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SageMath
E = EllipticCurve("gt1")
E.isogeny_class()
Elliptic curves in class 493680gt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
493680.gt4 | 493680gt1 | \([0, 1, 0, -6868000, -1138723852]\) | \(4937402992298041/2780405760000\) | \(20175496841590210560000\) | \([2]\) | \(26542080\) | \(2.9700\) | \(\Gamma_0(N)\)-optimal* |
493680.gt2 | 493680gt2 | \([0, 1, 0, -68820000, 218592629748]\) | \(4967657717692586041/29490113030400\) | \(213989515797497669222400\) | \([2, 2]\) | \(53084160\) | \(3.3166\) | \(\Gamma_0(N)\)-optimal* |
493680.gt1 | 493680gt3 | \([0, 1, 0, -1099546400, 14033212423668]\) | \(20260414982443110947641/720358602480\) | \(5227148108464419962880\) | \([4]\) | \(106168320\) | \(3.6631\) | \(\Gamma_0(N)\)-optimal* |
493680.gt3 | 493680gt4 | \([0, 1, 0, -29325600, 467770698228]\) | \(-384369029857072441/12804787777021680\) | \(-92915558969541854996398080\) | \([2]\) | \(106168320\) | \(3.6631\) |
Rank
sage: E.rank()
The elliptic curves in class 493680gt have rank \(0\).
Complex multiplication
The elliptic curves in class 493680gt do not have complex multiplication.Modular form 493680.2.a.gt
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.