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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 9600.by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9600.by1 | 9600bv1 | \([0, 1, 0, -7658, 247938]\) | \(24836849888/820125\) | \(1640250000000\) | \([2]\) | \(18432\) | \(1.1170\) | \(\Gamma_0(N)\)-optimal |
9600.by2 | 9600bv2 | \([0, 1, 0, 2467, 865563]\) | \(6483584/1265625\) | \(-324000000000000\) | \([2]\) | \(36864\) | \(1.4636\) |
Rank
sage: E.rank()
The elliptic curves in class 9600.by have rank \(1\).
Complex multiplication
The elliptic curves in class 9600.by do not have complex multiplication.Modular form 9600.2.a.by
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.