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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 69366.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69366.i1 | 69366i1 | \([1, 0, 1, -7906, -271180]\) | \(54640564143369625/6250708992\) | \(6250708992\) | \([2]\) | \(90048\) | \(0.90691\) | \(\Gamma_0(N)\)-optimal |
69366.i2 | 69366i2 | \([1, 0, 1, -7266, -316748]\) | \(-42415439299593625/18630677653632\) | \(-18630677653632\) | \([2]\) | \(180096\) | \(1.2535\) |
Rank
sage: E.rank()
The elliptic curves in class 69366.i have rank \(1\).
Complex multiplication
The elliptic curves in class 69366.i do not have complex multiplication.Modular form 69366.2.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.