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SageMath
E = EllipticCurve("fr1")
E.isogeny_class()
Elliptic curves in class 52800.fr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 52800.fr1 | 52800gr4 | \([0, 1, 0, -47233, -3966337]\) | \(5690357426/891\) | \(1824768000000\) | \([2]\) | \(131072\) | \(1.3637\) | |
| 52800.fr2 | 52800gr2 | \([0, 1, 0, -3233, -50337]\) | \(3650692/1089\) | \(1115136000000\) | \([2, 2]\) | \(65536\) | \(1.0171\) | |
| 52800.fr3 | 52800gr1 | \([0, 1, 0, -1233, 15663]\) | \(810448/33\) | \(8448000000\) | \([2]\) | \(32768\) | \(0.67056\) | \(\Gamma_0(N)\)-optimal |
| 52800.fr4 | 52800gr3 | \([0, 1, 0, 8767, -326337]\) | \(36382894/43923\) | \(-89954304000000\) | \([2]\) | \(131072\) | \(1.3637\) |
Rank
sage: E.rank()
The elliptic curves in class 52800.fr have rank \(0\).
Complex multiplication
The elliptic curves in class 52800.fr do not have complex multiplication.Modular form 52800.2.a.fr
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.
