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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 690.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
690.j1 | 690j2 | \([1, 0, 0, -3925, -94975]\) | \(6687281588245201/165600\) | \(165600\) | \([2]\) | \(480\) | \(0.51991\) | |
690.j2 | 690j1 | \([1, 0, 0, -245, -1503]\) | \(-1626794704081/8125440\) | \(-8125440\) | \([2]\) | \(240\) | \(0.17333\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 690.j have rank \(0\).
Complex multiplication
The elliptic curves in class 690.j do not have complex multiplication.Modular form 690.2.a.j
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.