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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 390390.ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 390390.ek1 | 390390ek8 | \([1, 0, 0, -17872184925, -919610233582743]\) | \(130796627670002750950880364889/4007004103295286093000\) | \(19341043468822616571267237000\) | \([2]\) | \(1003290624\) | \(4.5251\) | |
| 390390.ek2 | 390390ek6 | \([1, 0, 0, -1163999925, -13094289859743]\) | \(36134533748915083453404889/5565686539253841000000\) | \(26864505878849293023369000000\) | \([2, 2]\) | \(501645312\) | \(4.1785\) | |
| 390390.ek3 | 390390ek5 | \([1, 0, 0, -391050540, 938130422670]\) | \(1370131553911340548947529/714126686285699857170\) | \(3446953116503992641886870530\) | \([2]\) | \(334430208\) | \(3.9758\) | |
| 390390.ek4 | 390390ek3 | \([1, 0, 0, -318999925, 1993861140257]\) | \(743764321292317933404889/74603529000000000000\) | \(360096985208961000000000000\) | \([4]\) | \(250822656\) | \(3.8320\) | |
| 390390.ek5 | 390390ek2 | \([1, 0, 0, -310969890, 2108605219200]\) | \(688999042618248810121129/779639711718968100\) | \(3763171977282520695792900\) | \([2, 2]\) | \(167215104\) | \(3.6292\) | |
| 390390.ek6 | 390390ek1 | \([1, 0, 0, -310885390, 2109809530100]\) | \(688437529087783927489129/882972090000\) | \(4261937630760810000\) | \([4]\) | \(83607552\) | \(3.2826\) | \(\Gamma_0(N)\)-optimal |
| 390390.ek7 | 390390ek4 | \([1, 0, 0, -232241240, 3202004456130]\) | \(-286999819333751016766729/751553009101890965970\) | \(-3627602828310089231562689730\) | \([2]\) | \(334430208\) | \(3.9758\) | |
| 390390.ek8 | 390390ek7 | \([1, 0, 0, 2024185075, -72216630136743]\) | \(190026536708029086053555111/576736012771479654093000\) | \(-2783794577069492937692979237000\) | \([2]\) | \(1003290624\) | \(4.5251\) |
Rank
sage: E.rank()
The elliptic curves in class 390390.ek have rank \(0\).
Complex multiplication
The elliptic curves in class 390390.ek do not have complex multiplication.Modular form 390390.2.a.ek
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
