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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 51714.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51714.g1 | 51714g1 | \([1, -1, 0, -29067102, 58309856788]\) | \(771864882375147625/29358565696512\) | \(103305269876510219461632\) | \([2]\) | \(4300800\) | \(3.1842\) | \(\Gamma_0(N)\)-optimal |
51714.g2 | 51714g2 | \([1, -1, 0, 12060738, 210063360820]\) | \(55138849409108375/5449537181735712\) | \(-19175524958570059750892832\) | \([2]\) | \(8601600\) | \(3.5307\) |
Rank
sage: E.rank()
The elliptic curves in class 51714.g have rank \(1\).
Complex multiplication
The elliptic curves in class 51714.g do not have complex multiplication.Modular form 51714.2.a.g
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.