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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 69360cb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69360.q6 | 69360cb1 | \([0, -1, 0, -370016, -106318080]\) | \(-56667352321/16711680\) | \(-1652241732017848320\) | \([2]\) | \(884736\) | \(2.2102\) | \(\Gamma_0(N)\)-optimal |
| 69360.q5 | 69360cb2 | \([0, -1, 0, -6288736, -6067652864]\) | \(278202094583041/16646400\) | \(1645787662752153600\) | \([2, 2]\) | \(1769472\) | \(2.5568\) | |
| 69360.q4 | 69360cb3 | \([0, -1, 0, -6658656, -5313312000]\) | \(330240275458561/67652010000\) | \(6688583923153674240000\) | \([2, 2]\) | \(3538944\) | \(2.9033\) | |
| 69360.q2 | 69360cb4 | \([0, -1, 0, -100618336, -388442119424]\) | \(1139466686381936641/4080\) | \(403379329105920\) | \([2]\) | \(3538944\) | \(2.9033\) | |
| 69360.q7 | 69360cb5 | \([0, -1, 0, 14149344, -31897612800]\) | \(3168685387909439/6278181696900\) | \(-620708019828575584665600\) | \([2]\) | \(7077888\) | \(3.2499\) | |
| 69360.q3 | 69360cb6 | \([0, -1, 0, -33385376, 69542885376]\) | \(41623544884956481/2962701562500\) | \(292914845250566400000000\) | \([2, 2]\) | \(7077888\) | \(3.2499\) | |
| 69360.q8 | 69360cb7 | \([0, -1, 0, 30287104, 303399169920]\) | \(31077313442863199/420227050781250\) | \(-41546790641250000000000000\) | \([2]\) | \(14155776\) | \(3.5965\) | |
| 69360.q1 | 69360cb8 | \([0, -1, 0, -524685376, 4626055605376]\) | \(161572377633716256481/914742821250\) | \(90438319985363112960000\) | \([2]\) | \(14155776\) | \(3.5965\) |
Rank
sage: E.rank()
The elliptic curves in class 69360cb have rank \(0\).
Complex multiplication
The elliptic curves in class 69360cb do not have complex multiplication.Modular form 69360.2.a.cb
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
