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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 284592bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 284592.bd3 | 284592bd1 | \([0, -1, 0, -3480158964, 79022926981440]\) | \(87364831012240243408/1760913\) | \(93955501718953281792\) | \([2]\) | \(132710400\) | \(3.8159\) | \(\Gamma_0(N)\)-optimal |
| 284592.bd2 | 284592bd2 | \([0, -1, 0, -3480277544, 79017272707584]\) | \(21843440425782779332/3100814593569\) | \(661789857593708721200784384\) | \([2, 2]\) | \(265420800\) | \(4.1625\) | |
| 284592.bd4 | 284592bd3 | \([0, -1, 0, -3166514864, 93843186862944]\) | \(-8226100326647904626/4152140742401883\) | \(-1772337266679619373906549938176\) | \([2]\) | \(530841600\) | \(4.5090\) | |
| 284592.bd1 | 284592bd4 | \([0, -1, 0, -3795937504, 63829483128160]\) | \(14171198121996897746/4077720290568771\) | \(1740570968673386314456270067712\) | \([2]\) | \(530841600\) | \(4.5090\) |
Rank
sage: E.rank()
The elliptic curves in class 284592bd have rank \(1\).
Complex multiplication
The elliptic curves in class 284592bd do not have complex multiplication.Modular form 284592.2.a.bd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
