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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 25872.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25872.p1 | 25872bc2 | \([0, -1, 0, -4575048, 3768057840]\) | \(-448504189023625/135168\) | \(-3191671281942528\) | \([]\) | \(435456\) | \(2.3391\) | |
25872.p2 | 25872bc1 | \([0, -1, 0, -47448, 6889968]\) | \(-500313625/574992\) | \(-13577070414200832\) | \([]\) | \(145152\) | \(1.7898\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25872.p have rank \(0\).
Complex multiplication
The elliptic curves in class 25872.p do not have complex multiplication.Modular form 25872.2.a.p
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.