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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 32487.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32487.d1 | 32487n4 | \([1, 0, 0, -1212849, 514011504]\) | \(1677087406638588673/4641\) | \(546009009\) | \([2]\) | \(245760\) | \(1.7949\) | |
| 32487.d2 | 32487n2 | \([1, 0, 0, -75804, 8026479]\) | \(409460675852593/21538881\) | \(2534027810769\) | \([2, 2]\) | \(122880\) | \(1.4483\) | |
| 32487.d3 | 32487n3 | \([1, 0, 0, -71639, 8948610]\) | \(-345608484635233/94427721297\) | \(-11109326982870753\) | \([2]\) | \(245760\) | \(1.7949\) | |
| 32487.d4 | 32487n1 | \([1, 0, 0, -4999, 110480]\) | \(117433042273/22801233\) | \(2682542261217\) | \([2]\) | \(61440\) | \(1.1018\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32487.d have rank \(1\).
Complex multiplication
The elliptic curves in class 32487.d do not have complex multiplication.Modular form 32487.2.a.d
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.
