Demo: Elliptic Curves in Sage
October 19 at ECC 2010
William Stein
{{{id=8|
///
}}}
Elliptic Curves over $\QQ$
{{{id=7|
E = EllipticCurve([1,2,3,4,5]); E
///
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
}}}
{{{id=13|
E = E.minimal_model(); show(E)
///
\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + xy = x^3 - x^2 + 4x + 3
}}}
{{{id=172|
E.
///
}}}
Invariants
{{{id=12|
E.discriminant()
///
-10351
}}}
{{{id=6|
N = E.conductor(); N
///
10351
}}}
{{{id=22|
N.factor()
///
11 * 941
}}}
{{{id=5|
E.tamagawa_numbers()
///
[1, 1]
}}}
Rank
{{{id=4|
E.rank()
///
1
}}}
{{{id=125|
E.analytic_rank()
///
1
}}}
{{{id=11|
E.gens()
///
[(2 : 3 : 1)]
}}}
{{{id=10|
E.regulator()
///
1.26539379176449
}}}
{{{id=3|
E.simon_two_descent()
///
(1, 1, [(2 : 3 : 1)])
}}}
{{{id=173|
print E.mwrank()
///
Curve [1,-1,0,4,3] : Basic pair: I=-183, J=-6858
disc=-71546112
2-adic index bound = 2
By Lemma 5.1(b), 2-adic index = 1
2-adic index = 1
One (I,J) pair
*** BSD give two (I,J) pairs
Looking for quartics with I = -183, J = -6858
Looking for Type 3 quartics:
Trying positive a from 1 up to 6 (square a first...)
(1,2,-9,22,-11) --nontrivial...(x:y:z) = (1 : 1 : 0)
Point = [2:3:1]
height = 1.26539379176449
Rank of B=im(eps) increases to 1
(4,3,6,3,-4) --trivial
Trying positive a from 1 up to 6 (...then non-square a)
Trying negative a from -1 down to -5
(-4,3,6,3,4) --trivial
Finished looking for Type 3 quartics.
Mordell rank contribution from B=im(eps) = 1
Selmer rank contribution from B=im(eps) = 1
Sha rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer rank contribution from A=ker(eps) = 0
Sha rank contribution from A=ker(eps) = 0
Used full 2-descent via multiplication-by-2 map
Rank = 1
Rank of S^2(E) = 1
Searching for points (bound = 8)...done:
found points of rank 1
and regulator 1.26539379176449
Processing points found during 2-descent...done:
now regulator = 1.26539379176449
Saturating (bound = 100)...done:
points were already saturated.
Transferring points from minimal curve [1,-1,0,4,3] back to original curve [1,-1,0,4,3]
Generator 1 is [2:3:1]; height 1.26539379176449
Regulator = 1.26539379176449
The rank and full Mordell-Weil basis have been determined unconditionally.
(0.113583 seconds)
}}}
Integral and $S$-Integral Points (up to sign)
{{{id=1|
E.integral_points()
///
[(2 : 3 : 1)]
}}}
{{{id=14|
E.S_integral_points([2])
///
[(-39/64 : 279/512 : 1), (2 : 3 : 1)]
}}}
The Complex $L$-series
{{{id=15|
L = E.lseries(); L
///
Complex L-series of the Elliptic Curve defined by y^2 + x*y = x^3 - x^2 + 4*x + 3 over Rational Field
}}}
{{{id=17|
line([(s,L(s).real()) for s in [0.5,0.55,..,2]])
///
}}}
Evaluate anywhere:
{{{id=19|
L(2+3*I)
///
0.763030838302007 - 0.343385137835191*I
}}}
Taylor series about any point:
{{{id=16|
L.taylor_series(1)
///
3.51873114993151*z - 7.76698842523228*z^2 + 12.5750690885479*z^3 - 13.7330374564920*z^4 + 8.64019179676616*z^5 + O(z^6)
}}}
{{{id=133|
L.taylor_series(1+I)
///
-0.485502124065793 + 0.627256178203893*I + (8.10448376619248 + 3.38500260152220*I)*z + (-16.8659849368413 - 18.6176821545456*I)*z^2 + (19.6975963690651 + 43.4411765145958*I)*z^3 + (-10.7717354501570 - 67.9428401937405*I)*z^4 + (-8.01551313473187 + 81.0705067446227*I)*z^5 + O(z^6)
}}}
Imaginary parts of the zeros in the critical strip
{{{id=20|
L.zeros(10)
///
[0.000000000, 0.895959461, 2.61710659, 3.23646650, 3.80592295, 4.36500628, 5.27684817, 5.77700168, 6.37368526, 7.48823858]
}}}
{{{id=131|
points([(1/2,y) for y in L.zeros(20)]).show(xmin=0,xmax=1,figsize=[2,3])
///
}}}
A rank 4 curve
{{{id=144|
E = elliptic_curves.rank(4)[0]; E
///
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 79*x + 289 over Rational Field
}}}
{{{id=142|
time E.rank()
///
4
Time: CPU 0.00 s, Wall: 0.00 s
}}}
{{{id=151|
E.gens()
///
[(-9 : 19 : 1), (-8 : 23 : 1), (-7 : 25 : 1), (4 : -7 : 1)]
}}}
{{{id=145|
L = E.lseries()
///
}}}
{{{id=146|
L.taylor_series(1)
///
5.54631009473167e-24 + (-2.08951550639391e-23)*z + (-4.15704192504384e-22)*z^2 + (1.66720224204167e-21)*z^3 + 8.94384739590089*z^4 - 33.6950287693207*z^5 + O(z^6)
}}}
Open Problem: Prove that $\text{ord}_{s=1} L(E,s) = 4$.
{{{id=132|
///
}}}
$p$-adic Invariants Associated to the rank 2 curve "389a"
{{{id=26|
E = EllipticCurve('389a'); E
///
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
}}}
{{{id=174|
E.rank()
///
2
}}}
{{{id=21|
L5 = E.padic_lseries(5); L5
///
5-adic L-series of Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
}}}
{{{id=23|
show(L5.series(3))
///
\newcommand{\Bold}[1]{\mathbf{#1}}O(5^{5}) + O(5^{2})T + \left(4 + 4 \cdot 5 + O(5^{2})\right)T^{2} + \left(2 + 4 \cdot 5 + O(5^{2})\right)T^{3} + \left(3 + O(5^{2})\right)T^{4} + O(T^{5})
}}}
{{{id=137|
show(E.padic_regulator(5))
///
\newcommand{\Bold}[1]{\mathbf{#1}}5^{2} + 2 \cdot 5^{3} + 2 \cdot 5^{4} + 4 \cdot 5^{5} + 3 \cdot 5^{6} + 4 \cdot 5^{7} + 3 \cdot 5^{8} + 5^{9} + 2 \cdot 5^{12} + 4 \cdot 5^{13} + 5^{14} + 4 \cdot 5^{15} + 4 \cdot 5^{16} + 5^{17} + 5^{19} + 4 \cdot 5^{20} + O(5^{21})
}}}
{{{id=27|
S = E.sha(); S
///
Shafarevich-Tate group for the Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
}}}
{{{id=29|
S.an()
///
1.00000000000000
}}}
{{{id=138|
S.an_padic(5, prec=3)
///
1 + O(5^2)
}}}
{{{id=30|
S.p_primary_bound(5)
///
0
}}}
Prove that Sha[29] = 0, using explicit computation with $p$-adic $L$-series, regulator, results from Iwasawa theory, etc. (See Wuthrich-Stein.)
{{{id=32|
time S.p_primary_bound(29)
///
0
Time: CPU 1.93 s, Wall: 1.94 s
}}}
Open Problem: Prove that Sha is finite for the rank 2 curve $E$ above. (Using Sage I verified that Sha[p]=0 for p < 2466, except p=107, 599, 1049.)
{{{id=139|
///
}}}
Shafarevich-Tate Groups
Back to our rank 1 curve, whose Sha we prove trivial below (hence proving the full BSD conjecture for this curve).
{{{id=34|
E = EllipticCurve([1..5]).minimal_model()
S = E.sha()
S.an()
///
1
}}}
{{{id=31|
S.bound_kato() # no info, since rank is 1
///
False
}}}
{{{id=38|
E.heegner_discriminants_list(3)
///
[-24, -39, -43]
}}}
{{{id=36|
S.bound_kolyvagin(-39)
///
([2], 1)
}}}
{{{id=37|
E.selmer_rank()
///
1
}}}
Heegner Points: Find point over class field, then trace it down "by hand" (yes, there are better ways)
{{{id=41|
y = E.heegner_point(-39); y
///
Heegner point of discriminant -39 on elliptic curve of conductor 10351
}}}
{{{id=40|
y.numerical_approx()
///
(-0.888422002508970 - 0.788950457973954*I : -1.26899253838416 + 2.02424289524747*I : 1.00000000000000)
}}}
{{{id=43|
z = y.point_exact(100); z
///
(1/9*a : 1/10449*a^3 - 22/3483*a^2 + 23/387*a + 114/43 : 1)
}}}
{{{id=44|
z.parent()
///
Abelian group of points on Elliptic Curve defined by y^2 + x*y = x^3 + (-1)*x^2 + 4*x + 3 over Number Field in a with defining polynomial x^4 - 87*x^3 - 315*x^2 + 7695*x + 139239
}}}
{{{id=45|
PHI = z.curve().base_ring().complex_embeddings()
///
}}}
{{{id=46|
EC = E.change_ring(CC)
X = [EC.point((phi(z[0]),phi(z[1]), 1), check=False) for phi in PHI]; X
///
[(-0.888422002508657 - 0.788950457974162*I : 2.15741454089335 - 1.23529243727291*I : 1), (-0.888422002508657 + 0.788950457974162*I : 2.15741454089335 + 1.23529243727291*I : 1), (1.51393212680968 : 2.53038144982774 : 1), (9.92957854487430 : 25.8214561350522 : 1)]
}}}
{{{id=47|
sum(X)
///
(1.99999999999999 : 3.00000000000005 : 1.00000000000000)
}}}
{{{id=48|
E.gens()
///
[(2 : 3 : 1)]
}}}
Image of Galois
{{{id=51|
E.division_polynomial(5)
///
5*x^12 - 15*x^11 + 257*x^10 + 900*x^9 - 4245*x^8 + 9630*x^7 - 31215*x^6 - 2757*x^5 - 113225*x^4 + 5985*x^3 - 55160*x^2 - 202620*x - 33911
}}}
{{{id=50|
rho = E.galois_representation(); rho
///
Compatible family of Galois representations associated to the Elliptic Curve defined by y^2 + x*y = x^3 - x^2 + 4*x + 3 over Rational Field
}}}
{{{id=49|
rho.non_surjective()
///
[]
}}}
{{{id=53|
F = EllipticCurve([1..5])
E.is_isomorphic(F)
///
True
}}}
{{{id=54|
E.isomorphism_to(F)
///
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + x*y = x^3 - x^2 + 4*x + 3 over Rational Field
To: Abelian group of points on Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
Via: (u,r,s,t) = (1, 1, 0, 1)
}}}
Same mod 5 representation:
{{{id=105|
F = E.mod5family(); F
///
WARNING: Output truncated!
full_output.txt
Elliptic Curve defined by y^2 = x^3 + (-350100912848470557262178354914780276580297990571157449372901113371873512333562228749157572253063355118187409237207899324506760329921263064807247723908536755135051803613540977040182671638923846671210152426310283686085796678191332592937705890916935518557151598754822259731517510192150922826648559120159488597001572610400764151973851684392714878778723876501221425099461085590917912292724983607637645107524865556414431757422205058997175127024919679973700729626963052615689100194152515484014289804500994560339954531764384106772741292665339904*t^20-3076148796739621953443219074961560223321319021417154542217433481890039421403133361733468069441814085528551842954583515856927480386779263830554314459667991524379298105797089586641140433228801228410106047409502556922621995718848910503560837300872037541227759970904447681142751663853907541247685568093133174022760283293193431559118771450835627388250514425123655428923179973280471609300183262344137220027781119360322565674277599408304848530625914691755189946375875290182368565932293775457411843189471819619076958858710326520370180451981066240*t^19-9255805954515104139134796453109504875910216663589848919548603527609214366856591797450154219943513430001925852260771135809779164982607856099958139058201304192413720067175387955738378326497063933119902127963007753700718469307791073884476912050068709908984129857476607321136187279477936956595851146255586236255154101892736506944366171652485954524092360987277877800894175331125913263664938284780195683813723391967099337666658862104161796749655723114885475782299038021043653771944363976685021561398409853716872744762532370949276783161198510080*t^18-19397013449155320116122269126763868609414612020482748191142375841983183486810718516386492337645036452836314625194877805425950009935690179217819728655275526230940295981347238035715348090570054869224742868936210569879798936643390185822146593403579732595328213957213839826551749150139201891925648214323765604948255786188599198288718346287937814342642254971778504394083272770293660706608881106176260975328036488365620284230377669104532085302934447006783353481701070318387373697332532472683862310554447900899741133492764684208892853235713310720*t^17-33322256906608665762486911443172822768459203130474271370223873457806377892513528789543149965395147151715342596677812777521737511634301712627151152006957725950586006558570724133001395236816995945330205506557199514285483882700898891843101086409906178676688895195736930100074642928621217186443694319305476310619831090690829050530250069484681282703517083550557367748108912358016846278545232230216448743840545021747910766463724918861084223906996834238969444740431327386546034771810232842143352195031615390772584229096246184065652637527179264000*t^16-40609447907018192735213912804962068205670962757903062390340258313605635785312494812762898124562177207642532851803543035912414278122402260936404402918636161572059665503776313013887578694615712204187410242158865333126345394590734906115745861851339390901692201972986406091886158707627955085006216802645272011689291019520028838975822408470016814034447402524589041250488784069367624587509209999105986290571681058528555669881333790591650150554766795569132584732641423573742768663225954242244850336227462156532620220650264756353552932812582027264*t^15-31731901206502079507143908513980355856508398758627105660839562374610621021939981435759508452202444560181593622165934304607430156057005663506319818063785026055713655623777066128714741695000804228746783981925948781105147069396966508865777790939490722482041065835749288984045987520798603906530794297933289745146616203500035712653712212911683847391197615597426335260093161827629559683658564946467384115000672726092868187518169623808965646760662245986066750501648083311609291925866992781834914925432160951632497766724270759032756603288268308480*t^14-14685725012897922331847447917284722047275557120481621112109393023537797720853707757690175837152848274435994135583882219483685393780340893062866261352756565192013134147345673400530675569679843840597219480945722046209396906634395233846297238710337811475235567476848470647044976817649801638022513243645859771513766612644829446415715804280417969729933965865848717114167493509569554581735198004685603429709053817147863722301925058091615724760333618448110685907580440422954617841984650644692263134076766865008173924160321376901012785997398671360*t^13-6305671210764502404382844686599298061187190279662192697272491442184177009184745323105549734834664960742044636619467780798299989643839382498153413555330282920116492251253654690509789556887359262851284730283138844983071419228359335848121747023531694626521601533383744910289520937123303793189182255171057212431627171254075670215277112284201082493538520967028800404870013813867944164501532624713064770220983916943138517395428089730500175138267148816516019479781997913911038493174515845033960521716745460372256599456119525832443565013456650240*t^12-10229195092244162261088361237248472612713619672224323674757179234223081608786812289725432999279362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...
873347609011612102871371407056769886700078785604018364257096692936601233458166690171957164274383064073693351112306113328933136171380396875946689939176058771672133237006633723597222747334721038822586843103011113755599704145150773694367819195680730313644861166473486751609971217696707955903193878440298051804447704625636629160825603194709408473879545459691903123975021679368417612667276207505223015130669202408095925309080901003896168483619806722920172887685597084959059367175948360141515832389477386971645887679849661936435200*t^19+795806354924869266373325970184766841850378685832573550999713737587613924200236970461131719806794375057430310264527538750255870131083593879050870894145669816286934961416425356445429823566284817385573777022614893150741521883034201353187235131206579893164581576546186839261195619053483002449776636245583170832063572311855928084751029893228889160952320067104184024353317771172046774748829795137420956127030713359951511972643855329113972672350190254860133483173869532659882228013245649741577611160504600583356127695964609143926749438354809351794562352105502176389840044815948260918746497260094580642861395483545309141940718145133102838439897369662513941510441302862591570998800253687621240475996538864620545077975145982038651639542008145501684832659862244498555316130332857444075371070018276593512880941511475200*t^18+2615193020464744849319961868068658811722539502929127516529725582482184444809773327973385191116749751228108211261217693764616065516933099803073970993222294736230894655827062253576796313669765047422746950104458446716248153583810954033681207799635682689457647237681874696538057050831525124161202497533415479860813530693767724145132115596421032239869413839313713604663843992746774102744527322745640128992905846916726435826935224386579527793936949302668420517720943548401468618921848762803038610332166688337798839698071242364900914277579646676260908509922775793665688751177751836233960108796814101358305570031318342239710263543283900977057895265622921838926534686669267864553013480555945930591529747392186990115926017365417090460136483992807000256475100749175225339646839510007093241336889560055811761545779609600*t^17+2839430557851775749341813881127831406751553491933817070577136019767532181880165269836521603347158546506921794447872171033732707780527946458550444740322308673943938050681183801617457556464697483782739779199333176247109017779916617009223971836487061777193291256748736686143769181133358428800943673507771002897717922670209169063520031957807823849120552030066450468843526035141881090903466666206069411310923756193232580732838581011960855741262812061211648158410567382326693071790701723673547248153958700430905096512678883558579633328317826240411506069082953542772167487754357160422429464713424427178978249730056722685043266669367172091229353512977067088887458112097246786564379358176377086061928457733358405902605897991406025316884026774475186819647545153884581981666148650695903501843393472607947520304008396800*t^16+2099856227636950456288674268929434671236886429211352206981980460861283185079938533673134282289577582648616727474412702102385250336629924782041615901046389835289189165992286028296484779067918410982823894943798802599644260862189657766362598307114587667840560755747192541705539015581429127827550490812074370872535481865962358825702028087800823347952484302297224123914061257232405139249615546574617271374916084245765098875451306971105732043429791168098818442459718820484821941255664693459645676433062443018463695559973300595870800042869579710786160363135804021395600590682069427487191228756574777250959574104768325754766118209418025286396813063350150677376024567863420676902299693585338973277550130781747104408767522539235710407917922133302344315873821415462559360822314908513672363544827006286708473731513057280*t^15+1257998980616394386201251505949415749135490519886071621161635959064288959839318677530485519263701316712332284554802876656038195513970715942861315858740198909364688964034670018025361433022934094135791432898334599250944186352316492008827709305518178786816754070655990340345730502916562817535629723840770314531647827306673086952072514054837643862329991065526089180664067408385466890723696145747776715481129215841742193550662228500340821242222216440301337755061173918097987904101627251535329680613349572613860516357032595338164803285334671208893657217593352471726398158226958392594094677455445960862963567932232180659967872329129590918964752950958838250547449393463251728631072062866227880120081575180925504052509035442465515914263391189892178370874030455311467502293613240235678493299171167018466871480208588800*t^14+6974782328035688842310521666637810718735533400275571924985564502717666720889080683038035446142776015733801394424380451335495476870738125747370461255089094309042346348141149140024757783347954487972671805094697797320612498806183054383843632157565740776293798069730375811965392680482319972144416410692444503588681295378868651650820181716476681557569974769277821975221426716654108779039811082951767836939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78699862190047059159955939451487651687704666385337340515947150894363358226034575233141745631477821113345581207536344760320*t+2293140081272999525880457735045306060525868282393980175678905055643499661042037620011121678960611236555415388966528539008455164438681004548185750833175346387407676738839197859431069791043575439630761173504871678458866133404091360732231634033824998248213892682638261956830553391753468074009778838536521407287542973867114580046398016345735904437287294801628396023730079843290661314052657633669714875229666193231636703897296298062756517907025710898126952418506345838766905041393004115972708227939925988927409896884348575741774605332076432222834932791167924549508628304410623246027264392175361765635857274517912345798276489875577687079863687010967693046538138260774075298344948403219289995406334901971998531315049588389590155546177911350531571696478778607534485841104724854382594037111519373584544825344) over Fraction Field of Univariate Polynomial Ring in t over Rational Field
}}}
{{{id=108|
G = EllipticCurve([a(t=1) for a in F.a_invariants()]).minimal_model(); G
///
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 640804168893951905637456113348245371201487663923467637718566216386653978913277921550075928125742*x - 98770846242844128774558916352826645288991075442204178749699276893375664008383305935974004704968720083222472651422915574042236889700867998179209 over Rational Field
}}}
{{{id=109|
E.aplist(50)
///
[1, 0, -3, -1, -1, 1, 5, 4, -6, -2, 2, 7, 6, 8, 0]
}}}
{{{id=110|
G.aplist(50)
///
[1, 0, 0, -1, -1, 6, 5, -6, -6, -7, -8, -3, 6, 8, 0]
}}}
{{{id=152|
factor(G.conductor())
///
5^2 * 11 * 941 * 109425295591 * 315154685475065659 * 4706373428975316362621837
}}}
{{{id=153|
factor(E.conductor())
///
11 * 941
}}}
Plotting
{{{id=55|
plot(E, plot_points=300, thickness=2, color=hue(.75), xmax=2)
///
}}}
{{{id=156|
Es = [EllipticCurve('11a'), EllipticCurve('37a'), EllipticCurve('389a'),
EllipticCurve('5077a'), elliptic_curves.rank(4)[0]]
set_random_seed(1)
G = graphics_array([e.plot(thickness=4, color=hue(random()),plot_points=200) for e in Es])
G.show(figsize=[8,2],axes=False)
///
}}}
{{{id=155|
///
}}}
{{{id=58|
///
}}}
Elliptic Curves over Finite Fields
{{{id=63|
k. = GF(7^6); k
///
Finite Field in a of size 7^6
}}}
{{{id=62|
E = EllipticCurve(k, [a,a^2])
E.cardinality()
///
117837
}}}
{{{id=61|
time E.abelian_group()
///
Additive abelian group isomorphic to Z/117837 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + a*x + a^2 over Finite Field in a of size 7^6
Time: CPU 0.09 s, Wall: 0.10 s
}}}
Efficient computation of the point counts mod all small primes:
{{{id=60|
E = EllipticCurve([1..5])
time ap = E.aplist(10^6)
///
Time: CPU 2.28 s, Wall: 2.29 s
}}}
{{{id=56|
ap[:10]
///
[1, 0, -3, -1, -1, 1, 5, 4, -6, -2]
}}}
Visualize the Sato-Tate Distribution:
{{{id=64|
P = prime_range(10^6)
v = stats.TimeSeries([ap[i]/(2*math.sqrt(P[i])) for i in range(len(ap))])
v.plot_histogram(bins=200)
///
}}}
Remark: That the above distribution converges to a semicircle for any non-CM elliptic curve over $\QQ$ is a major recent theorem of Taylor et al.
{{{id=159|
///
}}}
Point counting over big fields
{{{id=66|
E = EllipticCurve(GF(next_prime(10^50)), [1..5])
time E.cardinality()
///
99999999999999999999999982865176588640916234222362
Time: CPU 0.03 s, Wall: 1.55 s
}}}
Example with non-prime base field (but $j$-invariant in the prime subfield).
{{{id=111|
p = next_prime(10^50)
q = p^5
E = EllipticCurve(GF(q,'a'), [1..5])
time E.cardinality()
///
10000000000000000000000000000000000000000000000075500000000000000000000000000000000000000000000228010000000000000000000000000181609431340990491938162471889163591442768033541398879013163210299401542467472841889372639005122561820006471215426251120821802
Time: CPU 0.36 s, Wall: 1.65 s
}}}
{{{id=112|
///
}}}
The Free Abelian Group on the Supersingular $j$-invariants.
{{{id=68|
X = SupersingularModule(389); X
///
Module of supersingular points on X_0(1)/F_389 over Integer Ring
}}}
{{{id=69|
T2 = X.hecke_matrix(2)
DiGraph(T2).show(frame=True, figsize=5)
///
}}}
{{{id=71|
graph_editor(DiGraph(T2)) # WARNING: converts to simple graph (now)
///
}}}
Brandt Modules: Free abelian group on enhanced elliptic supersingular curves (or right ideal classes in an Eichler order)
{{{id=76|
M = BrandtModule(389, 2); M
///
Brandt module of dimension 97 of level 389*2 of weight 2 over Rational Field
}}}
{{{id=75|
time T2 = M.hecke_matrix(2); T2
///
Time: CPU 10.62 s, Wall: 10.64 s
97 x 97 dense matrix over Rational Field (type 'print T2.str()' to see all of the entries)
}}}
{{{id=84|
M.quaternion_algebra()
///
Quaternion Algebra (-2, -389) with base ring Rational Field
}}}
{{{id=85|
M.right_ideals()[:2]
///
(Fractional ideal (2 + 2*j + 2*k, 2*i + 2*k, 4*j, 4*k), Fractional ideal (2 + 2*j + 6*k, 2*i + 10*k, 12*j, 12*k))
}}}
{{{id=82|
T2.charpoly().factor()
///
(x - 5) * x * (x + 4) * (x - 1)^16 * (x + 1)^16 * (x^2 - 4*x + 2) * (x^2 + 4*x + 2) * (x^3 - 6*x^2 + 8*x - 2) * (x^3 + 6*x^2 + 8*x - 2) * (x^6 - 9*x^5 + 28*x^4 - 32*x^3 + 2*x^2 + 12*x - 1) * (x^6 + 15*x^5 + 88*x^4 + 256*x^3 + 386*x^2 + 284*x + 79) * (x^20 - 43*x^19 + 845*x^18 - 10037*x^17 + 80272*x^16 - 455306*x^15 + 1876165*x^14 - 5640284*x^13 + 12155786*x^12 - 17733125*x^11 + 14442673*x^10 + 762763*x^9 - 16382696*x^8 + 16704432*x^7 - 3963105*x^6 - 5037274*x^5 + 4041921*x^4 - 621414*x^3 - 347474*x^2 + 141200*x - 14500) * (x^20 + 37*x^19 + 617*x^18 + 6115*x^17 + 39948*x^16 + 179958*x^15 + 565869*x^14 + 1217820*x^13 + 1656778*x^12 + 1017947*x^11 - 682987*x^10 - 1767421*x^9 - 958924*x^8 + 428912*x^7 + 612703*x^6 + 61198*x^5 - 117559*x^4 - 19606*x^3 + 8774*x^2 + 224*x - 4)
}}}
{{{id=74|
graph_editor(DiGraph(T2))
///
}}}
{{{id=117|
///
}}}
ECM: Elliptic Curve Factorization
{{{id=124|
print ecm((2^197 + 1)/3)
///
/Users/wstein/sage/install/sage-4.5.3/local/lib/python2.6/site-packages/sage/interfaces/ecm.py:108: DeprecationWarning: os.popen3 is deprecated. Use the subprocess module.
i,o,e = os.popen3(cmd)
GMP-ECM 6.2.1 [powered by GMP 4.2.1] [ECM]
Input number is 66955751844124594814248420514215108438425124740949701470891 (59 digits)
Using B1=10, B2=84, polynomial x^1, sigma=206979353
Step 1 took 0ms
Step 2 took 1ms
}}}
{{{id=121|
time ecm.factor((2^197 + 1)/3)
///
[197002597249, 1348959352853811313, 251951573867253012259144010843]
Time: CPU 0.08 s, Wall: 4.05 s
}}}
{{{id=123|
///
}}}
{{{id=116|
///
}}}
Plot of elliptic curve over finite field
{{{id=73|
@interact
def f(p=(389,primes(3,1000))):
show(EllipticCurve(GF(p), [1..5]).plot(), frame=True)
///
}}}
{{{id=79|
///
}}}
Elliptic Curves over Number Fields
{{{id=89|
K. = QuadraticField(2); K
///
Number Field in w with defining polynomial x^2 - 2
}}}
{{{id=88|
E = EllipticCurve([ 0, -1, 1, -3*w -4, 3*w + 4 ]); show(E)
///
\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + y = x^3 + \left(-1\right)x^2 + \left(-3 \sqrt{2} - 4\right)x + \left(3 \sqrt{2} + 4\right)
}}}
{{{id=102|
show(E.j_invariant())
///
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1732608}{1031} \sqrt{2} + \frac{266240}{1031}
}}}
{{{id=83|
N = E.conductor(); N
///
Fractional ideal (-24*w + 11)
}}}
{{{id=92|
N.norm().factor()
///
1031
}}}
{{{id=99|
E.tamagawa_numbers()
///
[1]
}}}
{{{id=93|
time T = E.simon_two_descent(); T
///
Time: CPU 0.11 s, Wall: 0.54 s
(2, 2, [(w + 3/2 : -1/4*w - 1 : 1), (-w - 1 : -w - 2 : 1)])
}}}
{{{id=91|
P, Q = T[2]
///
}}}
{{{id=90|
P.height()
///
0.933266708161421
}}}
{{{id=94|
Q.height()
///
0.183656956219703
}}}
{{{id=95|
E.height_pairing_matrix([P,Q]).det()
///
0.0235582958658636
}}}
{{{id=165|
embs = K.embeddings(CC)
Lambda = E.period_lattice(embs[0]); Lambda
///
Period lattice associated to Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-3*w-4)*x + (3*w+4) over Number Field in w with defining polynomial x^2 - 2 with respect to the embedding Ring morphism:
From: Number Field in w with defining polynomial x^2 - 2
To: Algebraic Field
Defn: w |--> -1.414213562373095?
}}}
{{{id=166|
Lambda.basis()
///
(10.7444017502464, 5.37220087512321 + 2.12681775202610*I)
}}}
{{{id=169|
///
}}}
Missing: But no $L$-series yet..., though this should be easy. What is the conjectural order of Sha? What are the nontrivial zeros of $L(E,s)$?
{{{id=170|
for p in primes(20):
for P, e in K.factor(p):
F = P.residue_field()
print p, P, E.change_ring(F).cardinality()
///
2 Fractional ideal (w) 5
3 Fractional ideal (3) 15
5 Fractional ideal (5) 29
7 Fractional ideal (2*w - 1) 13
7 Fractional ideal (2*w + 1) 12
11 Fractional ideal (11) 129
13 Fractional ideal (13) 187
17 Fractional ideal (-3*w + 1) 24
17 Fractional ideal (-3*w - 1) 24
19 Fractional ideal (19) 376
}}}
{{{id=77|
///
}}}
Sage has 3d plotting...
{{{id=72|
from scipy import io
x = io.loadmat(DATA + 'yodapose.mat', struct_as_record=True)
from sage.plot.plot3d.index_face_set import IndexFaceSet
V = x['V']; F3 = x['F3']-1; F4 = x['F4']-1
Y = (IndexFaceSet(F3, V, color = Color('#00aa00')) +
IndexFaceSet(F4, V, color = Color('#00aa00')))
Y = Y.rotateX(-1)
Y.show(aspect_ratio = [1,1,1], frame = False, figsize = 4, spin=True, zoom=1.3)
///
}}}
{{{id=81|
///
}}}
{{{id=171|
///
}}}