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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 4641.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4641.e1 | 4641d1 | \([1, 0, 1, -46, 107]\) | \(10431681625/710073\) | \(710073\) | \([2]\) | \(768\) | \(-0.12872\) | \(\Gamma_0(N)\)-optimal |
4641.e2 | 4641d2 | \([1, 0, 1, 39, 481]\) | \(6804992375/102626433\) | \(-102626433\) | \([2]\) | \(1536\) | \(0.21785\) |
Rank
sage: E.rank()
The elliptic curves in class 4641.e have rank \(1\).
Complex multiplication
The elliptic curves in class 4641.e do not have complex multiplication.Modular form 4641.2.a.e
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.