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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 346560ex
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
346560.ex2 | 346560ex1 | \([0, -1, 0, -33755425, 104391847105]\) | \(-50284268371/26542080\) | \(-2245211636227385673646080\) | \([2]\) | \(37355520\) | \(3.3768\) | \(\Gamma_0(N)\)-optimal |
346560.ex1 | 346560ex2 | \([0, -1, 0, -595644705, 5594836757697]\) | \(276288773643091/41990400\) | \(3551994971375356241510400\) | \([2]\) | \(74711040\) | \(3.7233\) |
Rank
sage: E.rank()
The elliptic curves in class 346560ex have rank \(1\).
Complex multiplication
The elliptic curves in class 346560ex do not have complex multiplication.Modular form 346560.2.a.ex
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 Cremona 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 Cremona labels.