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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1617.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1617.i1 | 1617f2 | \([1, 0, 1, -174466, -17340319]\) | \(14553591673375/5208653241\) | \(210187945886590287\) | \([2]\) | \(17920\) | \(2.0248\) | |
1617.i2 | 1617f1 | \([1, 0, 1, 33049, -1901203]\) | \(98931640625/96059601\) | \(-3876351387330807\) | \([2]\) | \(8960\) | \(1.6782\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1617.i have rank \(1\).
Complex multiplication
The elliptic curves in class 1617.i do not have complex multiplication.Modular form 1617.2.a.i
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.