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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 62475.cj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 62475.cj1 | 62475cg6 | \([1, 0, 1, -87881526, 316476489823]\) | \(40832710302042509761/91556816413125\) | \(168305748346683486328125\) | \([2]\) | \(9437184\) | \(3.3389\) | |
| 62475.cj2 | 62475cg4 | \([1, 0, 1, -7490901, 1023677323]\) | \(25288177725059761/14387797265625\) | \(26448593132867431640625\) | \([2, 2]\) | \(4718592\) | \(2.9923\) | |
| 62475.cj3 | 62475cg2 | \([1, 0, 1, -4789776, -4016621927]\) | \(6610905152742241/35128130625\) | \(64574834998447265625\) | \([2, 2]\) | \(2359296\) | \(2.6457\) | |
| 62475.cj4 | 62475cg1 | \([1, 0, 1, -4783651, -4027450927]\) | \(6585576176607121/187425\) | \(344536934765625\) | \([2]\) | \(1179648\) | \(2.2992\) | \(\Gamma_0(N)\)-optimal |
| 62475.cj5 | 62475cg3 | \([1, 0, 1, -2186651, -8363840677]\) | \(-629004249876241/16074715228425\) | \(-29549596436077700390625\) | \([2]\) | \(4718592\) | \(2.9923\) | |
| 62475.cj6 | 62475cg5 | \([1, 0, 1, 29681724, 8160821323]\) | \(1573196002879828319/926055908203125\) | \(-1702336742877960205078125\) | \([2]\) | \(9437184\) | \(3.3389\) |
Rank
sage: E.rank()
The elliptic curves in class 62475.cj have rank \(0\).
Complex multiplication
The elliptic curves in class 62475.cj do not have complex multiplication.Modular form 62475.2.a.cj
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
