[was@modular sharifi]$ [was@modular sharifi]$ Magma V2.8-10 Mon Apr 8 2002 17:35:11 on modular [Seed = 1338080889] Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/emacs.m" Loading "/home/was/magma/local/init.m" > Attach("unitcomp.m"); In file "/home/was/people/sharifi/unitcomp.m", line 235, column 4: >> function random_L() ^ User error: bad syntax >> Attach("unitcomp.m"); ^ Runtime error in 'Attach': Can't attach intrinsics of "unitcomp.m" > > Attach("unitcomp.m"); > f, L, K, del, tr_alpha := FindDegree37FieldMod(59); Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 59 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 28*z^35 + 6*z^34 + 38*z^33 + 18*z^32 + 18*z^30 + 45*z^29 + 44*z^28 + 57*z^27 + 45*z^26 + 28*z^25 + 38*z^24 + 57*z^23 + 8*z^22 + 8*z^21 + 44*z^20 + 6*z^19 + 6*z^18 + 44*z^17 + 8*z^16 + 8*z^15 + 57*z^14 + 38*z^13 + 28*z^12 + 45*z^11 + 57*z^10 + 44*z^9 + 45*z^8 + 18*z^7 + 18*z^5 + 38*z^4 + 6*z^3 + 28*z^2 + 21 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.830 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.870 > f; X^37 + 44*X^34 + 43*X^33 + 11*X^32 + 46*X^31 + 52*X^30 + 13*X^29 + 4*X^28 + 15*X^27 + 42*X^26 + 51*X^25 + 38*X^24 + 56*X^23 + 26*X^22 + 12*X^21 + 29*X^20 + 10*X^19 + 52*X^18 + 47*X^17 + 20*X^16 + 22*X^15 + 5*X^14 + 53*X^13 + 2*X^12 + 15*X^11 + 22*X^10 + 42*X^9 + 39*X^8 + 57*X^7 + 42*X^6 + 7*X^5 + 49*X^4 + 13*X^3 + 18*X^2 + 48*X + 30 > f, L, K, del, tr_alpha := FindDegree37FieldMod(NextPrime(10^15)); Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 1000000000000037 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 196452623054476*z^35 + 210499022289144*z^34 + 341125190400915*z^33 + 576384024628603*z^32 + 576384024628603*z^30 + 575360953172295*z^29 + 648085684668694*z^28 + 245276184420986*z^27 + 575360953172295*z^26 + 196452623054476*z^25 + 341125190400915*z^24 + 245276184420986*z^23 + 751904074777693*z^22 + 751904074777693*z^21 + 648085684668694*z^20 + 210499022289144*z^19 + 210499022289144*z^18 + 648085684668694*z^17 + 751904074777693*z^16 + 751904074777693*z^15 + 245276184420986*z^14 + 341125190400915*z^13 + 196452623054476*z^12 + 575360953172295*z^11 + 245276184420986*z^10 + 648085684668694*z^9 + 575360953172295*z^8 + 576384024628603*z^7 + 576384024628603*z^5 + 341125190400915*z^4 + 210499022289144*z^3 + 196452623054476*z^2 + 441011681677668 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 6.350 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] FindDegree37FieldMod( ell: 1000000000000037 ) In file "/home/was/people/sharifi/unitcomp.m", line 260, column 31: >> time f := MinimalPolynomial(tr_alpha); ^ Runtime error in 'MinimalPolynomial': Coefficient ring does not allow minimal polynomials > f := CyclotomicPolynomial(37); p := NextPrime(10^15); IsIrreducible(PolynomialRing(GF(p))!f); false > f := CyclotomicPolynomial(37); p := NextPrime(10^16); IsIrreducible(PolynomialRing(GF(p))!f); false > f := CyclotomicPolynomial(37); p := NextPrime(10^17); IsIrreducible(PolynomialRing(GF(p))!f); false > f := CyclotomicPolynomial(37); p := NextPrime(10^18); false > while not IsIrreducible(PolynomialRing(GF(p))!f) do p := NextPrime(p); end while; >> = NextPrime(p); end for; ^ User error: bad syntax > while not IsIrreducible(PolynomialRing(GF(p))!f) do p := NextPrime(p); end while; > p; 1000000000000000031 > f, L, K, del, tr_alpha := FindDegree37FieldMod(p); Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 1000000000000000031 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 548657296557764158*z^35 + 495584317882473229*z^34 + 266295994901438542*z^33 + 293130011958325300*z^32 + 293130011958325300*z^30 + 383324093084962560*z^29 + 444922831927656779*z^28 + 665258335664497340*z^27 + 383324093084962560*z^26 + 548657296557764158*z^25 + 266295994901438542*z^24 + 665258335664497340*z^23 + 766997184628794806*z^22 + 766997184628794806*z^21 + 444922831927656779*z^20 + 495584317882473229*z^19 + 495584317882473229*z^18 + 444922831927656779*z^17 + 766997184628794806*z^16 + 766997184628794806*z^15 + 665258335664497340*z^14 + 266295994901438542*z^13 + 548657296557764158*z^12 + 383324093084962560*z^11 + 665258335664497340*z^10 + 444922831927656779*z^9 + 383324093084962560*z^8 + 293130011958325300*z^7 + 293130011958325300*z^5 + 266295994901438542*z^4 + 495584317882473229*z^3 + 548657296557764158*z^2 + 143731750864412793 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 6.330 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 19.630 > f; X^37 + 999999999993516447*X^34 + 999999881765362207*X^33 + 999876664493234207*X^32 + 992105099726184479*X^31 + 415103858218976038*X^30 + 213333186008598275*X^29 + 929794330305950791*X^28 + 333708409564958358*X^27 + 809904762350584438*X^26 + 77713232566687440*X^25 + 623588101889907830*X^24 + 596848960287585005*X^23 + 860079413746770940*X^22 + 102598690249888810*X^21 + 710156130803949340*X^20 + 522431849060907962*X^19 + 373946866260944125*X^18 + 240253934319741163*X^17 + 499044008389735914*X^16 + 967280933551485936*X^15 + 825731621007033764*X^14 + 538750331864357824*X^13 + 947949891901554131*X^12 + 979778893502193521*X^11 + 841721005817950522*X^10 + 335170315729667362*X^9 + 419770299421759182*X^8 + 578491904232371518*X^7 + 399811716704899965*X^6 + 208118839766878691*X^5 + 163495229339950950*X^4 + 146327022411190080*X^3 + 773055976482771030*X^2 + 343992510416201349*X + 365226241779928687 > p > ; 1000000000000000031 > S := PolynomialRing(Z); > S!f; >> S!f; ^ Runtime error in '!': Illegal coercion > e := Eltseq(f); > e; [ 365226241779928687, 343992510416201349, 773055976482771030, 146327022411190080, 163495229339950950, 208118839766878691, 399811716704899965, 578491904232371518, 419770299421759182, 335170315729667362, 841721005817950522, 979778893502193521, 947949891901554131, 538750331864357824, 825731621007033764, 967280933551485936, 499044008389735914, 240253934319741163, 373946866260944125, 522431849060907962, 710156130803949340, 102598690249888810, 860079413746770940, 596848960287585005, 623588101889907830, 77713232566687440, 809904762350584438, 333708409564958358, 929794330305950791, 213333186008598275, 415103858218976038, 992105099726184479, 999876664493234207, 999999881765362207, 999999999993516447, 0, 0, 1 ] > [Integers()!i : i in e]; >> [Integers()!i : i in e]; ^ Runtime error in '!': Illegal coercion LHS: RngInt RHS: RngUPolResElt > e[1]; 365226241779928687 > a := e[1]; > Integers()!a; >> Integers()!a; ^ Runtime error in '!': Illegal coercion LHS: RngInt RHS: RngUPolResElt > Lift(a); >> Lift(a); ^ Runtime error in 'Lift': Bad argument types Argument types given: RngUPolResElt > Parent(a); Univariate Quotient Polynomial Algebra in z over Finite field of size 1000000000000000031 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 > ee := [Integers()!Coefficient(i,0) : i in e]; [ 365226241779928687, 343992510416201349, 773055976482771030, 146327022411190080, 163495229339950950, 208118839766878691, 399811716704899965, 578491904232371518, 419770299421759182, 335170315729667362, 841721005817950522, 979778893502193521, 947949891901554131, 538750331864357824, 825731621007033764, 967280933551485936, 499044008389735914, 240253934319741163, 373946866260944125, 522431849060907962, 710156130803949340, 102598690249888810, 860079413746770940, 596848960287585005, 623588101889907830, 77713232566687440, 809904762350584438, 333708409564958358, 929794330305950791, 213333186008598275, 415103858218976038, 992105099726184479, 999876664493234207, 999999881765362207, 999999999993516447, 0, 0, 1 ] > ee := $1; > eee := [ee[i]-p : i in [1..35]] cat [0,0,1]; > eee; [ -634773758220071344, -656007489583798682, -226944023517229001, -853672977588809951, -836504770660049081, -791881160233121340, -600188283295100066, -421508095767628513, -580229700578240849, -664829684270332669, -158278994182049509, -20221106497806510, -52050108098445900, -461249668135642207, -174268378992966267, -32719066448514095, -500955991610264117, -759746065680258868, -626053133739055906, -477568150939092069, -289843869196050691, -897401309750111221, -139920586253229091, -403151039712415026, -376411898110092201, -922286767433312591, -190095237649415593, -666291590435041673, -70205669694049240, -786666813991401756, -584896141781023993, -7894900273815552, -123335506765824, -118234637824, -6483584, 0, 0, 1 ] > p := NextPrime(p*1000); while not IsIrreducible(PolynomialRing(GF(p))!f) do p := NextPrime(p); end while; >> p := NextPrime(p*1000); while not IsIrreducible(PolynomialRing(GF(p))!f) do ^ Runtime error in '!': Illegal coercion > f := CyclotomicPolynomial(37); p := NextPrime(10^18); > > p := NextPrime(p*1000); while not IsIrreducible(PolynomialRing(GF(p))!f) do while> end while; >> end while; ^ User error: bad syntax > p := NextPrime(p*1000); while not IsIrreducible(PolynomialRing(GF(p))!f) do p := NextPrime(p); end while; > p; 1000000000000000003007287 > f, L, K, del, tr_alpha := FindDegree37FieldMod(p); Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 1000000000000000003007287 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 544385704402944097414362*z^35 + 523496753722038218464577*z^34 + 24914867522782452040934*z^33 + 763284525238972226610059*z^32 + 763284525238972226610059*z^30 + 964249577551633855025828*z^29 + 691362058458388945089540*z^28 + 11813509118943512663049*z^27 + 964249577551633855025828*z^26 + 544385704402944097414362*z^25 + 24914867522782452040934*z^24 + 11813509118943512663049*z^23 + 434207441634884895822849*z^22 + 434207441634884895822849*z^21 + 691362058458388945089540*z^20 + 523496753722038218464577*z^19 + 523496753722038218464577*z^18 + 691362058458388945089540*z^17 + 434207441634884895822849*z^16 + 434207441634884895822849*z^15 + 11813509118943512663049*z^14 + 24914867522782452040934*z^13 + 544385704402944097414362*z^12 + 964249577551633855025828*z^11 + 11813509118943512663049*z^10 + 691362058458388945089540*z^9 + 964249577551633855025828*z^8 + 763284525238972226610059*z^7 + 763284525238972226610059*z^5 + 24914867522782452040934*z^4 + 523496753722038218464577*z^3 + 544385704402944097414362*z^2 + 662811240856025005247910 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 8.290 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 23.530 > e := Eltseq(f); > ee := [Integers()!Coefficient(i,0) : i in e]; > ee; [ 551041596559177793844261, 607531487417351657977321, 664006641084584018157247, 899111004809094153587257, 626289531945333569221413, 939454695197277085635458, 255023373092349808791174, 408199371085809383330786, 270451416365675771458615, 547882197645938000179264, 660079223246049270055587, 916004569140654234978077, 364740042289535891797714, 624615550959552203370740, 853591301751009558398576, 549130500575575762425738, 482901650893223668129466, 440977040017191214828156, 650230424795256753739575, 993565475563617357645326, 976595051560919147062581, 887709615750557647696223, 678859425628810065908628, 507834661447066013431464, 790727925892117097718724, 573136014757547818517448, 302390809770237443181872, 985750333707468228571131, 778215929794330246097317, 980387213333186010997559, 999974415103858221982519, 999999992105099729191735, 999999999876664496241463, 999999999999881768369463, 999999999999999996523703, 0, 0, 1 ] > eee := [ee[i]-p : i in [1..35]] cat [0,0,1]; > eee; [ -448958403440822209163026, -392468512582648345029966, -335993358915415984850040, -100888995190905849420030, -373710468054666433785874, -60545304802722917371829, -744976626907650194216113, -591800628914190619676501, -729548583634324231548672, -452117802354062002828023, -339920776753950732951700, -83995430859345768029210, -635259957710464111209573, -375384449040447799636547, -146408698248990444608711, -450869499424424240581549, -517098349106776334877821, -559022959982808788179131, -349769575204743249267712, -6434524436382645361961, -23404948439080855944706, -112290384249442355311064, -321140574371189937098659, -492165338552933989575823, -209272074107882905288563, -426863985242452184489839, -697609190229762559825415, -14249666292531774436156, -221784070205669756909970, -19612786666813992009728, -25584896141781024768, -7894900273815552, -123335506765824, -118234637824, -6483584, 0, 0, 1 ] > 2^38; 274877906944 > f; X^37 + 999999999999999996523703*X^34 + 999999999999881768369463*X^33 + 999999999876664496241463*X^32 + 999999992105099729191735*X^31 + 999974415103858221982519*X^30 + 980387213333186010997559*X^29 + 778215929794330246097317*X^28 + 985750333707468228571131*X^27 + 302390809770237443181872*X^26 + 573136014757547818517448*X^25 + 790727925892117097718724*X^24 + 507834661447066013431464*X^23 + 678859425628810065908628*X^22 + 887709615750557647696223*X^21 + 976595051560919147062581*X^20 + 993565475563617357645326*X^19 + 650230424795256753739575*X^18 + 440977040017191214828156*X^17 + 482901650893223668129466*X^16 + 549130500575575762425738*X^15 + 853591301751009558398576*X^14 + 624615550959552203370740*X^13 + 364740042289535891797714*X^12 + 916004569140654234978077*X^11 + 660079223246049270055587*X^10 + 547882197645938000179264*X^9 + 270451416365675771458615*X^8 + 408199371085809383330786*X^7 + 255023373092349808791174*X^6 + 939454695197277085635458*X^5 + 626289531945333569221413*X^4 + 899111004809094153587257*X^3 + 664006641084584018157247*X^2 + 607531487417351657977321*X + 551041596559177793844261 > R := PolynomialRing(GF(p)); > Factorization(R!X^37 + 999999999999999996523703*X^34 + 999999999999881768369463*X^33 + 999999999876664496241463*X^32 + 999999992105099729191735*X^31 + 999974415103858221982519*X^30 + 980387213333186010997559*X^29 + 778215929794330246097317*X^28 + 985750333707468228571131*X^27 + 302390809770237443181872*X^26 + 573136014757547818517448*X^25 + 790727925892117097718724*X^24 + 507834661447066013431464*X^23 + 678859425628810065908628*X^22 + 887709615750557647696223*X^21 + 976595051560919147062581*X^20 + 993565475563617357645326*X^19 + 650230424795256753739575*X^18 + 440977040017191214828156*X^17 + 482901650893223668129466*X^16 + 549130500575575762425738*X^15 + 853591301751009558398576*X^14 + 624615550959552203370740*X^13 + 364740042289535891797714*X^12 + 916004569140654234978077*X^11 + 660079223246049270055587*X^10 + 547882197645938000179264*X^9 + 270451416365675771458615*X^8 + 408199371085809383330786*X^7 + 255023373092349808791174*X^6 + 939454695197277085635458*X^5 + 626289531945333569221413*X^4 + 899111004809094153587257*X^3 + 664006641084584018157247*X^2 + 607531487417351657977321*X + 551041596559177793844261); >> Factorization(R!f); ^ Runtime error in '!': Illegal coercion > Factorization(R!X^37 + 999999999999999996523703*X^34 + 999999999999881768369463*X^33 + 999999999876664496241463*X^32 + 999999992105099729191735*X^31 + 999974415103858221982519*X^30 + 980387213333186010997559*X^29 + 778215929794330246097317*X^28 + 985750333707468228571131*X^27 + 302390809770237443181872*X^26 + 573136014757547818517448*X^25 + 790727925892117097718724*X^24 + 507834661447066013431464*X^23 + 678859425628810065908628*X^22 + 887709615750557647696223*X^21 + 976595051560919147062581*X^20 + 993565475563617357645326*X^19 + 650230424795256753739575*X^18 + 440977040017191214828156*X^17 + 482901650893223668129466*X^16 + 549130500575575762425738*X^15 + 853591301751009558398576*X^14 + 624615550959552203370740*X^13 + 364740042289535891797714*X^12 + 916004569140654234978077*X^11 + 660079223246049270055587*X^10 + 547882197645938000179264*X^9 + 270451416365675771458615*X^8 + 408199371085809383330786*X^7 + 255023373092349808791174*X^6 + 939454695197277085635458*X^5 + 626289531945333569221413*X^4 + 899111004809094153587257*X^3 + 664006641084584018157247*X^2 + 607531487417351657977321*X + 551041596559177793844261); [ , ] > p; 1000000000000000003007287 > ; In file "/home/was/people/sharifi/unitcomp.m", line 226, column 21: >> if an ne 0 do ^ User error: bad syntax > f, L, K, del, tr_alpha := FindDegree37FieldMod(59); Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 59 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 28*z^35 + 6*z^34 + 38*z^33 + 18*z^32 + 18*z^30 + 45*z^29 + 44*z^28 + 57*z^27 + 45*z^26 + 28*z^25 + 38*z^24 + 57*z^23 + 8*z^22 + 8*z^21 + 44*z^20 + 6*z^19 + 6*z^18 + 44*z^17 + 8*z^16 + 8*z^15 + 57*z^14 + 38*z^13 + 28*z^12 + 45*z^11 + 57*z^10 + 44*z^9 + 45*z^8 + 18*z^7 + 18*z^5 + 38*z^4 + 6*z^3 + 28*z^2 + 21 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.210 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.880 > f; X^37 + 44*X^34 + 43*X^33 + 11*X^32 + 46*X^31 + 52*X^30 + 13*X^29 + 4*X^28 + 15*X^27 + 42*X^26 + 51*X^25 + 38*X^24 + 56*X^23 + 26*X^22 + 12*X^21 + 29*X^20 + 10*X^19 + 52*X^18 + 47*X^17 + 20*X^16 + 22*X^15 + 5*X^14 + 53*X^13 + 2*X^12 + 15*X^11 + 22*X^10 + 42*X^9 + 39*X^8 + 57*X^7 + 42*X^6 + 7*X^5 + 49*X^4 + 13*X^3 + 18*X^2 + 48*X + 30 > f, L, K, del, tr_alpha := FindDegree37FieldMod(59); Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 59 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 28*z^35 + 6*z^34 + 38*z^33 + 18*z^32 + 18*z^30 + 45*z^29 + 44*z^28 + 57*z^27 + 45*z^26 + 28*z^25 + 38*z^24 + 57*z^23 + 8*z^22 + 8*z^21 + 44*z^20 + 6*z^19 + 6*z^18 + 44*z^17 + 8*z^16 + 8*z^15 + 57*z^14 + 38*z^13 + 28*z^12 + 45*z^11 + 57*z^10 + 44*z^9 + 45*z^8 + 18*z^7 + 18*z^5 + 38*z^4 + 6*z^3 + 28*z^2 + 21 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.210 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.860 > f, L, K, del, tr_alpha := FindDegree37FieldMod(11); Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 11 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 8*z^35 + 2*z^34 + 7*z^33 + 10*z^32 + 10*z^30 + 3*z^29 + 5*z^28 + 9*z^27 + 3*z^26 + 8*z^25 + 7*z^24 + 9*z^23 + 7*z^22 + 7*z^21 + 5*z^20 + 2*z^19 + 2*z^18 + 5*z^17 + 7*z^16 + 7*z^15 + 9*z^14 + 7*z^13 + 8*z^12 + 3*z^11 + 9*z^10 + 5*z^9 + 3*z^8 + 10*z^7 + 10*z^5 + 7*z^4 + 2*z^3 + 8*z^2 + 6 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.200 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] [Interrupt twice in half a second; exiting] Total time: 119.979 seconds [was@modular sharifi]$ me Magma V2.8-10 Mon Apr 8 2002 17:53:51 on modular [Seed = 1051726699] Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/emacs.m" Loading "/home/was/magma/local/init.m" > ell := 59; p := 37; K := GF(ell^(p-1)); > K; Finite field of size 59^36 > a := K.1; > Order(a); 312895929945977139981054709770674547645517193154968407535645880 > z := a^(Order(a) div 37); > z; 14*K.1^35 + 37*K.1^34 + 49*K.1^33 + 19*K.1^32 + 15*K.1^31 + 17*K.1^30 + 14*K.1^29 + 45*K.1^28 + 45*K.1^27 + 52*K.1^26 + 42*K.1^25 + 12*K.1^24 + 34*K.1^23 + 20*K.1^22 + 25*K.1^21 + 56*K.1^20 + 50*K.1^19 + 45*K.1^18 + 7*K.1^17 + 4*K.1^16 + 42*K.1^15 + 38*K.1^14 + 17*K.1^13 + 39*K.1^12 + 26*K.1^11 + 16*K.1^10 + 10*K.1^9 + 12*K.1^8 + K.1^7 + 6*K.1^6 + 9*K.1^5 + 20*K.1^4 + 27*K.1^3 + 20*K.1^2 + 45*K.1 + 19 > Order(z); 37 > p := NextPrime(p*1000); while not IsIrreducible(PolynomialRing(GF(p))!f) do p := NextPrime(p); end while; >> p := NextPrime(p*1000); while not IsIrreducible(PolynomialRing(GF(p))!f) do ^ User error: Identifier 'f' has not been declared or assigned > p := NextPrime(p*1000000); while not IsIrreducible(PolynomialRing(GF(p))!f) do p := NextPrime(p); end while; >> p := NextPrime(p*1000000); while not IsIrreducible(PolynomialRing(GF(p))!f ^ User error: Identifier 'f' has not been declared or assigned > > ell := NextPrime(100000000000); while not IsIrreducible(PolynomialRing(GF(ell))!f) do ell := NextPrime(ell); end while; > ell := 59; > K; Finite field of size 59^36 > f := CyclotomicPolynomial(37); p := NextPrime(10^18); ell := NextPrime(100000000000); > f; $.1^36 + $.1^35 + $.1^34 + $.1^33 + $.1^32 + $.1^31 + $.1^30 + $.1^29 + $.1^28 + $.1^27 + $.1^26 + $.1^25 + $.1^24 + $.1^23 + $.1^22 + $.1^21 + $.1^20 + $.1^19 + $.1^18 + $.1^17 + $.1^16 + $.1^15 + $.1^14 + $.1^13 + $.1^12 + $.1^11 + $.1^10 + $.1^9 + $.1^8 + $.1^7 + $.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1 > p := 37; K := GF(ell^(p-1)); > K; Finite field of size 100000000063^36 > R := PolynomialRing(K); > g := R!f; > Roots(g); [Interrupt twice in half a second; exiting] Total time: 44.739 seconds [was@modular sharifi]$ me Magma V2.8-10 Mon Apr 8 2002 17:58:14 on modular [Seed = 445920342] Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/emacs.m" Loading "/home/was/magma/local/init.m" > > Attach("unitcomp.m"); > f, L, K, del, tr_alpha := FindDegree37FieldMod(59); Forming extension L. Time: 0.010 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 59 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 28*z^35 + 6*z^34 + 38*z^33 + 18*z^32 + 18*z^30 + 45*z^29 + 44*z^28 + 57*z^27 + 45*z^26 + 28*z^25 + 38*z^24 + 57*z^23 + 8*z^22 + 8*z^21 + 44*z^20 + 6*z^19 + 6*z^18 + 44*z^17 + 8*z^16 + 8*z^15 + 57*z^14 + 38*z^13 + 28*z^12 + 45*z^11 + 57*z^10 + 44*z^9 + 45*z^8 + 18*z^7 + 18*z^5 + 38*z^4 + 6*z^3 + 28*z^2 + 21 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.210 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.920 > f; X^37 + 44*X^34 + 43*X^33 + 11*X^32 + 46*X^31 + 52*X^30 + 13*X^29 + 4*X^28 + 15*X^27 + 42*X^26 + 51*X^25 + 38*X^24 + 56*X^23 + 26*X^22 + 12*X^21 + 29*X^20 + 10*X^19 + 52*X^18 + 47*X^17 + 20*X^16 + 22*X^15 + 5*X^14 + 53*X^13 + 2*X^12 + 15*X^11 + 22*X^10 + 42*X^9 + 39*X^8 + 57*X^7 + 42*X^6 + 7*X^5 + 49*X^4 + 13*X^3 + 18*X^2 + 48*X + 30 > > > > > f; In file "/home/was/people/sharifi/unitcomp.m", line 367, column 5: >> end function; ^ User error: bad syntax X^37 + 44*X^34 + 43*X^33 + 11*X^32 + 46*X^31 + 52*X^30 + 13*X^29 + 4*X^28 + 15*X^27 + 42*X^26 + 51*X^25 + 38*X^24 + 56*X^23 + 26*X^22 + 12*X^21 + 29*X^20 + 10*X^19 + 52*X^18 + 47*X^17 + 20*X^16 + 22*X^15 + 5*X^14 + 53*X^13 + 2*X^12 + 15*X^11 + 22*X^10 + 42*X^9 + 39*X^8 + 57*X^7 + 42*X^6 + 7*X^5 + 49*X^4 + 13*X^3 + 18*X^2 + 48*X + 30 > a := Coefficient(f,34); > a; 44 > Parent(a); Univariate Quotient Polynomial Algebra in z over Finite field of size 59 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 > GF(59)!a; >> GF(59)!a; ^ Runtime error in '!': Illegal coercion LHS: FldFin RHS: RngUPolResElt > Degree(a); 0 > Degree(Coefficient(f,36)); -1 > ; In file "/home/was/people/sharifi/unitcomp.m", line 373, column 5: >> end function; ^ User error: bad syntax > ; > time g, L, K, del, tr_alpha := FindDegree37FieldMod(59); > time g, L, K, del, tr_alpha := FindDegree37FieldMod(59); In file "/home/was/people/sharifi/unitcomp.m", line 326, column 30: >> S := PolynomialRing(GF(ell)); ^ Runtime error: Undefined reference 'ell' in package "/home/was/people/sharifi/unitcomp.m" > ; > time g, L, K, del, tr_alpha := FindDegree37FieldMod(59); Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 59 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 28*z^35 + 6*z^34 + 38*z^33 + 18*z^32 + 18*z^30 + 45*z^29 + 44*z^28 + 57*z^27 + 45*z^26 + 28*z^25 + 38*z^24 + 57*z^23 + 8*z^22 + 8*z^21 + 44*z^20 + 6*z^19 + 6*z^18 + 44*z^17 + 8*z^16 + 8*z^15 + 57*z^14 + 38*z^13 + 28*z^12 + 45*z^11 + 57*z^10 + 44*z^9 + 45*z^8 + 18*z^7 + 18*z^5 + 38*z^4 + 6*z^3 + 28*z^2 + 21 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.210 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.960 f = X^37 + 44*X^34 + 43*X^33 + 11*X^32 + 46*X^31 + 52*X^30 + 13*X^29 + 4*X^28 + 15*X^27 + 42*X^26 + 51*X^25 + 38*X^24 + 56*X^23 + 26*X^22 + 12*X^21 + 29*X^20 + 10*X^19 + 52*X^18 + 47*X^17 + 20*X^16 + 22*X^15 + 5*X^14 + 53*X^13 + 2*X^12 + 15*X^11 + 22*X^10 + 42*X^9 + 39*X^8 + 57*X^7 + 42*X^6 + 7*X^5 + 49*X^4 + 13*X^3 + 18*X^2 + 48*X + 30 g = T^37 + 44*T^34 + 43*T^33 + 11*T^32 + 46*T^31 + 52*T^30 + 13*T^29 + 4*T^28 + 15*T^27 + 42*T^26 + 51*T^25 + 38*T^24 + 56*T^23 + 26*T^22 + 12*T^21 + 29*T^20 + 10*T^19 + 52*T^18 + 47*T^17 + 20*T^16 + 22*T^15 + 5*T^14 + 53*T^13 + 2*T^12 + 15*T^11 + 22*T^10 + 42*T^9 + 39*T^8 + 57*T^7 + 42*T^6 + 7*T^5 + 49*T^4 + 13*T^3 + 18*T^2 + 48*T + 30 Time: 2.200 > Parent(g); Univariate Polynomial Ring in T over GF(59) > ; > time g := Find37FieldCRT(200); Find37FieldCRT( stop: 200 ) In file "/home/was/people/sharifi/unitcomp.m", line 353, column 29: >> f := FindDegree37Field(ell); ^ Runtime error in 'FindDegree37Field': Bad argument types Argument types given: RngIntElt > time g := Find37FieldCRT(200); Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 13 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 11*z^35 + z^34 + 11*z^33 + 3*z^32 + 3*z^30 + 4*z^29 + 12*z^28 + 4*z^26 + 11*z^25 + 11*z^24 + 5*z^22 + 5*z^21 + 12*z^20 + z^19 + z^18 + 12*z^17 + 5*z^16 + 5*z^15 + 11*z^13 + 11*z^12 + 4*z^11 + 12*z^9 + 4*z^8 + 3*z^7 + 3*z^5 + 11*z^4 + z^3 + 11*z^2 + 9 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.200 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.820 f = X^37 + 10*X^34 + 9*X^33 + 11*X^32 + 3*X^31 + 8*X^30 + 7*X^29 + 3*X^28 + X^27 + X^26 + 7*X^25 + 3*X^24 + 3*X^22 + 10*X^21 + 11*X^20 + 12*X^19 + 12*X^18 + 11*X^17 + 3*X^16 + 9*X^15 + 11*X^14 + 3*X^13 + 5*X^12 + 2*X^11 + 11*X^10 + 7*X^9 + X^7 + 8*X^6 + 5*X^5 + 3*X^4 + 3*X^3 + 5*X + 9 g = T^37 + 10*T^34 + 9*T^33 + 11*T^32 + 3*T^31 + 8*T^30 + 7*T^29 + 3*T^28 + T^27 + T^26 + 7*T^25 + 3*T^24 + 3*T^22 + 10*T^21 + 11*T^20 + 12*T^19 + 12*T^18 + 11*T^17 + 3*T^16 + 9*T^15 + 11*T^14 + 3*T^13 + 5*T^12 + 2*T^11 + 11*T^10 + 7*T^9 + T^7 + 8*T^6 + 5*T^5 + 3*T^4 + 3*T^3 + 5*T + 9 Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 17 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 16*z^35 + 6*z^34 + 6*z^33 + 10*z^32 + 10*z^30 + 9*z^29 + 12*z^28 + 13*z^27 + 9*z^26 + 16*z^25 + 6*z^24 + 13*z^23 + 13*z^22 + 13*z^21 + 12*z^20 + 6*z^19 + 6*z^18 + 12*z^17 + 13*z^16 + 13*z^15 + 13*z^14 + 6*z^13 + 16*z^12 + 9*z^11 + 13*z^10 + 12*z^9 + 9*z^8 + 10*z^7 + 10*z^5 + 6*z^4 + 6*z^3 + 16*z^2 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.230 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.840 f = X^37 + 12*X^34 + 8*X^33 + 11*X^32 + 11*X^31 + X^30 + 6*X^28 + 16*X^27 + 4*X^26 + 14*X^25 + 14*X^24 + 6*X^23 + 13*X^22 + 4*X^21 + 2*X^20 + 12*X^19 + 2*X^18 + 5*X^17 + 4*X^16 + 9*X^15 + 8*X^14 + 2*X^13 + 9*X^12 + 5*X^11 + 10*X^10 + 4*X^9 + 12*X^8 + 8*X^6 + 14*X^5 + 4*X^4 + 12*X^3 + 11*X^2 + 6*X + 3 g = T^37 + 12*T^34 + 8*T^33 + 11*T^32 + 11*T^31 + T^30 + 6*T^28 + 16*T^27 + 4*T^26 + 14*T^25 + 14*T^24 + 6*T^23 + 13*T^22 + 4*T^21 + 2*T^20 + 12*T^19 + 2*T^18 + 5*T^17 + 4*T^16 + 9*T^15 + 8*T^14 + 2*T^13 + 9*T^12 + 5*T^11 + 10*T^10 + 4*T^9 + 12*T^8 + 8*T^6 + 14*T^5 + 4*T^4 + 12*T^3 + 11*T^2 + 6*T + 3 Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 19 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 3*z^35 + 4*z^34 + 4*z^33 + 5*z^32 + 5*z^30 + 13*z^29 + 7*z^27 + 13*z^26 + 3*z^25 + 4*z^24 + 7*z^23 + 3*z^22 + 3*z^21 + 4*z^19 + 4*z^18 + 3*z^16 + 3*z^15 + 7*z^14 + 4*z^13 + 3*z^12 + 13*z^11 + 7*z^10 + 13*z^8 + 5*z^7 + 5*z^5 + 4*z^4 + 4*z^3 + 3*z^2 + 17 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.200 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.740 f = X^37 + 14*X^34 + X^33 + 16*X^32 + 4*X^31 + 7*X^30 + 18*X^29 + 16*X^28 + 8*X^26 + 18*X^25 + 12*X^23 + 18*X^22 + 13*X^21 + 3*X^20 + 5*X^19 + 10*X^18 + 2*X^17 + 15*X^16 + 9*X^15 + 10*X^14 + X^13 + 11*X^12 + 11*X^11 + 5*X^10 + 16*X^9 + 11*X^8 + 5*X^7 + 2*X^6 + 3*X^3 + 8*X^2 + 11*X + 1 g = T^37 + 14*T^34 + T^33 + 16*T^32 + 4*T^31 + 7*T^30 + 18*T^29 + 16*T^28 + 8*T^26 + 18*T^25 + 12*T^23 + 18*T^22 + 13*T^21 + 3*T^20 + 5*T^19 + 10*T^18 + 2*T^17 + 15*T^16 + 9*T^15 + 10*T^14 + T^13 + 11*T^12 + 11*T^11 + 5*T^10 + 16*T^9 + 11*T^8 + 5*T^7 + 2*T^6 + 3*T^3 + 8*T^2 + 11*T + 1 Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 59 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 28*z^35 + 6*z^34 + 38*z^33 + 18*z^32 + 18*z^30 + 45*z^29 + 44*z^28 + 57*z^27 + 45*z^26 + 28*z^25 + 38*z^24 + 57*z^23 + 8*z^22 + 8*z^21 + 44*z^20 + 6*z^19 + 6*z^18 + 44*z^17 + 8*z^16 + 8*z^15 + 57*z^14 + 38*z^13 + 28*z^12 + 45*z^11 + 57*z^10 + 44*z^9 + 45*z^8 + 18*z^7 + 18*z^5 + 38*z^4 + 6*z^3 + 28*z^2 + 21 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.210 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.900 f = X^37 + 44*X^34 + 43*X^33 + 11*X^32 + 46*X^31 + 52*X^30 + 13*X^29 + 4*X^28 + 15*X^27 + 42*X^26 + 51*X^25 + 38*X^24 + 56*X^23 + 26*X^22 + 12*X^21 + 29*X^20 + 10*X^19 + 52*X^18 + 47*X^17 + 20*X^16 + 22*X^15 + 5*X^14 + 53*X^13 + 2*X^12 + 15*X^11 + 22*X^10 + 42*X^9 + 39*X^8 + 57*X^7 + 42*X^6 + 7*X^5 + 49*X^4 + 13*X^3 + 18*X^2 + 48*X + 30 g = T^37 + 44*T^34 + 43*T^33 + 11*T^32 + 46*T^31 + 52*T^30 + 13*T^29 + 4*T^28 + 15*T^27 + 42*T^26 + 51*T^25 + 38*T^24 + 56*T^23 + 26*T^22 + 12*T^21 + 29*T^20 + 10*T^19 + 52*T^18 + 47*T^17 + 20*T^16 + 22*T^15 + 5*T^14 + 53*T^13 + 2*T^12 + 15*T^11 + 22*T^10 + 42*T^9 + 39*T^8 + 57*T^7 + 42*T^6 + 7*T^5 + 49*T^4 + 13*T^3 + 18*T^2 + 48*T + 30 Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 61 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 3*z^35 + 51*z^34 + 17*z^33 + 36*z^32 + 36*z^30 + 51*z^29 + 50*z^28 + 13*z^27 + 51*z^26 + 3*z^25 + 17*z^24 + 13*z^23 + 44*z^22 + 44*z^21 + 50*z^20 + 51*z^19 + 51*z^18 + 50*z^17 + 44*z^16 + 44*z^15 + 13*z^14 + 17*z^13 + 3*z^12 + 51*z^11 + 13*z^10 + 50*z^9 + 51*z^8 + 36*z^7 + 36*z^5 + 17*z^4 + 51*z^3 + 3*z^2 + 42 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.220 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.990 f = X^37 + 45*X^34 + 48*X^33 + 55*X^32 + 2*X^31 + X^30 + 39*X^29 + 60*X^28 + 55*X^27 + 53*X^26 + 46*X^25 + 37*X^24 + 19*X^23 + 43*X^22 + 51*X^21 + 43*X^20 + 54*X^19 + 19*X^18 + 47*X^17 + 27*X^16 + 42*X^15 + 18*X^14 + 26*X^13 + 25*X^12 + X^11 + 10*X^10 + 38*X^9 + 17*X^8 + 15*X^7 + 29*X^6 + 3*X^5 + 14*X^4 + 5*X^3 + 8*X^2 + 38*X + 27 g = T^37 + 45*T^34 + 48*T^33 + 55*T^32 + 2*T^31 + T^30 + 39*T^29 + 60*T^28 + 55*T^27 + 53*T^26 + 46*T^25 + 37*T^24 + 19*T^23 + 43*T^22 + 51*T^21 + 43*T^20 + 54*T^19 + 19*T^18 + 47*T^17 + 27*T^16 + 42*T^15 + 18*T^14 + 26*T^13 + 25*T^12 + T^11 + 10*T^10 + 38*T^9 + 17*T^8 + 15*T^7 + 29*T^6 + 3*T^5 + 14*T^4 + 5*T^3 + 8*T^2 + 38*T + 27 CRT step -- 10.52 So far: [ -4493233, 842509, -1752400, -24944, -2771948, -7176281, 5746229, 2916486, -2363855, 3012523, 1756688, -4332768, -1729996, 1956906, -3496807, 575272, 7437147, -7140369, -2725217, 7016762, 5020221, 2786287, -3680453, -5992987, -1650074, 2017438, 6425732, -5280898, 6969615, 1132931, 4437324, -4533396, 1499496, 3222800, -6483584, 0, 0, 1 ] Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 79 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 29*z^35 + 55*z^34 + 48*z^33 + 8*z^32 + 8*z^30 + 59*z^29 + 2*z^28 + 73*z^27 + 59*z^26 + 29*z^25 + 48*z^24 + 73*z^23 + 12*z^22 + 12*z^21 + 2*z^20 + 55*z^19 + 55*z^18 + 2*z^17 + 12*z^16 + 12*z^15 + 73*z^14 + 48*z^13 + 29*z^12 + 59*z^11 + 73*z^10 + 2*z^9 + 59*z^8 + 8*z^7 + 8*z^5 + 48*z^4 + 55*z^3 + 29*z^2 + 26 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.250 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.990 f = X^37 + 25*X^34 + 70*X^33 + 13*X^32 + 53*X^31 + 32*X^30 + 39*X^29 + 33*X^28 + 76*X^27 + 62*X^26 + 53*X^25 + 40*X^24 + 21*X^23 + 67*X^22 + 49*X^21 + 34*X^20 + 23*X^19 + 78*X^18 + 44*X^17 + 60*X^16 + 51*X^15 + 39*X^14 + 59*X^13 + 21*X^12 + 32*X^11 + 25*X^10 + 43*X^9 + 34*X^8 + 78*X^7 + 39*X^6 + 62*X^5 + 74*X^3 + 32*X^2 + 27*X + 60 g = T^37 + 25*T^34 + 70*T^33 + 13*T^32 + 53*T^31 + 32*T^30 + 39*T^29 + 33*T^28 + 76*T^27 + 62*T^26 + 53*T^25 + 40*T^24 + 21*T^23 + 67*T^22 + 49*T^21 + 34*T^20 + 23*T^19 + 78*T^18 + 44*T^17 + 60*T^16 + 51*T^15 + 39*T^14 + 59*T^13 + 21*T^12 + 32*T^11 + 25*T^10 + 43*T^9 + 34*T^8 + 78*T^7 + 39*T^6 + 62*T^5 + 74*T^3 + 32*T^2 + 27*T + 60 Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 89 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 40*z^35 + z^34 + 21*z^33 + 3*z^32 + 3*z^30 + 35*z^29 + 77*z^27 + 35*z^26 + 40*z^25 + 21*z^24 + 77*z^23 + 32*z^22 + 32*z^21 + z^19 + z^18 + 32*z^16 + 32*z^15 + 77*z^14 + 21*z^13 + 40*z^12 + 35*z^11 + 77*z^10 + 35*z^8 + 3*z^7 + 3*z^5 + 21*z^4 + z^3 + 40*z^2 + 63 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.110 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 0.960 f = X^37 + 66*X^34 + 29*X^33 + 4*X^32 + 5*X^31 + 2*X^30 + 46*X^29 + 69*X^28 + 55*X^27 + 47*X^26 + 79*X^25 + 78*X^24 + 2*X^23 + 24*X^22 + 83*X^21 + 64*X^20 + 33*X^19 + 42*X^18 + 80*X^17 + 35*X^16 + 56*X^15 + 42*X^14 + 29*X^13 + 34*X^12 + 2*X^11 + X^10 + 40*X^9 + 47*X^8 + 84*X^7 + 55*X^6 + 16*X^5 + 61*X^4 + 38*X^3 + 31*X^2 + 63*X + 84 g = T^37 + 66*T^34 + 29*T^33 + 4*T^32 + 5*T^31 + 2*T^30 + 46*T^29 + 69*T^28 + 55*T^27 + 47*T^26 + 79*T^25 + 78*T^24 + 2*T^23 + 24*T^22 + 83*T^21 + 64*T^20 + 33*T^19 + 42*T^18 + 80*T^17 + 35*T^16 + 56*T^15 + 42*T^14 + 29*T^13 + 34*T^12 + 2*T^11 + T^10 + 40*T^9 + 47*T^8 + 84*T^7 + 55*T^6 + 16*T^5 + 61*T^4 + 38*T^3 + 31*T^2 + 63*T + 84 Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 109 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 13*z^35 + 67*z^34 + 106*z^33 + 3*z^32 + 3*z^30 + 11*z^29 + 38*z^28 + 52*z^27 + 11*z^26 + 13*z^25 + 106*z^24 + 52*z^23 + 74*z^22 + 74*z^21 + 38*z^20 + 67*z^19 + 67*z^18 + 38*z^17 + 74*z^16 + 74*z^15 + 52*z^14 + 106*z^13 + 13*z^12 + 11*z^11 + 52*z^10 + 38*z^9 + 11*z^8 + 3*z^7 + 3*z^5 + 106*z^4 + 67*z^3 + 13*z^2 + 22 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.110 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 0.970 f = X^37 + 63*X^34 + 8*X^33 + 40*X^32 + 100*X^31 + 67*X^30 + 89*X^29 + 20*X^28 + 80*X^27 + 66*X^26 + X^25 + 56*X^24 + 103*X^23 + 6*X^22 + 15*X^21 + 30*X^20 + 91*X^19 + 11*X^18 + 69*X^17 + 64*X^16 + 26*X^15 + 19*X^14 + 2*X^13 + 43*X^12 + 68*X^11 + 67*X^10 + 83*X^9 + 41*X^8 + 38*X^7 + 27*X^6 + 38*X^5 + 81*X^4 + 3*X^3 + X^2 + 75*X + 33 g = T^37 + 63*T^34 + 8*T^33 + 40*T^32 + 100*T^31 + 67*T^30 + 89*T^29 + 20*T^28 + 80*T^27 + 66*T^26 + T^25 + 56*T^24 + 103*T^23 + 6*T^22 + 15*T^21 + 30*T^20 + 91*T^19 + 11*T^18 + 69*T^17 + 64*T^16 + 26*T^15 + 19*T^14 + 2*T^13 + 43*T^12 + 68*T^11 + 67*T^10 + 83*T^9 + 41*T^8 + 38*T^7 + 27*T^6 + 38*T^5 + 81*T^4 + 3*T^3 + T^2 + 75*T + 33 Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 113 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 58*z^35 + 24*z^34 + 91*z^33 + 49*z^32 + 49*z^30 + 86*z^29 + 102*z^28 + 56*z^27 + 86*z^26 + 58*z^25 + 91*z^24 + 56*z^23 + 56*z^22 + 56*z^21 + 102*z^20 + 24*z^19 + 24*z^18 + 102*z^17 + 56*z^16 + 56*z^15 + 56*z^14 + 91*z^13 + 58*z^12 + 86*z^11 + 56*z^10 + 102*z^9 + 86*z^8 + 49*z^7 + 49*z^5 + 91*z^4 + 24*z^3 + 58*z^2 + 90 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.130 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 0.960 f = X^37 + 17*X^34 + 53*X^33 + 61*X^32 + 67*X^31 + 73*X^30 + 66*X^29 + 92*X^28 + 28*X^27 + 46*X^26 + 40*X^25 + 75*X^24 + X^23 + 58*X^22 + 3*X^21 + 20*X^20 + 63*X^19 + 63*X^18 + 100*X^17 + 55*X^16 + 7*X^15 + 81*X^14 + 86*X^13 + 87*X^12 + 99*X^11 + 79*X^10 + 74*X^9 + 23*X^8 + 100*X^7 + 109*X^6 + 89*X^5 + 75*X^4 + 95*X^3 + 81*X^2 + 22*X + 73 g = T^37 + 17*T^34 + 53*T^33 + 61*T^32 + 67*T^31 + 73*T^30 + 66*T^29 + 92*T^28 + 28*T^27 + 46*T^26 + 40*T^25 + 75*T^24 + T^23 + 58*T^22 + 3*T^21 + 20*T^20 + 63*T^19 + 63*T^18 + 100*T^17 + 55*T^16 + 7*T^15 + 81*T^14 + 86*T^13 + 87*T^12 + 99*T^11 + 79*T^10 + 74*T^9 + 23*T^8 + 100*T^7 + 109*T^6 + 89*T^5 + 75*T^4 + 95*T^3 + 81*T^2 + 22*T + 73 Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 131 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 83*z^35 + 70*z^34 + 100*z^33 + 61*z^32 + 61*z^30 + 90*z^29 + 102*z^28 + 100*z^27 + 90*z^26 + 83*z^25 + 100*z^24 + 100*z^23 + 16*z^22 + 16*z^21 + 102*z^20 + 70*z^19 + 70*z^18 + 102*z^17 + 16*z^16 + 16*z^15 + 100*z^14 + 100*z^13 + 83*z^12 + 90*z^11 + 100*z^10 + 102*z^9 + 90*z^8 + 61*z^7 + 61*z^5 + 100*z^4 + 70*z^3 + 83*z^2 + 48 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.120 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 0.990 f = X^37 + 130*X^34 + 104*X^33 + 36*X^32 + 64*X^31 + 42*X^30 + 130*X^29 + 96*X^28 + 32*X^27 + 86*X^26 + 72*X^25 + 86*X^24 + 3*X^23 + 58*X^22 + 113*X^21 + X^20 + 11*X^19 + 13*X^18 + 41*X^17 + 119*X^16 + 24*X^15 + 106*X^14 + 99*X^13 + 113*X^12 + 107*X^11 + 7*X^10 + 90*X^9 + 33*X^8 + 103*X^7 + 74*X^6 + 49*X^5 + 88*X^4 + 10*X^3 + 40*X^2 + 118*X + 123 g = T^37 + 130*T^34 + 104*T^33 + 36*T^32 + 64*T^31 + 42*T^30 + 130*T^29 + 96*T^28 + 32*T^27 + 86*T^26 + 72*T^25 + 86*T^24 + 3*T^23 + 58*T^22 + 113*T^21 + T^20 + 11*T^19 + 13*T^18 + 41*T^17 + 119*T^16 + 24*T^15 + 106*T^14 + 99*T^13 + 113*T^12 + 107*T^11 + 7*T^10 + 90*T^9 + 33*T^8 + 103*T^7 + 74*T^6 + 49*T^5 + 88*T^4 + 10*T^3 + 40*T^2 + 118*T + 123 CRT step -- 6.75 So far: [ -80836533286084697, 66975563345854857, 11386057713100772, -71805543500898880, -17719410116834202, -85606231403767368, -81015051182520668, -28049887250629581, 54808687005980427, -17903549832058169, 82545037454991174, -40280302088041351, -60413319556376730, 31306766626319811, -34924345508252449, 81856318539096058, -64127040240441591, -82701333283509215, 84917620727847933, -45619682822865711, 62947702402846571, 16391888204680362, 68382141997723193, -17342592435595375, 69744392611215208, -6953203077586616, 63129662250715074, -40796532230514306, 71847730592145342, 8296194355663598, -39812753188183955, -7894900273815552, -123335506765824, -118234637824, -6483584, 0, 0, 1 ] Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 163 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 135*z^35 + 143*z^34 + 37*z^33 + 144*z^32 + 144*z^30 + 27*z^29 + 139*z^27 + 27*z^26 + 135*z^25 + 37*z^24 + 139*z^23 + 109*z^22 + 109*z^21 + 143*z^19 + 143*z^18 + 109*z^16 + 109*z^15 + 139*z^14 + 37*z^13 + 135*z^12 + 27*z^11 + 139*z^10 + 27*z^8 + 144*z^7 + 144*z^5 + 37*z^4 + 143*z^3 + 135*z^2 + 59 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.110 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.000 f = X^37 + 67*X^34 + 127*X^33 + 69*X^32 + 100*X^31 + 58*X^30 + 160*X^29 + 40*X^28 + 2*X^27 + 89*X^26 + 112*X^25 + 9*X^24 + 25*X^23 + 9*X^22 + 131*X^21 + 74*X^20 + 17*X^19 + 97*X^18 + X^17 + 64*X^16 + 76*X^15 + 16*X^14 + 41*X^13 + 78*X^12 + 132*X^11 + 43*X^10 + 7*X^9 + 135*X^8 + 96*X^7 + 25*X^6 + 76*X^5 + 9*X^4 + 148*X^3 + 158*X^2 + 18*X + 123 g = T^37 + 67*T^34 + 127*T^33 + 69*T^32 + 100*T^31 + 58*T^30 + 160*T^29 + 40*T^28 + 2*T^27 + 89*T^26 + 112*T^25 + 9*T^24 + 25*T^23 + 9*T^22 + 131*T^21 + 74*T^20 + 17*T^19 + 97*T^18 + T^17 + 64*T^16 + 76*T^15 + 16*T^14 + 41*T^13 + 78*T^12 + 132*T^11 + 43*T^10 + 7*T^9 + 135*T^8 + 96*T^7 + 25*T^6 + 76*T^5 + 9*T^4 + 148*T^3 + 158*T^2 + 18*T + 123 Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 167 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 145*z^35 + 102*z^34 + 66*z^33 + 11*z^32 + 11*z^30 + 165*z^29 + 19*z^28 + 16*z^27 + 165*z^26 + 145*z^25 + 66*z^24 + 16*z^23 + 79*z^22 + 79*z^21 + 19*z^20 + 102*z^19 + 102*z^18 + 19*z^17 + 79*z^16 + 79*z^15 + 16*z^14 + 66*z^13 + 145*z^12 + 165*z^11 + 16*z^10 + 19*z^9 + 165*z^8 + 11*z^7 + 11*z^5 + 66*z^4 + 102*z^3 + 145*z^2 + 21 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.120 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 1.000 f = X^37 + 24*X^34 + 124*X^33 + 62*X^32 + 120*X^31 + 36*X^30 + 127*X^29 + 41*X^28 + 87*X^27 + 52*X^26 + 57*X^25 + 94*X^24 + 126*X^23 + 10*X^22 + 139*X^21 + 127*X^20 + 118*X^19 + 63*X^18 + 55*X^17 + 121*X^16 + 87*X^15 + 166*X^14 + 133*X^13 + 14*X^12 + 78*X^11 + 110*X^10 + 14*X^9 + 150*X^8 + 130*X^7 + 27*X^6 + 42*X^5 + 17*X^4 + 164*X^3 + 79*X^2 + 92*X + 134 g = T^37 + 24*T^34 + 124*T^33 + 62*T^32 + 120*T^31 + 36*T^30 + 127*T^29 + 41*T^28 + 87*T^27 + 52*T^26 + 57*T^25 + 94*T^24 + 126*T^23 + 10*T^22 + 139*T^21 + 127*T^20 + 118*T^19 + 63*T^18 + 55*T^17 + 121*T^16 + 87*T^15 + 166*T^14 + 133*T^13 + 14*T^12 + 78*T^11 + 110*T^10 + 14*T^9 + 150*T^8 + 130*T^7 + 27*T^6 + 42*T^5 + 17*T^4 + 164*T^3 + 79*T^2 + 92*T + 134 Forming extension L. Time: 0.000 L = Univariate Quotient Polynomial Algebra in w over Univariate Quotient Polynomial Algebra in z over Finite field of size 227 with modulus z^36 + z^35 + z^34 + z^33 + z^32 + z^31 + z^30 + z^29 + z^28 + z^27 + z^26 + z^25 + z^24 + z^23 + z^22 + z^21 + z^20 + z^19 + z^18 + z^17 + z^16 + z^15 + z^14 + z^13 + z^12 + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 with modulus w^37 + 108*z^35 + 206*z^34 + 5*z^33 + 190*z^32 + 190*z^30 + 31*z^29 + 143*z^28 + 142*z^27 + 31*z^26 + 108*z^25 + 5*z^24 + 142*z^23 + 185*z^22 + 185*z^21 + 143*z^20 + 206*z^19 + 206*z^18 + 143*z^17 + 185*z^16 + 185*z^15 + 142*z^14 + 5*z^13 + 108*z^12 + 31*z^11 + 142*z^10 + 143*z^9 + 31*z^8 + 190*z^7 + 190*z^5 + 5*z^4 + 206*z^3 + 108*z^2 + 74 Computing trace of random element (alpha = w ) 34 steps: 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, Time: 0.110 Finished computing trace of random element. Computing minimal polynomial of trace(alpha), as an element of K[X] Time: 0.990 f = X^37 + 217*X^34 + 148*X^33 + 2*X^32 + 15*X^31 + 133*X^30 + 138*X^29 + 192*X^28 + 8*X^27 + 124*X^26 + 218*X^25 + 79*X^24 + 199*X^23 + 99*X^22 + 193*X^21 + 212*X^20 + 185*X^19 + 226*X^18 + 112*X^17 + 144*X^16 + 22*X^15 + 171*X^14 + 98*X^13 + 213*X^12 + 208*X^11 + 13*X^10 + 150*X^9 + 191*X^8 + 126*X^7 + 93*X^6 + 74*X^5 + 57*X^4 + 197*X^3 + 203*X^2 + 59*X + 101 g = T^37 + 217*T^34 + 148*T^33 + 2*T^32 + 15*T^31 + 133*T^30 + 138*T^29 + 192*T^28 + 8*T^27 + 124*T^26 + 218*T^25 + 79*T^24 + 199*T^23 + 99*T^22 + 193*T^21 + 212*T^20 + 185*T^19 + 226*T^18 + 112*T^17 + 144*T^16 + 22*T^15 + 171*T^14 + 98*T^13 + 213*T^12 + 208*T^11 + 13*T^10 + 150*T^9 + 191*T^8 + 126*T^7 + 93*T^6 + 74*T^5 + 57*T^4 + 197*T^3 + 203*T^2 + 59*T + 101 Found it!!! Time: 20.690 > g; $.1^37 - 6483584*$.1^34 - 118234637824*$.1^33 - 123335506765824*$.1^32 - 7894900273815552*$.1^31 - 39812753188183955*$.1^30 + 8296194355663598*$.1^29 + 71847730592145342*$.1^28 - 40796532230514306*$.1^27 + 63129662250715074*$.1^26 - 6953203077586616*$.1^25 + 69744392611215208*$.1^24 - 17342592435595375*$.1^23 + 68382141997723193*$.1^22 + 16391888204680362*$.1^21 + 62947702402846571*$.1^20 - 45619682822865711*$.1^19 + 84917620727847933*$.1^18 - 82701333283509215*$.1^17 - 64127040240441591*$.1^16 + 81856318539096058*$.1^15 - 34924345508252449*$.1^14 + 31306766626319811*$.1^13 - 60413319556376730*$.1^12 - 40280302088041351*$.1^11 + 82545037454991174*$.1^10 - 17903549832058169*$.1^9 + 54808687005980427*$.1^8 - 28049887250629581*$.1^7 - 81015051182520668*$.1^6 - 85606231403767368*$.1^5 - 17719410116834202*$.1^4 - 71805543500898880*$.1^3 + 11386057713100772*$.1^2 + 66975563345854857*$.1 - 80836533286084697 > R := PolynomialRing(Q); > g; $.1^37 - 6483584*$.1^34 - 118234637824*$.1^33 - 123335506765824*$.1^32 - 7894900273815552*$.1^31 - 39812753188183955*$.1^30 + 8296194355663598*$.1^29 + 71847730592145342*$.1^28 - 40796532230514306*$.1^27 + 63129662250715074*$.1^26 - 6953203077586616*$.1^25 + 69744392611215208*$.1^24 - 17342592435595375*$.1^23 + 68382141997723193*$.1^22 + 16391888204680362*$.1^21 + 62947702402846571*$.1^20 - 45619682822865711*$.1^19 + 84917620727847933*$.1^18 - 82701333283509215*$.1^17 - 64127040240441591*$.1^16 + 81856318539096058*$.1^15 - 34924345508252449*$.1^14 + 31306766626319811*$.1^13 - 60413319556376730*$.1^12 - 40280302088041351*$.1^11 + 82545037454991174*$.1^10 - 17903549832058169*$.1^9 + 54808687005980427*$.1^8 - 28049887250629581*$.1^7 - 81015051182520668*$.1^6 - 85606231403767368*$.1^5 - 17719410116834202*$.1^4 - 71805543500898880*$.1^3 + 11386057713100772*$.1^2 + 66975563345854857*$.1 - 80836533286084697 > Parent(g); Univariate Polynomial Ring over Integer Ring > R := PolynomialRing(Z); > Parent(g); Univariate Polynomial Ring in x over Integer Ring > g; x^37 - 6483584*x^34 - 118234637824*x^33 - 123335506765824*x^32 - 7894900273815552*x^31 - 39812753188183955*x^30 + 8296194355663598*x^29 + 71847730592145342*x^28 - 40796532230514306*x^27 + 63129662250715074*x^26 - 6953203077586616*x^25 + 69744392611215208*x^24 - 17342592435595375*x^23 + 68382141997723193*x^22 + 16391888204680362*x^21 + 62947702402846571*x^20 - 45619682822865711*x^19 + 84917620727847933*x^18 - 82701333283509215*x^17 - 64127040240441591*x^16 + 81856318539096058*x^15 - 34924345508252449*x^14 + 31306766626319811*x^13 - 60413319556376730*x^12 - 40280302088041351*x^11 + 82545037454991174*x^10 - 17903549832058169*x^9 + 54808687005980427*x^8 - 28049887250629581*x^7 - 81015051182520668*x^6 - 85606231403767368*x^5 - 17719410116834202*x^4 - 71805543500898880*x^3 + 11386057713100772*x^2 + 66975563345854857*x - 80836533286084697 > h := Discriminant(g); > factor(h); [Interrupt twice in half a second; exiting] Total time: 33.590 seconds [was@modular sharifi]$ [was@modular sharifi]$ me Magma V2.8-10 Mon Apr 8 2002 18:20:00 on modular [Seed = 93007534] Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/emacs.m" Loading "/home/was/magma/local/init.m" > R := PolynomialRing(Z); > f := X^37 - 6483584*X^34 - 118234637824*X^33 - 123335506765824*X^32 - 7894900273815552*X^31 - 25584896141781024768*X^30 - 19612786666813992009728*X^29 - 2221784070205669762924544*X^28 - 33628014249666292632903483392*X^27 - 4805711697609190244214712041472*X^26 - 2249002615426863992005848511545344*X^25 - 13099755496539209311468832290825568256*X^24 - 3171787436319383501703813676940597919744*X^23 + 476259323830076662111107898811789814530048*X^22 - 1396232608839552259966984463923520026947092480*X^21 - 331493134727514939719441018060252656606965137408*X^20 - 80268638062435074559599184759300711777564488630272*X^19 - 872057565672136492561824204817812097995282872168087552*X^18 + 1772659418875854490177280483057352783210247369401565184*X^17 + 37244222236334875481641252538596552828631758622687299108864*X^16 - 20651404785477501467881895153357983415526349942938256921329664*X^15 + 31183544125608715763774641955998078374374445370791241228146966528*X^14 + 2854705449484624416795330612386811215415869973011706932441160613888*X^13 - 18557314583560485308211477301528775481854373440798991639264756844462080*X^12 + 3087405021478910646130093242279350919332930043815268747163999299543498752*X^11 - 844861169134880185162881813189113039529594781451540816736263726469132320768*X^10 - 181630591887896963687470296480916555113983363630481468702049951552077809319936*X^9 + 484965911395764970871665544609840479278207020589886844109688505883361242049937408*X^8 + 55114049776782199582622334540957461483624433957263207123073326516074293876028866560*X^7 - 106850589825632789894896612887329721094911179135082410100519870323032421196184294522880*X^6 + 122531389798606603051301724324273450024230408425919996998908148191119982817601008959488*X^5 + 1982469259694895457314935195126430029297012127795195501396552566165464092269185291272585216*X^4 - 660677952891702989779583125409354140514319272374345508868647000437128182296246506433818918912*X^3 - 264988194757774598059997009109229291894782867188883433765067934878438826901051317961493560950784*X^2 + 672876206080541961605226903062733805192789531352250137517709918865110475053588494379165834704584704*X - 262293029201456827937357306747978653209062535979993123331515899375253384527718903616471614294706880512; > f; X^37 - 6483584*X^34 - 118234637824*X^33 - 123335506765824*X^32 - 7894900273815552*X^31 - 25584896141781024768*X^30 - 19612786666813992009728*X^29 - 2221784070205669762924544*X^28 - 33628014249666292632903483392*X^27 - 4805711697609190244214712041472*X^26 - 2249002615426863992005848511545344*X^25 - 13099755496539209311468832290825568256*X^24 - 3171787436319383501703813676940597919744*X^23 + 476259323830076662111107898811789814530048*X^22 - 1396232608839552259966984463923520026947092480*X^21 - 331493134727514939719441018060252656606965137408*X^20 - 80268638062435074559599184759300711777564488630272*X^19 - 872057565672136492561824204817812097995282872168087552*X^18 + 1772659418875854490177280483057352783210247369401565184*X^17 + 37244222236334875481641252538596552828631758622687299108864*X^16 - 20651404785477501467881895153357983415526349942938256921329664*X^15 + 31183544125608715763774641955998078374374445370791241228146966528*X^14 + 2854705449484624416795330612386811215415869973011706932441160613888*X^13 - 18557314583560485308211477301528775481854373440798991639264756844462080*X^12 + 3087405021478910646130093242279350919332930043815268747163999299543498752*X^11 - 844861169134880185162881813189113039529594781451540816736263726469132320768*X^10 - 181630591887896963687470296480916555113983363630481468702049951552077809319936*X^9 + 484965911395764970871665544609840479278207020589886844109688505883361242049937408*X^8 + 55114049776782199582622334540957461483624433957263207123073326516074293876028866560*X^7 - 106850589825632789894896612887329721094911179135082410100519870323032421196184294522880*X^6 + 122531389798606603051301724324273450024230408425919996998908148191119982817601008959488*X^5 + 1982469259694895457314935195126430029297012127795195501396552566165464092269185291272585216*X^4 - 660677952891702989779583125409354140514319272374345508868647000437128182296246506433818918912*X^3 - 264988194757774598059997009109229291894782867188883433765067934878438826901051317961493560950784*X^2 + 672876206080541961605226903062733805192789531352250137517709918865110475053588494379165834704584704*X - 262293029201456827937357306747978653209062535979993123331515899375253384527718903616471614294706880512 > d := Discriminant(f); > Valuation(d,37); 755 > 37^755 - d; -63795197874674975243969494936156320891595744675183250915956724832773570844941015351120306019591652451706450836740501794223610527093699039187719971037352505740974748297023022846770041021960511737944753611834268816633066134720660619625169844033196057376310617597830888944979267563801724780016005402238206329080193462876320562461384501970465319030958814882091573640137807547429923007922156596987390438654461505151607267652946554975892148254336518836678578030383771923751354993486428555469867623527229171152388552005734071044809193150488097258799737483543048948685161052096600039645528194008438336899102963155012053864539631681259351508744204578089032005130357880744689195115780429891809131641757854554497327634572915226221814232836981024379766482166620866740352072595758623335353935015284026756024606427857493533160889545609384946550419760206268368987078782471911883566276821922570052473889427138930612871935573634725253247244660924165255878428870527269771792086036677435426235416020810501421475544943882617983766653996693136720447184137140772636469240607776690221726721164177973999740405764411653033395637933395612654233002028175758729968916245933665463457977207889677962045995256701950622369262739476967540805016865591338011282237677203283371799279962168436719401857890543582267868629515837292931723321228277781654071966596713836514762565117673828365580191064248232381371186079969999027163123606631796899213021184911215185384288533838425634484345598104575203093927701843241808263254392887780619577255183146357483111655615397403704656607704409947001653875573229800650902451860207291603643657210606621517253814377002028492677340527529142635217416734060366269407160177173266607559748846769079237153539669419816639657041244436611993645920207283825774162353398053547829527324932558551567956415747409211912333446828540151965425485647846642709916914521799737868731390151414820171357566945346626477872989282633311516421793313675333274249547928068701106298704917171342698918298209598635193232242516907311878895020132532292490153543748264319890333768728106670059300358929820662801607038024938196376888960616432591112845568711358138333331068629905408555280824213340532016242825434934592273304093947135396458286883520110015158923779565525608067890021488011316448688461198425561611057729279100390373098156095932075015332512398235677920332328048557412677269496503728303524526874396727107675308223188839177658730820675087217833647232125655518596281766815679380298838698697964959970510462917560165430677505853004841226785574289423888221744694277708753440315527232158113247246556013421997177041563308722627184841310332509442523815908127229144184774792338134294780257705471023020206715857034796011948858756305215265831535295081865539303180307313430585617647486134172347125575585894066732967199308157202519482029030135049367473460339922368343791906130730903348394575014448034795904910501457229872158217114085917062016587204585467846689369951048978687312150060925529684983389112591907941376478755845478237095312163250110635386635554367294523212194759791462789812476784152871306221815467248414234375502118492911710860405119710714266562928093335347168009289674626415419915808022335772279413672595271894639093007903378529293293676719751902536884402625333534767100596535688634786278699398406172895676215597202777635356732018786670768775148264239310385817970644910896876536096680929266736229496666670419274578800626801465561456246716814891407408444902237859804044069233163941949015516636794320138807440940003531271867095723748781072688043408428154678962403808821811379459716149698073280516233831731335460116945036650740841522230726249588962688669956234484122673423601810932469156536079524929022853458310980253678075911883892016333593002576869561366426559779319615603567614238123084490784960739 > Log(d)/Log(10); 3766.804787988830542784130135 > factor(755); [ <5, 1>, <151, 1> ] 1 > 755-35; 720 > f; X^37 - 6483584*X^34 - 118234637824*X^33 - 123335506765824*X^32 - 7894900273815552*X^31 - 25584896141781024768*X^30 - 19612786666813992009728*X^29 - 2221784070205669762924544*X^28 - 33628014249666292632903483392*X^27 - 4805711697609190244214712041472*X^26 - 2249002615426863992005848511545344*X^25 - 13099755496539209311468832290825568256*X^24 - 3171787436319383501703813676940597919744*X^23 + 476259323830076662111107898811789814530048*X^22 - 1396232608839552259966984463923520026947092480*X^21 - 331493134727514939719441018060252656606965137408*X^20 - 80268638062435074559599184759300711777564488630272*X^19 - 872057565672136492561824204817812097995282872168087552*X^18 + 1772659418875854490177280483057352783210247369401565184*X^17 + 37244222236334875481641252538596552828631758622687299108864*X^16 - 20651404785477501467881895153357983415526349942938256921329664*X^15 + 31183544125608715763774641955998078374374445370791241228146966528*X^14 + 2854705449484624416795330612386811215415869973011706932441160613888*X^13 - 18557314583560485308211477301528775481854373440798991639264756844462080*X^12 + 3087405021478910646130093242279350919332930043815268747163999299543498752*X^11 - 844861169134880185162881813189113039529594781451540816736263726469132320768*X^10 - 181630591887896963687470296480916555113983363630481468702049951552077809319936*X^9 + 484965911395764970871665544609840479278207020589886844109688505883361242049937408*X^8 + 55114049776782199582622334540957461483624433957263207123073326516074293876028866560*X^7 - 106850589825632789894896612887329721094911179135082410100519870323032421196184294522880*X^6 + 122531389798606603051301724324273450024230408425919996998908148191119982817601008959488*X^5 + 1982469259694895457314935195126430029297012127795195501396552566165464092269185291272585216*X^4 - 660677952891702989779583125409354140514319272374345508868647000437128182296246506433818918912*X^3 - 264988194757774598059997009109229291894782867188883433765067934878438826901051317961493560950784*X^2 + 672876206080541961605226903062733805192789531352250137517709918865110475053588494379165834704584704*X - 262293029201456827937357306747978653209062535979993123331515899375253384527718903616471614294706880512 > Log(Coefficient(f,0))/Log(10); 101.4187867487696528721503350 + 1.364376353841841347485783625*i > e := d/37^755; > e; 649361174172301757705743957153561885486679985982386529632477749891138648659328104310297016180951851333961367406969498942456124675456572650818683340501423370877370328202976815078678124503543527325500483100366669896029305005848343518555139101275345289065828252409694750416381887308528910333067944326078581009494279234798437936646242534523972453392652805244418966306553998629242174245596276741497523510693968229208205358738428822943279604528085546176699428848614472919815873331349181159412705520034395220946800976982216610581007697577585788582207187569786166902147747576579508854887254976208087847114091222760153771718073121777843345112246098161590107448043882550535327977016539291574941568789051559432453670739281655256854892621467823815905080038240170070211972582825249914130784179342093574258009541892828223599770229756468275335912436676296921358353614950336990128299959386301948129214385379290857155535625394424281246755890340447305213718006519164044398519246311157936416308470759538437456029508802143202271140616673535755603842363584011321291043705195287779547376722863323564071793595387161633391262812883860070902332619406533777899742702562084700612917774707162880316345479141698738196435656689320245385582580395052757040562729317762286293826841775874263962201318945336587856212043269324541761293008172274856347599001380616938456554346590050224125925613100490008423778159939349467812299109219652792867602879729149303085566466586465947468898599205053063199194384679367347627639585211731112747324671278149606966020642852969794243681549789482771268763178523150585462844810685784420603203257156328330686762584129443840083992545600323483687779715216115537409963902691680205028704902550873958004016561956474881431041710214544094739455697492471553052497050155364812089731479765679783286408847898515939384405521362703977819239349774290636667296599516298874199722416702383126169204903545344921694816060308461498348503163688937467629586712200003173875699428795337866176251429826423219886803754719644277132211532670380181859098152466593366384425770069413320341030308370410598050583765841230055423137778946553192613642076860429776388860193803244257178735407061907840520664505343062138928049118337867555465199104038675925484978215950223969045847638691001312461606061701736198112024677779690236973429562996540479914190942268479181162822993151938243720234295557590915574777885768308304069147166768650311556656222393432090021217071932584357378049614745073269011656818429234438105486895749089189276709088812004006434124200981709824723236715730345582218696249759379192114588993578069966874449281024 > Sqrt(e); 2.548256608295761405707294601E1291 > f; X^37 - 6483584*X^34 - 118234637824*X^33 - 123335506765824*X^32 - 7894900273815552*X^31 - 25584896141781024768*X^30 - 19612786666813992009728*X^29 - 2221784070205669762924544*X^28 - 33628014249666292632903483392*X^27 - 4805711697609190244214712041472*X^26 - 2249002615426863992005848511545344*X^25 - 13099755496539209311468832290825568256*X^24 - 3171787436319383501703813676940597919744*X^23 + 476259323830076662111107898811789814530048*X^22 - 1396232608839552259966984463923520026947092480*X^21 - 331493134727514939719441018060252656606965137408*X^20 - 80268638062435074559599184759300711777564488630272*X^19 - 872057565672136492561824204817812097995282872168087552*X^18 + 1772659418875854490177280483057352783210247369401565184*X^17 + 37244222236334875481641252538596552828631758622687299108864*X^16 - 20651404785477501467881895153357983415526349942938256921329664*X^15 + 31183544125608715763774641955998078374374445370791241228146966528*X^14 + 2854705449484624416795330612386811215415869973011706932441160613888*X^13 - 18557314583560485308211477301528775481854373440798991639264756844462080*X^12 + 3087405021478910646130093242279350919332930043815268747163999299543498752*X^11 - 844861169134880185162881813189113039529594781451540816736263726469132320768*X^10 - 181630591887896963687470296480916555113983363630481468702049951552077809319936*X^9 + 484965911395764970871665544609840479278207020589886844109688505883361242049937408*X^8 + 55114049776782199582622334540957461483624433957263207123073326516074293876028866560*X^7 - 106850589825632789894896612887329721094911179135082410100519870323032421196184294522880*X^6 + 122531389798606603051301724324273450024230408425919996998908148191119982817601008959488*X^5 + 1982469259694895457314935195126430029297012127795195501396552566165464092269185291272585216*X^4 - 660677952891702989779583125409354140514319272374345508868647000437128182296246506433818918912*X^3 - 264988194757774598059997009109229291894782867188883433765067934878438826901051317961493560950784*X^2 + 672876206080541961605226903062733805192789531352250137517709918865110475053588494379165834704584704*X - 262293029201456827937357306747978653209062535979993123331515899375253384527718903616471614294706880512 > K := NumberField(f); > K; Number Field with defining polynomial X^37 - 6483584*X^34 - 118234637824*X^33 - 123335506765824*X^32 - 7894900273815552*X^31 - 25584896141781024768*X^30 - 19612786666813992009728*X^29 - 2221784070205669762924544*X^28 - 33628014249666292632903483392*X^27 - 4805711697609190244214712041472*X^26 - 2249002615426863992005848511545344*X^25 - 13099755496539209311468832290825568256*X^24 - 3171787436319383501703813676940597919744*X^23 + 476259323830076662111107898811789814530048*X^22 - 1396232608839552259966984463923520026947092480*X^21 - 331493134727514939719441018060252656606965137408*X^20 - 80268638062435074559599184759300711777564488630272*X^19 - 872057565672136492561824204817812097995282872168087552*X^18 + 1772659418875854490177280483057352783210247369401565184*X^17 + 37244222236334875481641252538596552828631758622687299108864*X^16 - 20651404785477501467881895153357983415526349942938256921329664*X^15 + 31183544125608715763774641955998078374374445370791241228146966528*X^14 + 2854705449484624416795330612386811215415869973011706932441160613888*X^13 - 18557314583560485308211477301528775481854373440798991639264756844462080*X^12 + 3087405021478910646130093242279350919332930043815268747163999299543498752*X^11 - 844861169134880185162881813189113039529594781451540816736263726469132320768*X^10 - 181630591887896963687470296480916555113983363630481468702049951552077809319936*X^9 + 484965911395764970871665544609840479278207020589886844109688505883361242049937408*X^8 + 55114049776782199582622334540957461483624433957263207123073326516074293876028866560*X^7 - 106850589825632789894896612887329721094911179135082410100519870323032421196184294522880*X^6 + 122531389798606603051301724324273450024230408425919996998908148191119982817601008959488*X^5 + 1982469259694895457314935195126430029297012127795195501396552566165464092269185291272585216*X^4 - 660677952891702989779583125409354140514319272374345508868647000437128182296246506433818918912*X^3 - 264988194757774598059997009109229291894782867188883433765067934878438826901051317961493560950784*X^2 + 672876206080541961605226903062733805192789531352250137517709918865110475053588494379165834704584704*X - 262293029201456827937357306747978653209062535979993123331515899375253384527718903616471614294706880512 over the Rational Field > UnitGroup(NumberField(x^2+17)); Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*$.1 = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*$.1 = 0 to Maximal Equation Order with defining polynomial x^2 + 17 over Z >