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SageMath
E = EllipticCurve("ga1")
E.isogeny_class()
Elliptic curves in class 284592.ga
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
284592.ga1 | 284592ga1 | \([0, 1, 0, -47528840, 126104103684]\) | \(55635379958596/24057\) | \(5134353610548298752\) | \([2]\) | \(29030400\) | \(2.9326\) | \(\Gamma_0(N)\)-optimal |
284592.ga2 | 284592ga2 | \([0, 1, 0, -47291680, 127425084884]\) | \(-27403349188178/578739249\) | \(-247034289617920846153728\) | \([2]\) | \(58060800\) | \(3.2792\) |
Rank
sage: E.rank()
The elliptic curves in class 284592.ga have rank \(2\).
Complex multiplication
The elliptic curves in class 284592.ga do not have complex multiplication.Modular form 284592.2.a.ga
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.