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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 458640.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
458640.s1 | 458640s2 | \([0, 0, 0, -1756083, -589328782]\) | \(584759426925367/191909250000\) | \(196552016833536000000\) | \([2]\) | \(11796480\) | \(2.5970\) | |
458640.s2 | 458640s1 | \([0, 0, 0, -707763, 222280562]\) | \(38282975119927/1314144000\) | \(1345936444489728000\) | \([2]\) | \(5898240\) | \(2.2504\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 458640.s have rank \(2\).
Complex multiplication
The elliptic curves in class 458640.s do not have complex multiplication.Modular form 458640.2.a.s
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.