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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 2142.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2142.g1 | 2142d3 | \([1, -1, 0, -45711, -3750251]\) | \(14489843500598257/6246072\) | \(4553386488\) | \([2]\) | \(6144\) | \(1.1959\) | |
2142.g2 | 2142d4 | \([1, -1, 0, -6111, 98725]\) | \(34623662831857/14438442312\) | \(10525624445448\) | \([2]\) | \(6144\) | \(1.1959\) | |
2142.g3 | 2142d2 | \([1, -1, 0, -2871, -57443]\) | \(3590714269297/73410624\) | \(53516344896\) | \([2, 2]\) | \(3072\) | \(0.84928\) | |
2142.g4 | 2142d1 | \([1, -1, 0, 9, -2723]\) | \(103823/4386816\) | \(-3197988864\) | \([2]\) | \(1536\) | \(0.50271\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2142.g have rank \(0\).
Complex multiplication
The elliptic curves in class 2142.g do not have complex multiplication.Modular form 2142.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.