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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 31200.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31200.s1 | 31200bo1 | \([0, -1, 0, -264758, 52428012]\) | \(2052450196928704/4317958125\) | \(4317958125000000\) | \([2]\) | \(221184\) | \(1.8852\) | \(\Gamma_0(N)\)-optimal |
31200.s2 | 31200bo2 | \([0, -1, 0, -173633, 88969137]\) | \(-9045718037056/48125390625\) | \(-3080025000000000000\) | \([2]\) | \(442368\) | \(2.2318\) |
Rank
sage: E.rank()
The elliptic curves in class 31200.s have rank \(1\).
Complex multiplication
The elliptic curves in class 31200.s do not have complex multiplication.Modular form 31200.2.a.s
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.