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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 257600.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
257600.p1 | 257600p1 | \([0, 1, 0, -5033, -136937]\) | \(220348864/4025\) | \(257600000000\) | \([2]\) | \(344064\) | \(0.98448\) | \(\Gamma_0(N)\)-optimal |
257600.p2 | 257600p2 | \([0, 1, 0, -33, -391937]\) | \(-8/129605\) | \(-66357760000000\) | \([2]\) | \(688128\) | \(1.3311\) |
Rank
sage: E.rank()
The elliptic curves in class 257600.p have rank \(0\).
Complex multiplication
The elliptic curves in class 257600.p do not have complex multiplication.Modular form 257600.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.