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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 240240.l
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 240240.l1 | 240240l7 | \([0, -1, 0, -1692041176, 26789348552176]\) | \(130796627670002750950880364889/4007004103295286093000\) | \(16412688807097491836928000\) | \([2]\) | \(143327232\) | \(3.9358\) | |
| 240240.l2 | 240240l6 | \([0, -1, 0, -110201176, 381478856176]\) | \(36134533748915083453404889/5565686539253841000000\) | \(22797052064783732736000000\) | \([2, 2]\) | \(71663616\) | \(3.5892\) | |
| 240240.l3 | 240240l4 | \([0, -1, 0, -37022536, -27316941200]\) | \(1370131553911340548947529/714126686285699857170\) | \(2925062907026226614968320\) | \([2]\) | \(47775744\) | \(3.3865\) | |
| 240240.l4 | 240240l3 | \([0, -1, 0, -30201176, -58073143824]\) | \(743764321292317933404889/74603529000000000000\) | \(305576054784000000000000\) | \([2]\) | \(35831808\) | \(3.2426\) | |
| 240240.l5 | 240240l2 | \([0, -1, 0, -29440936, -61415945360]\) | \(688999042618248810121129/779639711718968100\) | \(3193404259200893337600\) | \([2, 2]\) | \(23887872\) | \(3.0399\) | |
| 240240.l6 | 240240l1 | \([0, -1, 0, -29432936, -61451030160]\) | \(688437529087783927489129/882972090000\) | \(3616653680640000\) | \([2]\) | \(11943936\) | \(2.6933\) | \(\Gamma_0(N)\)-optimal |
| 240240.l7 | 240240l5 | \([0, -1, 0, -21987336, -93269650320]\) | \(-286999819333751016766729/751553009101890965970\) | \(-3078361125281345396613120\) | \([2]\) | \(47775744\) | \(3.3865\) | |
| 240240.l8 | 240240l8 | \([0, -1, 0, 191638824, 2103657160176]\) | \(190026536708029086053555111/576736012771479654093000\) | \(-2362310708311980663164928000\) | \([2]\) | \(143327232\) | \(3.9358\) |
Rank
sage: E.rank()
The elliptic curves in class 240240.l have rank \(1\).
Complex multiplication
The elliptic curves in class 240240.l do not have complex multiplication.Modular form 240240.2.a.l
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.
