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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 259920.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 259920.e1 | 259920e4 | \([0, 0, 0, -1560603, -696339398]\) | \(11968836484/961875\) | \(33780628682375040000\) | \([2]\) | \(5898240\) | \(2.4904\) | |
| 259920.e2 | 259920e2 | \([0, 0, 0, -325983, 59001118]\) | \(436334416/81225\) | \(713146605516806400\) | \([2, 2]\) | \(2949120\) | \(2.1438\) | |
| 259920.e3 | 259920e1 | \([0, 0, 0, -309738, 66347107]\) | \(5988775936/285\) | \(156391799455440\) | \([2]\) | \(1474560\) | \(1.7972\) | \(\Gamma_0(N)\)-optimal |
| 259920.e4 | 259920e3 | \([0, 0, 0, 648717, 344198338]\) | \(859687196/1954815\) | \(-68652246557751229440\) | \([2]\) | \(5898240\) | \(2.4904\) |
Rank
sage: E.rank()
The elliptic curves in class 259920.e have rank \(1\).
Complex multiplication
The elliptic curves in class 259920.e do not have complex multiplication.Modular form 259920.2.a.e
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.
