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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 62400.ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 62400.ef1 | 62400gu4 | \([0, 1, 0, -125633, -17179137]\) | \(428320044872/73125\) | \(37440000000000\) | \([2]\) | \(294912\) | \(1.6110\) | |
| 62400.ef2 | 62400gu3 | \([0, 1, 0, -53633, 4600863]\) | \(33324076232/1285245\) | \(658045440000000\) | \([2]\) | \(294912\) | \(1.6110\) | |
| 62400.ef3 | 62400gu2 | \([0, 1, 0, -8633, -214137]\) | \(1111934656/342225\) | \(21902400000000\) | \([2, 2]\) | \(147456\) | \(1.2645\) | |
| 62400.ef4 | 62400gu1 | \([0, 1, 0, 1492, -21762]\) | \(367061696/426465\) | \(-426465000000\) | \([2]\) | \(73728\) | \(0.91789\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 62400.ef have rank \(1\).
Complex multiplication
The elliptic curves in class 62400.ef do not have complex multiplication.Modular form 62400.2.a.ef
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
