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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 30345.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 30345.c1 | 30345c6 | \([1, 1, 1, -20732866, -36274177462]\) | \(40832710302042509761/91556816413125\) | \(2209958973592137193125\) | \([2]\) | \(2359296\) | \(2.9778\) | |
| 30345.c2 | 30345c4 | \([1, 1, 1, -1767241, -118109962]\) | \(25288177725059761/14387797265625\) | \(347286449257034765625\) | \([2, 2]\) | \(1179648\) | \(2.6312\) | |
| 30345.c3 | 30345c2 | \([1, 1, 1, -1129996, 459743804]\) | \(6610905152742241/35128130625\) | \(847907676801950625\) | \([2, 2]\) | \(589824\) | \(2.2847\) | |
| 30345.c4 | 30345c1 | \([1, 1, 1, -1128551, 460985348]\) | \(6585576176607121/187425\) | \(4523983869825\) | \([4]\) | \(294912\) | \(1.9381\) | \(\Gamma_0(N)\)-optimal |
| 30345.c5 | 30345c3 | \([1, 1, 1, -515871, 958167654]\) | \(-629004249876241/16074715228425\) | \(-388004547981459198825\) | \([2]\) | \(1179648\) | \(2.6312\) | |
| 30345.c6 | 30345c5 | \([1, 1, 1, 7002464, -931938586]\) | \(1573196002879828319/926055908203125\) | \(-22352738382110595703125\) | \([2]\) | \(2359296\) | \(2.9778\) |
Rank
sage: E.rank()
The elliptic curves in class 30345.c have rank \(1\).
Complex multiplication
The elliptic curves in class 30345.c do not have complex multiplication.Modular form 30345.2.a.c
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.
