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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 86640by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86640.s2 | 86640by1 | \([0, -1, 0, 8544, -7767744]\) | \(357911/135375\) | \(-26086752830976000\) | \([2]\) | \(552960\) | \(1.8291\) | \(\Gamma_0(N)\)-optimal |
86640.s1 | 86640by2 | \([0, -1, 0, -540176, -148679040]\) | \(90458382169/2671875\) | \(514870121664000000\) | \([2]\) | \(1105920\) | \(2.1757\) |
Rank
sage: E.rank()
The elliptic curves in class 86640by have rank \(1\).
Complex multiplication
The elliptic curves in class 86640by do not have complex multiplication.Modular form 86640.2.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.