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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 77616by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
77616.gs1 | 77616by1 | \([0, 0, 0, -3535203, 2558406130]\) | \(55635379958596/24057\) | \(2112794186646528\) | \([2]\) | \(1935360\) | \(2.2830\) | \(\Gamma_0(N)\)-optimal |
77616.gs2 | 77616by2 | \([0, 0, 0, -3517563, 2585201290]\) | \(-27403349188178/578739249\) | \(-101654979496311048192\) | \([2]\) | \(3870720\) | \(2.6295\) |
Rank
sage: E.rank()
The elliptic curves in class 77616by have rank \(1\).
Complex multiplication
The elliptic curves in class 77616by do not have complex multiplication.Modular form 77616.2.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 Cremona 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 Cremona labels.