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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 31713.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31713.c1 | 31713h2 | \([1, 0, 0, -138404, -18070431]\) | \(11089567/1089\) | \(28792748532970719\) | \([2]\) | \(222208\) | \(1.8951\) | |
31713.c2 | 31713h1 | \([1, 0, 0, 10551, -1357680]\) | \(4913/33\) | \(-872507531302143\) | \([2]\) | \(111104\) | \(1.5486\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 31713.c have rank \(0\).
Complex multiplication
The elliptic curves in class 31713.c do not have complex multiplication.Modular form 31713.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.