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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 171462.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
171462.e1 | 171462n1 | \([1, 0, 1, -4238, 67052]\) | \(1771561/612\) | \(2907063795492\) | \([2]\) | \(532480\) | \(1.0933\) | \(\Gamma_0(N)\)-optimal |
171462.e2 | 171462n2 | \([1, 0, 1, 12572, 470492]\) | \(46268279/46818\) | \(-222390380355138\) | \([2]\) | \(1064960\) | \(1.4398\) |
Rank
sage: E.rank()
The elliptic curves in class 171462.e have rank \(1\).
Complex multiplication
The elliptic curves in class 171462.e do not have complex multiplication.Modular form 171462.2.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.