:47:37 on modular [Seed = 603168440] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 105 2 [] Total time: 0.220 seconds, Total memory usage: 4.62MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:45:45 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 03:45:24 on modular [Seed = 1005291053] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 420 2 Errors: /bin/sh: line 1: 28937 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:45:36 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 03:45:16 on modular [Seed = 887392538] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 420 2 Errors: /bin/sh: line 1: 28934 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:45:29 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 03:45:08 on modular [Seed = 836865047] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 420 2 Errors: /bin/sh: line 1: 28931 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 02:37:53 2005 Input: G :=DirichletGroup(168); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[5],20);Parent($1); Output: Magma V2.11-10 Mon Dec 5 2005 02:37:52 on modular [Seed = 281066862] ------------------------------------- Group of Dirichlet characters of modulus 168 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 168 2 [ Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 8 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 48 over Rational Field ] q + (-9799/5016464987212800*a^7 - 661127/4459079988633600*a^6 + 5969/573638077440*a^5 + 16758991/65574705715200*a^4 - 603775387/13934624964480*a^3 + 14610182343/30965833254400*a^2 + 65387258923/11612187470400*a - 19476177313999/7741458313600)*q^2 + (-266142103/19906741199783116800*a^7 + 149817191/311042831246611200*a^6 + 173029313/2845446140620800*a^5 - 22099254151/5184047187443520*a^4 - 9483376629317/138241258331827200*a^3 + 334575867229/43200393228696*a^2 - 246875651449831/2304020972197120*a - 235675861161782/45000409613225)*q^3 + (1/58023161856*a^7 - 43/348138971136*a^6 - 41/439569408*a^5 + 25841/9670526976*a^4 + 70519/402938624*a^3 - 4371163/805877248*a^2 + 2739201/100734656*a + 1629936657/201469312)*q^4 + (-1399/21560302961664*a^7 + 6661/2695037870208*a^6 + 85841/305098626816*a^5 - 215599/13210969952*a^4 - 535770697/1347518935104*a^3 + 1685781297/56146622296*a^2 - 17695656925/112293244592*a - 307734345945/14036655574)*q^5 + (380413753/15129123311835168768*a^7 - 121715656291/94557020698969804800*a^6 - 1531167499/18021158890598400*a^5 + 6353210557501/875527969434905600*a^4 + 4581423310679/154504935782630400*a^3 - 7802577696362677/656645977076179200*a^2 + 55036834534786031/218881992358726400*a + 376624072046688339/54720498089681600)*q^6 + (4273/569634332748800*a^7 - 2640299/13671223985971200*a^6 - 8790163/233032227033600*a^5 + 1161831817/539653578393600*a^4 + 120073871/8900536449200*a^3 - 205016482211/44971131532800*a^2 + 4504955976967/17801072898400*a + 30549650870213/14240858318720)*q^7 + (-32959/627058123401600*a^7 + 5657203/6688619982950400*a^6 + 329969/1434095193600*a^5 - 6274717/728607841280*a^4 - 15539903573/34836562411200*a^3 + 137223833609/9289749976320*a^2 - 28854144049/232243749408*a - 70644368356359/3870729156800)*q^8 + (-2141/29336052556800*a^7 + 50267/29336052556800*a^6 + 3949/12579782400*a^5 - 1967903/143804179200*a^4 - 109833567/203722587200*a^3 + 4225583033/203722587200*a^2 - 10351853423/50930646800*a - 232538089563/10186129360)*q^9 + (984493/15380126984217600*a^7 - 22965527/10253417989478400*a^6 - 33425539/93212890813440*a^5 + 57365051147/2563354497369600*a^4 + 24398075569/42722574956160*a^3 - 3408111602657/71204291593600*a^2 + 15923119891739/35602145796800*a + 599172767199101/17801072898400)*q^10 + (2141/44004078835200*a^7 - 50267/44004078835200*a^6 - 3949/18869673600*a^5 + 1967903/215706268800*a^4 + 36611189/101861293600*a^3 - 4225583033/305583880800*a^2 + 10351853423/76395970200*a + 67326567161/5093064680)*q^11 + (86939533/42025342532875468800*a^7 + 6590802397/23639255174742451200*a^6 - 746191843/54063476671795200*a^5 - 8095928922437/1969937931228537600*a^4 + 405858460990847/2626583908304716800*a^3 + 1207679029054651/164161494269044800*a^2 - 102554420323546167/218881992358726400*a - 24688973293588019/13680124522420400)*q^12 + (11741/603142234675200*a^7 + 62981/452356676006400*a^6 - 1522997/13707778060800*a^5 + 39145307/37696389667200*a^4 + 1353058567/4188487740800*a^3 - 5436188881/3141365805600*a^2 - 473047637677/1047121935200*a + 308381226277/52356096760)*q^13 + (-1547688293/45387369935505506304*a^7 + 237574576913/189114041397939609600*a^6 + 11306818819/162190430015385600*a^5 - 132731396923/16263677450803200*a^4 + 1064613417972889/7879751724914150400*a^3 + 264583332386069/69120629165913600*a^2 - 388090243377087737/656645977076179200*a + 22152894165870723/109440996179363200)*q^14 + (-180143/3344309991475200*a^7 - 6134779/2229539994316800*a^6 + 59051/159343910400*a^5 + 26703019/37158999905280*a^4 - 23941714277/23224374940800*a^3 + 10816103935/619316665088*a^2 + 41676710529/387072915680*a - 158322600633729/3870729156800)*q^15 + (1/9670526976*a^7 - 43/58023161856*a^6 - 41/73261568*a^5 + 25841/1611754496*a^4 + 211557/201469312*a^3 - 13113489/402938624*a^2 + 8217603/50367328*a + 3278055475/100734656)*q^16 + (44257/1951641294096384*a^7 - 37219/27106129084672*a^6 - 11017/836895923712*a^5 + 53766363/6776532271168*a^4 - 11728498927/40659193627008*a^3 - 10618010781/1694133067792*a^2 + 2047200601843/3388266135584*a + 27085206645/423533266948)*q^17 + (365423/3344309991475200*a^7 - 1321169/4459079988633600*a^6 - 112979/191212692480*a^5 + 5672739703/371589999052800*a^4 + 2740611011/2322437494080*a^3 - 1051602176679/30965833254400*a^2 + 566135531627/3870729156800*a + 248415428686947/7741458313600)*q^18 + (-7/332230809600*a^7 + 103/166115404800*a^6 + 91/769052800*a^5 - 18113/2768590080*a^4 - 482923/2307158400*a^3 + 642433/46143168*a^2 - 1216047/19226320*a - 1005382699/96131600)*q^19 + O(q^20) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^8 - 5328*a^6 + 96768*a^5 + 13200480*a^4 - 183472128*a^3 - 7304649984*a^2 + 365769105408*a + 8863960211712 Total time: 1.110 seconds, Total memory usage: 6.62MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 02:37:17 2005 Input: G :=DirichletGroup(168); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[5],44);Parent($1); Output: Magma V2.11-10 Mon Dec 5 2005 02:37:16 on modular [Seed = 398965269] ------------------------------------- Group of Dirichlet characters of modulus 168 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 168 2 [ Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 8 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 48 over Rational Field ] q + (-366161/49852747597086720*a^7 - 2797469/49852747597086720*a^6 + 158391289/2492637379854336*a^5 + 8901662149/12463186899271680*a^4 - 96803494055/623159344963584*a^3 - 12765707507471/3115796724817920*a^2 + 9452998877071/778949181204480*a + 799379725775941/259649727068160)*q^2 + (-133704061/3556162661925519360*a^7 - 722881927/1778081330962759680*a^6 + 20493849955/59269377698758656*a^5 + 115827613183/49391148082298880*a^4 - 4414687931579/4939114808229888*a^3 - 318224077585391/37043361061724160*a^2 + 5196394685395351/55565041592586240*a + 136453130538773749/27782520796293120)*q^3 + (-1/42785832960*a^7 - 17/21392916480*a^6 + 461/2139291648*a^5 + 31537/5348229120*a^4 - 305647/534822912*a^3 - 21474203/1337057280*a^2 + 15212051/668528640*a + 879076513/111421440)*q^4 + (106591/806589313440768*a^7 + 40379/50411832090048*a^6 - 76382663/67215776120064*a^5 - 8079881/1400328669168*a^4 + 5136411875/1867104892224*a^3 + 33800872687/1050246501876*a^2 - 60881185195/12602958022512*a - 12267009810503/787684876407)*q^5 + (-45302249/533424399288827904*a^7 - 646673957/1066848798577655808*a^6 + 33290024101/44452033274068992*a^5 + 120943483541/29634688849379328*a^4 - 2320917648437/1234778702057472*a^3 - 450321482240941/22226016637034496*a^2 + 700967914085105/8334756238887936*a + 174868732364074967/16669512477775872)*q^6 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514334695/96171558912*a^3 + 34216014163/1081930037760*a^2 - 47465191101 ** WARNING: Output too long, hence truncated. '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 02:36:32 2005 Input: G :=DirichletGroup(168); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[5],90);Parent($1); Output: Magma V2.11-10 Mon Dec 5 2005 02:36:31 on modular [Seed = 500020172] ------------------------------------- Group of Dirichlet characters of modulus 168 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 168 2 [ Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 8 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 48 over Rational Field ] q + (317/173203430400*a^7 - 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147/19676800*a^5 + 2853/39353600*a^4 + 58849/19676800*a^3 + 222341/9838400*a^2 + 10248889/7378800*a - 15283/17200)*q^19 + (144803/3972132003840*a^7 + 441/2149422080*a^6 - 2120691/66202200064*a^5 + 673813/5910910720*a^4 + 1273273011/82752750080*a^3 + 407651341/1477727680*a^2 + 66517853527/8866366080*a + 712507313/33584720)*q^20 + (-19247/472872857600*a^7 - 29929/354654643200*a^6 + 5034671/118218214400*a^5 - 24527361/59109107200*a^4 - 119652341/5910910720*a^3 - 349257509/3694319200*a^2 - 43458287687/7388638400*a - 1218787157/1007541600)*q^21 + (1/46126080*a^7 + 1/13178880*a^6 - 53/2306304*a^5 + 613/3294720*a^4 + 32111/2882880*a^3 + 63211/823680*a^2 + 308719/102960*a + 409/160)*q^22 + (-1/605606400*a^7 + 11/173030400*a^6 + 101/75700800*a^5 - 2239/43257600*a^4 + 13879/37850400*a^3 - 171883/10814400*a^2 + 5041/168975*a + 4437783/300400)*q^23 + (9467/794426400768*a^7 - 956143/4965165004800*a^6 - 22940193/1655055001600*a^5 + 160175209/413763750400*a^4 + 1370346727/413763750400*a^3 - 11793248219/103440937600*a^2 + 50196143149/44331830400*a - 26243615287/1007541600)*q^24 - 59*q^25 + (307511/11916396011520*a^7 - 65579/212792785920*a^6 - 1564845/66202200064*a^5 + 10404001/17732732160*a^4 + 1687795847/248258250240*a^3 - 252223303/4433183040*a^2 + 74671076059/26599098240*a - 8232535577/302262480)*q^26 + (292471/13240440012800*a^7 - 56689/206881875200*a^6 - 71564517/3310110003200*a^5 + 28429461/51720468800*a^4 + 4822759341/827527500800*a^3 - 277065927/3232529300*a^2 + 68854988087/29554553600*a - 23333182/807325)*q^27 + (193/13222809600*a^7 - 691/3777945600*a^6 - 6359/413212800*a^5 + 354649/944486400*a^4 + 2953637/826425600*a^3 - 20432513/236121600*a^2 + 41789051/14757600*a - 2467087/82560)*q^28 + (1/242242560*a^7 - 11/69212160*a^6 - 101/30280320*a^5 + 2239/17303040*a^4 - 13879/15140160*a^3 + 171883/4325760*a^2 - 5041/67590*a - 4437783/120160)*q^29 + (-547/5498521600*a^7 - 219/1099704320*a^6 + 13753/137463040*a^5 - 28863/34365760*a^4 - 16308011/343657600*a^3 - 119331099/343657600*a^2 - 55323101/4295720*a + 143667699/3905200)*q^30 + (-1/305141760*a^7 + 7/54489600*a^6 + 183/42380800*a^5 - 791/4540800*a^4 - 30941/31785600*a^3 + 81791/1135200*a^2 - 497557/3405600*a + 1076131/77400)*q^31 + (139/1968220800*a^7 + 2243/31491532800*a^6 - 271937/3936441600*a^5 + 4741733/7872883200*a^4 + 15797837/492055200*a^3 + 97513441/393644160*a^2 + 308292737/35146800*a + 27393021/7810400)*q^32 + (-102653/4965165004800*a^7 + 153017/1241291251200*a^6 + 770937/37614886400*a^5 - 3574881/9403721600*a^4 - 62978211/9403721600*a^3 + 480839973/25860234400*a^2 - 32243595151/11082957600*a + 319324289/19375800)*q^33 + (-53/2479276800*a^7 + 31/141672960*a^6 + 1299/55095040*a^5 - 1165/2361216*a^4 - 819031/103303200*a^3 + 1459543/14757600*a^2 - 247133/80496*a + 2289809/77400)*q^34 + (-27737/189149143040*a^7 + 36731/177327321600*a^6 + 33781299/236436428800*a^5 - 11901339/7388638400*a^4 - 3676303591/59109107200*a^3 - 1390742347/3694319200*a^2 - 250432670383/14777276800*a + 1043336056/31485675)*q^35 + (5561/76979302400*a^7 + 28411/153958604800*a^6 - 107033/1480371200*a^5 + 22164511/38489651200*a^4 + 168499519/4811206400*a^3 + 179789341/740185600*a^2 + 1625176379/171828800*a + 55626669/6248320)*q^36 + (1/9225216*a^7 + 1/2635776*a^6 - 265/2306304*a^5 + 613/658944*a^4 + 32111/576576*a^3 + 63211/164736*a^2 + 308719/20592*a - 231/32)*q^37 + (-15277/1083308728320*a^7 - 21739/1489549501440*a^6 + 829461/66202200064*a^5 - 14446879/124129125120*a^4 - 1332490559/248258250240*a^3 - 2193684743/31032281280*a^2 - 67682599603/26599098240*a - 656085151/302262480)*q^38 + (43/687315200*a^7 + 111/343657600*a^6 - 46009/687315200*a^5 + 330273/687315200*a^4 + 2918131/85914400*a^3 + 17731503/85914400*a^2 + 388335583/42957200*a + 5355567/300400)*q^39 + (-1/28173600*a^7 - 1099/1416729600*a^6 + 16917/550950400*a^5 + 42917/118060800*a^4 - 3319819/206606400*a^3 - 15301991/29515200*a^2 - 398112133/44272800*a - 9798481/154800)*q^40 + (-447/25462384640*a^7 + 1541/3182798080*a^6 + 21213/1273119232*a^5 - 561357/795699520*a^4 - 3416397/1591399040*a^3 + 3460881/18084080*a^2 - 22398963/56835680*a + 28489169/645860)*q^41 + (3009109/39721320038400*a^7 - 170821/2837237145600*a^6 - 238842831/3310110003200*a^5 + 14512241/18187417600*a^4 + 25473257153/827527500800*a^3 + 12039490211/59109107200*a^2 + 880428043283/88663660800*a + 71585213903/4030166400)*q^42 + 24*q^43 + (-101/2706303600*a^7 - 14737/173203430400*a^6 + 794803/21650428800*a^5 - 12595207/43300857600*a^4 - 48115219/2706303600*a^3 - 1337570791/10825214400*a^2 - 929072323/193307400*a - 9885609/781040)*q^44 + (-2237/82752750080*a^7 - 20497/29554553600*a^6 + 37251/2350930400*a^5 + 349053/671694400*a^4 - 25385811/2350930400*a^3 - 1101878883/1847159600*a^2 - 3165829207/461789900*a - 3312508801/41980900)*q^45 + (1/46126080*a^7 + 1/13178880*a^6 - 53/2306304*a^5 + 613/3294720*a^4 + 32111/2882880*a^3 + 63211/823680*a^2 + 308719/102960*a - 4711/160)*q^46 + (4637/74477475072*a^7 - 137/1329954912*a^6 - 115173/2068818752*a^5 + 77659/110829576*a^4 + 34422593/1551614064*a^3 + 606 ** WARNING: Output too long, hence truncated. '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 02:35:21 2005 Input: G :=DirichletGroup(168); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],90);Parent($1); Output: Magma V2.11-10 Mon Dec 5 2005 02:35:20 on modular [Seed = 13676182] ------------------------------------- Group of Dirichlet characters of modulus 168 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 168 2 [ Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 8 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 48 over Rational Field ] q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^6 - 7*q^7 - 8*q^8 + 9*q^9 - 12*q^12 - 2*q^13 + 14*q^14 + 16*q^16 + 22*q^17 - 18*q^18 + 21*q^21 + 38*q^23 + 24*q^24 + 25*q^25 + 4*q^26 - 27*q^27 - 28*q^28 + 26*q^29 + 34*q^31 - 32*q^32 - 44*q^34 + 36*q^36 + 6*q^39 - 26*q^41 - 42*q^42 - 82*q^43 - 76*q^46 - 48*q^48 + 49*q^49 - 50*q^50 - 66*q^51 - 8*q^52 - 22*q^53 + 54*q^54 + 56*q^56 - 52*q^58 + 106*q^59 + 94*q^61 - 68*q^62 - 63*q^63 + 64*q^64 - 34*q^67 + 88*q^68 - 114*q^69 - 58*q^71 - 72*q^72 - 75*q^75 - 12*q^78 + 81*q^81 + 52*q^82 + 58*q^83 + 84*q^84 + 164*q^86 - 78*q^87 - 122*q^89 + O(q^90) Power series ring in q over Rational Field Total time: 1.090 seconds, Total memory usage: 6.62MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 02:32:56 2005 Input: G :=DirichletGroup(168); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Norm(Coefficient(qEigenform(D[1],12),11)); Norm(Coefficient(qEigenform(D[2],12),11)); Norm(Coefficient(qEigenform(D[3],12),11)); Norm(Coefficient(qEigenform(D[4],12),11)); Norm(Coefficient(qEigenform(D[5],12),11)); Norm(Coefficient(qEigenform(D[6],12),11)); Norm(Coefficient(qEigenform(D[7],12),11)); Norm(Coefficient(qEigenform(D[8],12),11)); Norm(Coefficient(qEigenform(D[9],12),11)); Norm(Coefficient(qEigenform(D[10],12),11)); Norm(Coefficient(qEigenform(D[11],12),11)); Norm(Coefficient(qEigenform(D[12],12),11)); Norm(Coefficient(qEigenform(D[13],12),11)); Norm(Coefficient(qEigenform(D[14],12),11)); Norm(Coefficient(qEigenform(D[15],12),11)); Norm(Coefficient(qEigenform(D[16],12),11)); Norm(Coefficient(qEigenform(D[17],12),11)); Norm(Coefficient(qEigenform(D[18],12),11)); Norm(Coefficient(qEigenform(D[19],12),11)); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 02:32:35 on modular [Seed = 131574074] ------------------------------------- Group of Dirichlet characters of modulus 168 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 168 2 [ Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 8 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 48 over Rational Field ] 0 0 0 0 1048576 Errors: /bin/sh: line 1: 27945 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Sun Dec 4 21:21:42 2005 Input: Factorization(777); Factorization(777777); Output: Magma V2.11-10 Sun Dec 4 2005 21:21:41 on modular [Seed = 3077001384] ------------------------------------- [ <3, 1>, <7, 1>, <37, 1> ] [ <3, 1>, <7, 2>, <11, 1>, <13, 1>, <37, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Sun Dec 4 21:20:37 2005 Input: Factorization(4444); Output: Magma V2.11-10 Sun Dec 4 2005 21:20:36 on modular [Seed = 2975945068] ------------------------------------- [ <2, 2>, <11, 1>, <101, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.135' ************** MAGMA ***************** Host 128.135.197.33 (128.135.197.33) Time: Sun Dec 4 18:22:47 2005 Input: 3*7*691/(98*288); Output: Magma V2.11-10 Sun Dec 4 2005 18:22:47 on modular [Seed = 2927531971] ------------------------------------- 691/1344 Total time: 0.190 seconds, Total memory usage: 3.24MB '128.135' ************** MAGMA ***************** Host 128.135.197.33 (128.135.197.33) Time: Sun Dec 4 18:20:02 2005 Input: Factorization (945); Output: Magma V2.11-10 Sun Dec 4 2005 18:20:02 on modular [Seed = 1417965276] ------------------------------------- [ <3, 3>, <5, 1>, <7, 1> ] Total time: 0.180 seconds, Total memory usage: 3.24MB '128.135' ************** MAGMA ***************** Host 128.135.197.33 (128.135.197.33) Time: Sun Dec 4 18:19:48 2005 Input: Factorization (691); Output: Magma V2.11-10 Sun Dec 4 2005 18:19:48 on modular [Seed = 1401121261] ------------------------------------- [ <691, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.135' ************** MAGMA ***************** Host 128.135.197.33 (128.135.197.33) Time: Sun Dec 4 18:19:24 2005 Input: Factorization (638512875); Output: Magma V2.11-10 Sun Dec 4 2005 18:19:23 on modular [Seed = 1586390031] ------------------------------------- [ <3, 6>, <5, 3>, <7, 2>, <11, 1>, <13, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.135' ************** MAGMA ***************** Host 128.135.197.33 (128.135.197.33) Time: Sun Dec 4 18:19:18 2005 Input: Factorize (638512875); Output: Magma V2.11-10 Sun Dec 4 2005 18:19:17 on modular [Seed = 1167425306] ------------------------------------- >> Factorize (638512875);; ^ User error: Identifier 'Factorize' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sun Dec 4 18:17:48 2005 Input: G :=DirichletGroup(1176); G; X :=Elements(G); X; Output: Magma V2.11-10 Sun Dec 4 2005 18:17:47 on modular [Seed = 1990609991] ------------------------------------- Group of Dirichlet characters of modulus 1176 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '69.165.' ************** MAGMA ***************** Host 69.165.160.197 (69.165.160.197) Time: Sun Dec 4 12:38:06 2005 Input: 233*7 Output: Magma V2.11-10 Sun Dec 4 2005 12:38:05 on modular [Seed = 4281308580] ------------------------------------- 1631 Total time: 0.190 seconds, Total memory usage: 3.24MB '82.32.2' ************** MAGMA ***************** Host 82.32.210.219 (82.32.210.219) Time: Sun Dec 4 08:46:05 2005 Input: SetVerbose("BestCode",2); BKLC(GF(2),144,72); Output: Magma V2.11-10 Sun Dec 4 2005 08:46:04 on modular [Seed = 1451746925] ------------------------------------- Construction of a [ 144 , 72 , 22 ] Code: [1]: [15, 11, 3] "BCH code (d = 3, b = 1)" Cyclic Linear Code over GF(2) BCHCode with parameters 15 3 [2]: [11, 7, 3] Linear Code over GF(2) Shortening of [1] at { 12 .. 15 } [3]: [8, 7, 2] Linear Code over GF(2) Dual of the RepetitionCode of length 8 [4]: [127, 70] "BCH code (d = 20, b = 127)" Cyclic Linear Code over GF(2) BCHCode with parameters 127 20 0 [5]: [127, 70] "BCH code (d = 18, b = 125)" Cyclic Linear Code over GF(2) BCHCode with parameters 127 18 125 [6]: [127, 77] "BCH code (d = 16, b = 127)" Cyclic Linear Code over GF(2) BCHCode with parameters 127 16 0 [7]: [146, 77, 21] Linear Code over GF(2) ConstructionXX using [6] [5] [4] [3] and [2] [8]: [147, 77, 22] Linear Code over GF(2) ExtendCode [7] by 1 [9]: [144, 74, 22] Linear Code over GF(2) Shortening of [8] at { 145 .. 147 } [10]: [144, 72, 22] Linear Code over GF(2) Subcode of [9] [144, 72, 22] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 1] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 1 0 0 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 0 0 0 1 1 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 1 0 0 0 1 0] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 1 0 1 1 0 1 0 0 0 1] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 1 0 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 0 0 1 1 1 1 1 0 1 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 1 1 1 1 1 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 0 0 0 1 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 0 ** WARNING: Output too long, hence truncated. '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:39:50 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; P:=IsConjugate(D,F); Output: Magma V2.11-10 Sun Dec 4 2005 05:39:50 on modular [Seed = 620077168] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:33:36 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; IsConjugate(D,F); Output: Magma V2.11-10 Sun Dec 4 2005 05:33:36 on modular [Seed = 3506459890] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:32:42 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; [T,T1]:=IsConjugate(D,F); Output: Magma V2.11-10 Sun Dec 4 2005 05:32:42 on modular [Seed = 3458156583] ------------------------------------- >> [T,T1]:=IsConjugate(D,F); ^ User error: Illegal left hand side of an assignment statement Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:28:42 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; T:=IsConjugate(D,F); WhatType(T) ; Output: Magma V2.11-10 Sun Dec 4 2005 05:28:42 on modular [Seed = 3289727805] ------------------------------------- >> WhatType(T) ^ User error: Identifier 'WhatType' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:28:30 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; T:=IsConjugate(D,F); whattype(T) ; Output: Magma V2.11-10 Sun Dec 4 2005 05:28:30 on modular [Seed = 3239069228] ------------------------------------- >> whattype(T) ^ User error: Identifier 'whattype' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:28:10 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; T:=IsConjugate(D,F); T[2]; Output: Magma V2.11-10 Sun Dec 4 2005 05:28:10 on modular [Seed = 4264524135] ------------------------------------- >> T[2];; ^ Runtime error in '[]': Bad argument types Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:26:26 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; T:=IsConjugate(D,F); T; Output: Magma V2.11-10 Sun Dec 4 2005 05:26:26 on modular [Seed = 4096095109] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:26:04 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; T:=isConjugate(D,F); T; Output: Magma V2.11-10 Sun Dec 4 2005 05:26:03 on modular [Seed = 4045436649] ------------------------------------- >> T:=isConjugate(D,F); ^ User error: Identifier 'isConjugate' has not been declared or assigned >> T;; ^ User error: Identifier 'T' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:25:06 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; IsConjugate(D,F); Output: Magma V2.11-10 Sun Dec 4 2005 05:25:05 on modular [Seed = 3997133260] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:23:28 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; PositiveConjugates(F); Output: Magma V2.11-10 Sun Dec 4 2005 05:23:28 on modular [Seed = 3828704359] ------------------------------------- {@ Br.3 * Br.2 * Br.1 * Br.3 * Br.2 * Br.3 @} Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:22:41 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2 PositiveConjugates(F); Output: Magma V2.11-10 Sun Dec 4 2005 05:22:41 on modular [Seed = 3811730353] ------------------------------------- >> PositiveConjugates(F);; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:20:29 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2 Output: Magma V2.11-10 Sun Dec 4 2005 05:20:27 on modular [Seed = 2588613458] ------------------------------------- Total time: 0.210 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sun Dec 4 03:52:23 2005 Input: G :=DirichletGroup(1575); G; X :=Elements(G); X; Output: Magma V2.11-10 Sun Dec 4 2005 03:52:23 on modular [Seed = 1022218584] ------------------------------------- Group of Dirichlet characters of modulus 1575 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '161.246' ************** MAGMA ***************** Host 161.246.1.33 (161.246.1.33) Time: Sun Dec 4 00:04:53 2005 Input: 2+3 Output: Magma V2.11-10 Sun Dec 4 2005 00:04:53 on modular [Seed = 3963475716] ------------------------------------- 5 Total time: 0.190 seconds, Total memory usage: 3.24MB '24.158.' ************** MAGMA ***************** Host 24.158.163.183 (24.158.163.183) Time: Sat Dec 3 18:38:20 2005 Input: PR := PolynomialRing(RationalField(), 3); I := ideal; Groebner(I); I; Output: Magma V2.11-10 Sat Dec 3 2005 18:38:16 on modular [Seed = 4146501764] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: t, x, y Groebner basis: [ t^2 - 881468632448381366573236683482557819679940281233931949107407297230523\ 37978943342401657863550929248859858595742780052545132851789925614198409\ 65890428329870646111856827881383710690971227741142932604027197561650409\ 25125402837936120450078542683149044830295192323970601909312468153468633\ 17346114209676594823743330674408214872353594733791189252806684672260108\ 65314395800559554891665664695515627595412817451848634505655029814098086\ 92447418381951844984510128713578056169221035834151173057293367433802441\ 97075996481196562838811518958219547140650952459081317638824504086936336\ 55152613270231007054672967701333022471880356816564488615430985913568499\ 98570945175162427123089113181839479329642472288509258049451274606899357\ 28414398652671582064162705276077004712071357172940983407119239602167142\ 12999714874773499806153074422110353722732488435357865689492709594950307\ 30808404071291471503917195000430875596466072590511545554397895074227701\ 46034403785570756512070180529374318768027967388400817352217972292078597\ 91696334544813285540938583462482240808871396102407074256550837427950467\ 09735082904846880928527285442476134233199439150000434248805256638931115\ 89774991874125772137238696716377221916929512319307870907621750917716496\ 37139211239427677846623309126743035897594089798452080331446824730256827\ 08452218475563544058655156301436681367830088748581810382619892722065167\ 07830552127579233741552908333263803841217520198941798485221423490302726\ 79540838347047859896944806667811403078660627987895478387587446126966013\ 71874460248114450040247006389038947283901274967378884778607074651170457\ 99586587910168448769709071451982042527742478665959276648202933181800673\ 94479353222423158444939898338951071566402247484018736009761036912959334\ 32134487204849408042423399572163349123228744124467011646920896987768583\ 22597174914422331486208611697295360000000000000000000000000/63822184766\ 98146611041098502826768428796825644905358012592084487394159829130713495\ 94963064187513259274115849765916400834353984800651053031725085689974625\ 03184090809290171475661354908761031599878047109000389326662909672526579\ 27867767871688853654694467344956205338778533868535733150953120661302700\ 37207745689687723470660814991612589582045557321511428493321119529764850\ 98154590941344244780254538650948294910196862766695014577658780179988149\ 54966984908957311683888793606532081614327151239575683487577938491305483\ 68336288304810842410690685663348623309539171423763405981496755410688780\ 17955920029370648645824037653907268279005576593653515599440687072737018\ 73744729172045035750061750500622902635389741405423640332711573610711384\ 89077177988972000701819365903901035006980304514091608256038370125552216\ 86312348001218791640380256499773842719118987572373430757265348912296056\ 52110089873220366581669884277300336820032041916273777027755211082257840\ 17895697857374573792883671174985035120831738171863261544320376370166358\ 96094952321554437633716797099229735263889237468336374605438940120956401\ 99604989730451052911730057502689676856848690110587740550720575460663894\ 13908243001521415535883777575546378252090031231133833516190912070289361\ 07328206503316814241801578251068311358187148224259453418539700840613495\ 66241580552102911114782410893058457300090010395044006701700164671381111\ 68105729379047870965348755551196342848778469476921284915700483724259215\ 12547056707888135359641449136674161620347169187357029120681787923062733\ 18557569854864613756414163939228945179085520320433491949650508195075923\ 35630935291353352230042736857770910023520647057744049846769505208490963\ 76006341993878304540398659341962407354877354735783814430380291546166019\ 41887484408009637009000276945891737381414902602785939809168697925848270\ 9464085301804841363301053624633610152767772169833911*t*y^27 + 16145893244588019376321463759147101976054935655561059823010489353963009\ 95857280876045537358073066293646780347476751437396739060058377866891614\ 56958696459339321764451415803140190302531452764875464930045906699429392\ 93322197711290533346477885997704462519826824238395287070127382054334825\ 12474154678496780636450590167845652712038994355838278984888207707401136\ 25793101228875854173302043860918791935103024875827289424685640482585888\ 58527872842853282182062781647541561385354893199020090099382245944071376\ 17512817248742682206013705633610232054322954082279994063896441253448449\ 85346152522942705767309622364519937441451848751162340430863262042777544\ 81613073723888803675696796843708677322713387739455901933990027946795462\ 03413907973996855287381187409138404706105632484713921107131351330810291\ 30882218866024701152824647805101133029445682306957060509514785413913927\ 02975991009369511431508267835319244302544318312708912057705591174954092\ 56754545807622466250514459683223972238742211592013008987780034165014005\ 20762734024413972758306388541659428870424914285449729061264865229054252\ 38843378207817525535905207426582748723137040977502926977634751768186698\ 49581209374388602372388115673748962542027564878465454029491751240952477\ 34612791431978069681060826807695029958268938415100506426028189406456810\ 47151753004300858542276117825497020675952427736778915349393009710275069\ 48984179044091308012297661610223584745791147499482444973386598700462218\ 22606684483968114187424005870778280377989242995451388532501416851196355\ 01833124158085873190081333983264290540365607375139933318620476903777326\ 80107573193720458309561560284258689641907871095592925772353970575124512\ 12475404115735794731369583795917400222525265781613763422453149992626793\ 68082207701480540682384020108812882191767165651145406631366951396502112\ 07997057420021592274019931910213468160000000000000000000000/63822184766\ 98146611041098502826768428796825644905358012592084487394159829130713495\ 94963064187513259274115849765916400834353984800651053031725085689974625\ 03184090809290171475661354908761031599878047109000389326662909672526579\ 27867767871688853654694467344956205338778533868535733150953120661302700\ 37207745689687723470660814991612589582045557321511428493321119529764850\ 98154590941344244780254538650948294910196862766695014577658780179988149\ 54966984908957311683888793606532081614327151239575683487577938491305483\ 68336288304810842410690685663348623309539171423763405981496755410688780\ 17955920029370648645824037653907268279005576593653515599440687072737018\ 73744729172045035750061750500622902635389741405423640332711573610711384\ 89077177988972000701819365903901035006980304514091608256038370125552216\ 86312348001218791640380256499773842719118987572373430757265348912296056\ 52110089873220366581669884277300336820032041916273777027755211082257840\ 17895697857374573792883671174985035120831738171863261544320376370166358\ 96094952321554437633716797099229735263889237468336374605438940120956401\ 99604989730451052911730057502689676856848690110587740550720575460663894\ 13908243001521415535883777575546378252090031231133833516190912070289361\ 07328206503316814241801578251068311358187148224259453418539700840613495\ 66241580552102911114782410893058457300090010395044006701700164671381111\ 68105729379047870965348755551196342848778469476921284915700483724259215\ 12547056707888135359641449136674161620347169187357029120681787923062733\ 18557569854864613756414163939228945179085520320433491949650508195075923\ 35630935291353352230042736857770910023520647057744049846769505208490963\ 76006341993878304540398659341962407354877354735783814430380291546166019\ 41887484408009637009000276945891737381414902602785939809168697925848270\ 9464085301804841363301053624633610152767772169833911*t*y^26 - 14189569054767755841931853466183156132446347954453444573569145856951694\ 97594498858653122916161244563504680207863736830617783868905678588623263\ 58149999570175871115229328171082776321630175215040653265091559032547797\ 31430174135118414507556058351671949932506942478301266480284529031139437\ 41948941211604196291522896965189054469466259155503783241197069335082147\ 73147912721291181077513656885507356175832838774546070403733244183896350\ 95446525802377648670029602668889796922504929178979896629976088075503016\ 66367885038783727071910279103789092599463229739996561041273969939960124\ 90504451432947103273023964424762583695568813765769913557391797055569260\ 03266758660192688706865894970120436086965615126630219052628167651298978\ 43492852102361245459176793134050457018607409028604703270481579914732705\ 81011795630745919129853669628298786502591595974677665329154884586273146\ 96521801683268382183066190908691653150183855602060625946786621458606013\ 92698455171727954285104993759171409390867910757794957084321809070738001\ 46478087269697971501738074571113565736127070688036023597767204985658452\ 17140623587337134427872758514246847269965626360127204036526844463226741\ 6141667452863041499240889640760710761960227057997382487953994 ** WARNING: Output too long, hence truncated. '24.158.' ************** MAGMA ***************** Host 24.158.163.183 (24.158.163.183) Time: Sat Dec 3 18:37:19 2005 Input: PR := PolynomialRing(RationalField(), 3); I := ideal; Groebner(I); Output: Magma V2.11-10 Sat Dec 3 2005 18:37:17 on modular [Seed = 4264530376] ------------------------------------- Total time: 1.690 seconds, Total memory usage: 7.06MB '24.158.' ************** MAGMA ***************** Host 24.158.163.183 (24.158.163.183) Time: Sat Dec 3 18:37:03 2005 Input: PR := PolynomialRing(RationalField(), 3); I := ideal; Groebner(I); Output: Magma V2.11-10 Sat Dec 3 2005 18:37:01 on modular [Seed = 4213871869] ------------------------------------- Total time: 1.700 seconds, Total memory usage: 6.97MB '24.158.' ************** MAGMA ***************** Host 24.158.163.183 (24.158.163.183) Time: Sat Dec 3 18:35:08 2005 Input: PR := PolynomialRing(RationalField(), 3); I := ideal; time Groebner(I); Output: Magma V2.11-10 Sat Dec 3 2005 18:35:07 on modular [Seed = 3239094641] ------------------------------------- Time: 0.010 Total time: 0.200 seconds, Total memory usage: 3.43MB '24.158.' ************** MAGMA ***************** Host 24.158.163.183 (24.158.163.183) Time: Sat Dec 3 18:34:34 2005 Input: PR := PolynomialRing(RationalField(), 3); I := ideal; time Groebner(I); Output: Magma V2.11-10 Sat Dec 3 2005 18:34:34 on modular [Seed = 3624513609] ------------------------------------- Time: 0.010 Total time: 0.200 seconds, Total memory usage: 3.43MB '128.196' ************** MAGMA ***************** Host 128.196.20.120 (128.196.20.120) Time: Sat Dec 3 15:19:02 2005 Input: rand(23984793287429832938) Output: Magma V2.11-10 Sat Dec 3 2005 15:19:02 on modular [Seed = 47234208] ------------------------------------- >> rand(23984793287429832938); ^ User error: Identifier 'rand' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '128.196' ************** MAGMA ***************** Host 128.196.20.120 (128.196.20.120) Time: Sat Dec 3 15:18:50 2005 Input: fft(1283129847219873192873192873) Output: Magma V2.11-10 Sat Dec 3 2005 15:18:49 on modular [Seed = 98024376] ------------------------------------- >> fft(1283129847219873192873192873) ^ User error: Identifier 'fft' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '193.249' ************** MAGMA ***************** Host 193.249.239.125 (193.249.239.125) Time: Sat Dec 3 13:23:45 2005 Input: P := PolynomialRing(RationalField(), 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); C := Coordinates(I, P!1); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 3 2005 13:23:24 on modular [Seed = 1268378305] ------------------------------------- Errors: /bin/sh: line 1: 21853 Alarm clock nice -n 19 /usr/local/bin/magma '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:43:19 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); g:=1; C := Coordinates(I, -16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 3 2005 12:42:59 on modular [Seed = 1022273033] ------------------------------------- Errors: /bin/sh: line 1: 21802 Alarm clock nice -n 19 /usr/local/bin/magma '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:42:28 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); g:=1; C := Coordinates(I, g); Output: Magma V2.11-10 Sat Dec 3 2005 12:42:27 on modular [Seed = 870426918] ------------------------------------- >> C := Coordinates(I, g);; ^ Runtime error in 'Coordinates': Bad argument types Argument types given: RngMPol, RngIntElt Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:40:57 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); I; Output: Magma V2.11-10 Sat Dec 3 2005 12:40:56 on modular [Seed = 232756915] ------------------------------------- Ideal of Polynomial ring of rank 4 over Rational Field Lexicographical Order Variables: x, y, z, a User basis: [ 16*x^2*y^2 - 16*x^2*y + 4*x^2 + 96*x*y^2 - 48*x*y + 144*y^2, 64*x^2*z^2 - 64*x^2*z + 16*x^2 + 288*x*z^2 - 144*x*z + 324*z^2, 144*x^2*a^2 + 48*x^2*a + 4*x^2 + 576*x*a^2 + 96*x*a + 16*y^2*z^2 - 64*y^2*z + 64*y^2 + 48*y*z^2 - 96*y*z + 36*z^2 + 576*a^2, 16*z^2*a^2 + 48*z^2*a + 36*z^2 + 128*z*a^2 + 192*z*a + 256*a^2, 64*y^2*a^2 + 64*y^2*a + 16*y^2 + 128*y*a^2 + 64*y*a + 64*a^2, x^2 + 2*y^2 + 3*z^2 + 4*a^2 - 10, x^2 + 3*x + y^2 + y + z^2 + 2*z + a^2 - a - 15 ] Basis: [ 16*x^2*y^2 - 16*x^2*y + 4*x^2 + 96*x*y^2 - 48*x*y + 144*y^2, 64*x^2*z^2 - 64*x^2*z + 16*x^2 + 288*x*z^2 - 144*x*z + 324*z^2, 144*x^2*a^2 + 48*x^2*a + 4*x^2 + 576*x*a^2 + 96*x*a + 16*y^2*z^2 - 64*y^2*z + 64*y^2 + 48*y*z^2 - 96*y*z + 36*z^2 + 576*a^2, 16*z^2*a^2 + 48*z^2*a + 36*z^2 + 128*z*a^2 + 192*z*a + 256*a^2, 64*y^2*a^2 + 64*y^2*a + 16*y^2 + 128*y*a^2 + 64*y*a + 64*a^2, x^2 + 2*y^2 + 3*z^2 + 4*a^2 - 10, x^2 + 3*x + y^2 + y + z^2 + 2*z + a^2 - a - 15 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:40:35 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); Coordinates(I,1); Output: Magma V2.11-10 Sat Dec 3 2005 12:40:34 on modular [Seed = 13802372] ------------------------------------- >> Coordinates(I,1);; ^ Runtime error in 'Coordinates': Bad argument types Argument types given: RngMPol, RngIntElt Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:40:13 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); Output: Magma V2.11-10 Sat Dec 3 2005 12:40:13 on modular [Seed = 131439724] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:39:52 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); Coordinates(I, -16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 3 2005 12:39:31 on modular [Seed = 500139608] ------------------------------------- Errors: /bin/sh: line 1: 21771 Alarm clock nice -n 19 /usr/local/bin/magma '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:38:18 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); Coordinates(I, 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:38:18 on modular [Seed = 4013782258] ------------------------------------- >> Coordinates(I, 1);; ^ Runtime error in 'Coordinates': Bad argument types Argument types given: RngMPol, RngIntElt Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:36:46 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis; Coordinates(I, 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:36:46 on modular [Seed = 3862196831] ------------------------------------- >> I:=IdealWithFixedBasis> Coordinates(I, 1);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:34:53 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=ideal; Coordinates(I, 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:34:52 on modular [Seed = 4079052760] ------------------------------------- >> Coordinates(I, 1);; ^ Runtime error in 'Coordinates': Bad argument types Argument types given: RngMPol, RngIntElt Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:31:45 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); Coordinates(ideal, 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:31:44 on modular [Seed = 3222426967] ------------------------------------- >> Coordinates(ideal := PolynomialRing(Q, 4); Coordinates([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15], 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:29:47 on modular [Seed = 2691839420] ------------------------------------- >> Coordinates([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2- ^ Runtime error in 'Coordinates': Bad argument types Argument types given: SeqEnum[RngMPolElt], RngIntElt Total time: 0.200 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.15 (193.250.127.15) Time: Sat Dec 3 12:26:32 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); Coordinates([x^2+36*y^2-4*x^2*y+4*x^2*y^2+24*y^2*x-12*x*y, 9*z^2+128*z*a^2+48*a*z^2+64*a^2*z^2+64*a^2+48*z*a, 81*z^2-16*x^2*z+16*x^2*z^2-36*x*z+4*x^2+72*z^2*x, 16*y^2+9*z^2+24*x*a+144*a^2+16*x^2*a-16*y^2*z+x^2+64*x^2*a^2+4*y^2*z^2+12*z^2*y+192*x*a^2-24*y*z, 16*a^2+64*a^2*y^2+64*a^2*y+32*a*y^2+16*a*y+4*y^2, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+t^2+3*x+y+2*z-a-15], 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:26:32 on modular [Seed = 704364406] ------------------------------------- >> +64*a^2*y+32*a*y^2+16*a*y+4*y^2, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+t^2+ ^ User error: Identifier 't' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.15 (193.250.127.15) Time: Sat Dec 3 12:24:49 2005 Input: Coordinates([x^2+36*y^2-4*x^2*y+4*x^2*y^2+24*y^2*x-12*x*y, 9*z^2+128*z*a^2+48*a*z^2+64*a^2*z^2+64*a^2+48*z*a, 81*z^2-16*x^2*z+16*x^2*z^2-36*x*z+4*x^2+72*z^2*x, 16*y^2+9*z^2+24*x*a+144*a^2+16*x^2*a-16*y^2*z+x^2+64*x^2*a^2+4*y^2*z^2+12*z^2*y+192*x*a^2-24*y*z, 16*a^2+64*a^2*y^2+64*a^2*y+32*a*y^2+16*a*y+4*y^2, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+t^2+3*x+y+2*z-a-15], 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:24:49 on modular [Seed = 805157081] ------------------------------------- >> Coordinates([x^2+36*y^2-4*x^2*y+4*x^2*y^2+24*y^2*x-12*x*y, 9*z^2+128*z*a^2+ ^ User error: Identifier 'x' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:47:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:46:59 on modular [Seed = 3239201833] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] {@ @} Total time: 1.639 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:46:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:46:52 on modular [Seed = 3458024217] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] {@ <116034, 1, 5, 1>, <57537, 1, 5, 1> @} Total time: 1.570 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:46:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,7); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:46:44 on modular [Seed = 3407365652] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] {@ <1322, 1, 4, 1>, <0, 1, 4, 1>, <1047, 1, 4, 1> @} Total time: 1.669 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:43:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:43:37 on modular [Seed = 2976005283] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] 1.44778617658396039921843136573580142519427894255151861457351 3.654849499758574762261498916478491203095 2.811849734125242389181859855656213507312 Total time: 1.330 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:43:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:43:27 on modular [Seed = 3194827706] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] 1.44778617658396039921843136573580142519427894255151861457351 4.923736996348681348058582185502237552710 2.811849734125242389181859855656213507312 Total time: 1.330 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:43:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:43:18 on modular [Seed = 3144169142] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 4.923736996348681348058582185502237552710 2.811849734125242389181859855656213507312 Total time: 1.280 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:42:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:42:17 on modular [Seed = 2691903074] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] Total time: 1.229 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:37:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=3000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:37:33 on modular [Seed = 2639139387] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] >> Chabauty(P,11); ^ User error: Identifier 'P' has not been declared or assigned Total time: 1.870 seconds, Total memory usage: 38.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:36:48 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri D