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SageMath
E = EllipticCurve("fg1")
E.isogeny_class()
Elliptic curves in class 221760.fg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.fg1 | 221760db4 | \([0, 0, 0, -177708, -28833968]\) | \(12990838708516/144375\) | \(6897623040000\) | \([2]\) | \(786432\) | \(1.6170\) | |
| 221760.fg2 | 221760db2 | \([0, 0, 0, -11388, -426512]\) | \(13674725584/1334025\) | \(15933509222400\) | \([2, 2]\) | \(393216\) | \(1.2704\) | |
| 221760.fg3 | 221760db1 | \([0, 0, 0, -2568, 42712]\) | \(2508888064/396165\) | \(295735587840\) | \([2]\) | \(196608\) | \(0.92383\) | \(\Gamma_0(N)\)-optimal |
| 221760.fg4 | 221760db3 | \([0, 0, 0, 13812, -2049392]\) | \(6099383804/41507235\) | \(-1983039033507840\) | \([2]\) | \(786432\) | \(1.6170\) |
Rank
sage: E.rank()
The elliptic curves in class 221760.fg have rank \(1\).
Complex multiplication
The elliptic curves in class 221760.fg do not have complex multiplication.Modular form 221760.2.a.fg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
