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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 363726.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
363726.p1 | 363726p2 | \([1, -1, 0, -762867, 256630369]\) | \(38017791015625/3681348\) | \(4754343024742212\) | \([2]\) | \(3440640\) | \(2.0441\) | |
363726.p2 | 363726p1 | \([1, -1, 0, -44127, 4640125]\) | \(-7357983625/2909808\) | \(-3757923827939952\) | \([2]\) | \(1720320\) | \(1.6975\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 363726.p have rank \(1\).
Complex multiplication
The elliptic curves in class 363726.p do not have complex multiplication.Modular form 363726.2.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.