Demo: Elliptic Curves in Sage
October 19 at ECC 2010
William Stein
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Elliptic Curves over \QQ
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 Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field |
E = E.minimal_model(); show(E)
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E.
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Invariants
E.discriminant()
-10351 -10351 |
N = E.conductor(); N
10351 10351 |
N.factor()
11 * 941 11 * 941 |
E.tamagawa_numbers()
[1, 1] [1, 1] |
Rank
E.rank()
1 1 |
E.analytic_rank()
1 1 |
E.gens()
[(2 : 3 : 1)] [(2 : 3 : 1)] |
E.regulator()
1.26539379176449 1.26539379176449 |
E.simon_two_descent()
(1, 1, [(2 : 3 : 1)]) (1, 1, [(2 : 3 : 1)]) |
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) 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)
E.integral_points()
[(2 : 3 : 1)] [(2 : 3 : 1)] |
E.S_integral_points([2])
[(-39/64 : 279/512 : 1), (2 : 3 : 1)] [(-39/64 : 279/512 : 1), (2 : 3 : 1)] |
The Complex L-series
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 Complex L-series of the Elliptic Curve defined by y^2 + x*y = x^3 - x^2 + 4*x + 3 over Rational Field |
line([(s,L(s).real()) for s in [0.5,0.55,..,2]])
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Evaluate anywhere:
L(2+3*I)
0.763030838302007 - 0.343385137835191*I 0.763030838302007 - 0.343385137835191*I |
Taylor series about any point:
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) 3.51873114993151*z - 7.76698842523228*z^2 + 12.5750690885479*z^3 - 13.7330374564920*z^4 + 8.64019179676616*z^5 + O(z^6) |
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) -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
L.zeros(10)
[0.000000000, 0.895959461, 2.61710659, 3.23646650, 3.80592295, 4.36500628, 5.27684817, 5.77700168, 6.37368526, 7.48823858] [0.000000000, 0.895959461, 2.61710659, 3.23646650, 3.80592295, 4.36500628, 5.27684817, 5.77700168, 6.37368526, 7.48823858] |
points([(1/2,y) for y in L.zeros(20)]).show(xmin=0,xmax=1,figsize=[2,3])
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A rank 4 curve
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 Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 79*x + 289 over Rational Field |
time E.rank()
4 Time: CPU 0.00 s, Wall: 0.00 s 4 Time: CPU 0.00 s, Wall: 0.00 s |
E.gens()
[(-9 : 19 : 1), (-8 : 23 : 1), (-7 : 25 : 1), (4 : -7 : 1)] [(-9 : 19 : 1), (-8 : 23 : 1), (-7 : 25 : 1), (4 : -7 : 1)] |
L = E.lseries()
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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) 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.
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p-adic Invariants Associated to the rank 2 curve "389a"
E = EllipticCurve('389a'); E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field |
E.rank()
2 2 |
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 5-adic L-series of Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field |
show(L5.series(3))
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show(E.padic_regulator(5))
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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 Shafarevich-Tate group for the Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field |
S.an()
1.00000000000000 1.00000000000000 |
S.an_padic(5, prec=3)
1 + O(5^2) 1 + O(5^2) |
S.p_primary_bound(5)
0 0 |
Prove that Sha[29] = 0, using explicit computation with p-adic L-series, regulator, results from Iwasawa theory, etc. (See Wuthrich-Stein.)
time S.p_primary_bound(29)
0 Time: CPU 1.93 s, Wall: 1.94 s 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.)
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Shafarevich-Tate Groups
Back to our rank 1 curve, whose Sha we prove trivial below (hence proving the full BSD conjecture for this curve).
E = EllipticCurve([1..5]).minimal_model()
S = E.sha()
S.an()
1 1 |
S.bound_kato() # no info, since rank is 1
False False |
E.heegner_discriminants_list(3)
[-24, -39, -43] [-24, -39, -43] |
S.bound_kolyvagin(-39)
([2], 1) ([2], 1) |
E.selmer_rank()
1 1 |
Heegner Points: Find point over class field, then trace it down "by hand" (yes, there are better ways)
y = E.heegner_point(-39); y
Heegner point of discriminant -39 on elliptic curve of conductor 10351 Heegner point of discriminant -39 on elliptic curve of conductor 10351 |
y.numerical_approx()
(-0.888422002508970 - 0.788950457973954*I : -1.26899253838416 + 2.02424289524747*I : 1.00000000000000) (-0.888422002508970 - 0.788950457973954*I : -1.26899253838416 + 2.02424289524747*I : 1.00000000000000) |
z = y.point_exact(100); z
(1/9*a : 1/10449*a^3 - 22/3483*a^2 + 23/387*a + 114/43 : 1) (1/9*a : 1/10449*a^3 - 22/3483*a^2 + 23/387*a + 114/43 : 1) |
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 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 |
PHI = z.curve().base_ring().complex_embeddings()
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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)] [(-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)] |
sum(X)
(1.99999999999999 : 3.00000000000005 : 1.00000000000000) (1.99999999999999 : 3.00000000000005 : 1.00000000000000) |
E.gens()
[(2 : 3 : 1)] [(2 : 3 : 1)] |
Image of Galois
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 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 |
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 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 |
rho.non_surjective()
[] [] |
F = EllipticCurve([1..5])
E.is_isomorphic(F)
True True |
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) 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:
F = E.mod5family(); F
WARNING: Output truncated! full_output.txt Elliptic Curve defined by y^2 = x^3 + (-3501009128484705572621783549147802765802979905711574493729011133718735\ 123335622287491575722530633551181874092372078993245067603299212630648072\ 477239085367551350518036135409770401826716389238466712101524263102836860\ 857966781913325929377058909169355185571515987548222597315175101921509228\ 266485591201594885970015726104007641519738516843927148787787238765012214\ 250994610855909179122927249836076376451075248655564144317574222050589971\ 751270249196799737007296269630526156891001941525154840142898045009945603\ 39954531764384106772741292665339904*t^20-3076148796739621953443219074961\ 560223321319021417154542217433481890039421403133361733468069441814085528\ 551842954583515856927480386779263830554314459667991524379298105797089586\ 641140433228801228410106047409502556922621995718848910503560837300872037\ 541227759970904447681142751663853907541247685568093133174022760283293193\ 431559118771450835627388250514425123655428923179973280471609300183262344\ 137220027781119360322565674277599408304848530625914691755189946375875290\ 182368565932293775457411843189471819619076958858710326520370180451981066\ 240*t^19-925580595451510413913479645310950487591021666358984891954860352\ 760921436685659179745015421994351343000192585226077113580977916498260785\ 609995813905820130419241372006717538795573837832649706393311990212796300\ 775370071846930779107388447691205006870990898412985747660732113618727947\ 793695659585114625558623625515410189273650694436617165248595452409236098\ 727787780089417533112591326366493828478019568381372339196709933766665886\ 210416179674965572311488547578229903802104365377194436397668502156139840\ 9853716872744762532370949276783161198510080*t^18-19397013449155320116122\ 269126763868609414612020482748191142375841983183486810718516386492337645\ 036452836314625194877805425950009935690179217819728655275526230940295981\ 347238035715348090570054869224742868936210569879798936643390185822146593\ 403579732595328213957213839826551749150139201891925648214323765604948255\ 786188599198288718346287937814342642254971778504394083272770293660706608\ 881106176260975328036488365620284230377669104532085302934447006783353481\ 701070318387373697332532472683862310554447900899741133492764684208892853\ 235713310720*t^17-333222569066086657624869114431728227684592031304742713\ 702238734578063778925135287895431499653951471517153425966778127775217375\ 116343017126271511520069577259505860065585707241330013952368169959453302\ 055065571995142854838827008988918431010864099061786766888951957369301000\ 746429286212171864436943193054763106198310906908290505302500694846812827\ 035170835505573677481089123580168462785452322302164487438405450217479107\ 664637249188610842239069968342389694447404313273865460347718102328421433\ 52195031615390772584229096246184065652637527179264000*t^16-4060944790701\ 819273521391280496206820567096275790306239034025831360563578531249481276\ 289812456217720764253285180354303591241427812240226093640440291863616157\ 205966550377631301388757869461571220418741024215886533312634539459073490\ 611574586185133939090169220197298640609188615870762795508500621680264527\ 201168929101952002883897582240847001681403444740252458904125048878406936\ 762458750920999910598629057168105852855566988133379059165015055476679556\ 913258473264142357374276866322595424224485033622746215653262022065026475\ 6353552932812582027264*t^15-31731901206502079507143908513980355856508398\ 758627105660839562374610621021939981435759508452202444560181593622165934\ 304607430156057005663506319818063785026055713655623777066128714741695000\ 804228746783981925948781105147069396966508865777790939490722482041065835\ 749288984045987520798603906530794297933289745146616203500035712653712212\ 911683847391197615597426335260093161827629559683658564946467384115000672\ 726092868187518169623808965646760662245986066750501648083311609291925866\ 992781834914925432160951632497766724270759032756603288268308480*t^14-146\ 857250128979223318474479172847220472755571204816211121093930235377977208\ 537077576901758371528482744359941355838822194836853937803408930628662613\ 527565651920131341473456734005306755696798438405972194809457220462093969\ 066343952338462972387103378114752355674768484706470449768176498016380225\ 132436458597715137666126448294464157158042804179697299339658658487171141\ 674935095695545817351980046856034297090538171478637223019250580916157247\ 603336184481106859075804404229546178419846506446922631340767668650081739\ 24160321376901012785997398671360*t^13-6305671210764502404382844686599298\ 061187190279662192697272491442184177009184745323105549734834664960742044\ 636619467780798299989643839382498153413555330282920116492251253654690509\ 789556887359262851284730283138844983071419228359335848121747023531694626\ 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78699862190047059159955939451487651687704666385337340515947150894363358226034575233141745631477821113345581207536344760320*t+2293140081272999525880457735045306060525868282393980175678905055643499661042037620011121678960611236555415388966528539008455164438681004548185750833175346387407676738839197859431069791043575439630761173504871678458866133404091360732231634033824998248213892682638261956830553391753468074009778838536521407287542973867114580046398016345735904437287294801628396023730079843290661314052657633669714875229666193231636703897296298062756517907025710898126952418506345838766905041393004115972708227939925988927409896884348575741774605332076432222834932791167924549508628304410623246027264392175361765635857274517912345798276489875577687079863687010967693046538138260774075298344948403219289995406334901971998531315049588389590155546177911350531571696478778607534485841104724854382594037111519373584544825344) over Fraction Field of Univariate Polynomial Ring in t over Rational Field |
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 - 640804168893951905637456113348245371201487663923467637718566216386653978\ 913277921550075928125742*x - 987708462428441287745589163528266452889910754422041787496992768933756640\ 08383305935974004704968720083222472651422915574042236889700867998179209 over Rational Field Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 640804168893951905637456113348245371201487663923467637718566216386653978913277921550075928125742*x - 98770846242844128774558916352826645288991075442204178749699276893375664008383305935974004704968720083222472651422915574042236889700867998179209 over Rational Field |
E.aplist(50)
[1, 0, -3, -1, -1, 1, 5, 4, -6, -2, 2, 7, 6, 8, 0] [1, 0, -3, -1, -1, 1, 5, 4, -6, -2, 2, 7, 6, 8, 0] |
G.aplist(50)
[1, 0, 0, -1, -1, 6, 5, -6, -6, -7, -8, -3, 6, 8, 0] [1, 0, 0, -1, -1, 6, 5, -6, -6, -7, -8, -3, 6, 8, 0] |
factor(G.conductor())
5^2 * 11 * 941 * 109425295591 * 315154685475065659 * 4706373428975316362621837 5^2 * 11 * 941 * 109425295591 * 315154685475065659 * 4706373428975316362621837 |
factor(E.conductor())
11 * 941 11 * 941 |
Plotting
plot(E, plot_points=300, thickness=2, color=hue(.75), xmax=2)
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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)
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Elliptic Curves over Finite Fields
k.<a> = GF(7^6); k
Finite Field in a of size 7^6 Finite Field in a of size 7^6 |
E = EllipticCurve(k, [a,a^2])
E.cardinality()
117837 117837 |
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 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:
E = EllipticCurve([1..5])
time ap = E.aplist(10^6)
Time: CPU 2.28 s, Wall: 2.29 s Time: CPU 2.28 s, Wall: 2.29 s |
ap[:10]
[1, 0, -3, -1, -1, 1, 5, 4, -6, -2] [1, 0, -3, -1, -1, 1, 5, 4, -6, -2] |
Visualize the Sato-Tate Distribution:
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)
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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.
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Point counting over big fields
E = EllipticCurve(GF(next_prime(10^50)), [1..5])
time E.cardinality()
99999999999999999999999982865176588640916234222362 Time: CPU 0.03 s, Wall: 1.55 s 99999999999999999999999982865176588640916234222362 Time: CPU 0.03 s, Wall: 1.55 s |
Example with non-prime base field (but j-invariant in the prime subfield).
p = next_prime(10^50)
q = p^5
E = EllipticCurve(GF(q,'a'), [1..5])
time E.cardinality()
100000000000000000000000000000000000000000000000755000000000000000000000\ 000000000000000000000002280100000000000000000000000001816094313409904919\ 381624718891635914427680335413988790131632102994015424674728418893726390\ 05122561820006471215426251120821802 Time: CPU 0.36 s, Wall: 1.65 s 10000000000000000000000000000000000000000000000075500000000000000000000000000000000000000000000228010000000000000000000000000181609431340990491938162471889163591442768033541398879013163210299401542467472841889372639005122561820006471215426251120821802 Time: CPU 0.36 s, Wall: 1.65 s |
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The Free Abelian Group on the Supersingular j-invariants.
X = SupersingularModule(389); X
Module of supersingular points on X_0(1)/F_389 over Integer Ring Module of supersingular points on X_0(1)/F_389 over Integer Ring |
T2 = X.hecke_matrix(2)
DiGraph(T2).show(frame=True, figsize=5)
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graph_editor(DiGraph(T2)) # WARNING: converts to simple graph (now)
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Brandt Modules: Free abelian group on enhanced elliptic supersingular curves (or right ideal classes in an Eichler order)
M = BrandtModule(389, 2); M
Brandt module of dimension 97 of level 389*2 of weight 2 over Rational Field Brandt module of dimension 97 of level 389*2 of weight 2 over Rational Field |
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) 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) |
M.quaternion_algebra()
Quaternion Algebra (-2, -389) with base ring Rational Field Quaternion Algebra (-2, -389) with base ring Rational Field |
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)) (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)) |
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) (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) |
graph_editor(DiGraph(T2))
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ECM: Elliptic Curve Factorization
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 /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 |
time ecm.factor((2^197 + 1)/3)
[197002597249, 1348959352853811313, 251951573867253012259144010843] Time: CPU 0.08 s, Wall: 4.05 s [197002597249, 1348959352853811313, 251951573867253012259144010843] Time: CPU 0.08 s, Wall: 4.05 s |
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Plot of elliptic curve over finite field
@interact
def f(p=(389,primes(3,1000))):
show(EllipticCurve(GF(p), [1..5]).plot(), frame=True)
Click to the left again to hide and once more to show the dynamic interactive window |
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Elliptic Curves over Number Fields
K.<w> = QuadraticField(2); K
Number Field in w with defining polynomial x^2 - 2 Number Field in w with defining polynomial x^2 - 2 |
E = EllipticCurve([ 0, -1, 1, -3*w -4, 3*w + 4 ]); show(E)
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show(E.j_invariant())
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N = E.conductor(); N
Fractional ideal (-24*w + 11) Fractional ideal (-24*w + 11) |
N.norm().factor()
1031 1031 |
E.tamagawa_numbers()
[1] [1] |
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)]) Time: CPU 0.11 s, Wall: 0.54 s (2, 2, [(w + 3/2 : -1/4*w - 1 : 1), (-w - 1 : -w - 2 : 1)]) |
P, Q = T[2]
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P.height()
0.933266708161421 0.933266708161421 |
Q.height()
0.183656956219703 0.183656956219703 |
E.height_pairing_matrix([P,Q]).det()
0.0235582958658636 0.0235582958658636 |
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? 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? |
Lambda.basis()
(10.7444017502464, 5.37220087512321 + 2.12681775202610*I) (10.7444017502464, 5.37220087512321 + 2.12681775202610*I) |
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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)?
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 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 |
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Sage has 3d plotting...
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)
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