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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1012.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1012.c1 | 1012b2 | \([0, 1, 0, -2278, -42635]\) | \(-81743931616000/40745903\) | \(-651934448\) | \([]\) | \(648\) | \(0.64420\) | |
1012.c2 | 1012b1 | \([0, 1, 0, 22, -223]\) | \(70304000/1472207\) | \(-23555312\) | \([3]\) | \(216\) | \(0.094893\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1012.c have rank \(1\).
Complex multiplication
The elliptic curves in class 1012.c do not have complex multiplication.Modular form 1012.2.a.c
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.