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SageMath
E = EllipticCurve("gc1")
E.isogeny_class()
Elliptic curves in class 296450gc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
296450.gc4 | 296450gc1 | \([1, 0, 0, 1479162, 5659296292]\) | \(109902239/4312000\) | \(-14042457858496375000000\) | \([2]\) | \(26542080\) | \(2.9296\) | \(\Gamma_0(N)\)-optimal |
296450.gc2 | 296450gc2 | \([1, 0, 0, -40023838, 93272129292]\) | \(2177286259681/105875000\) | \(344792492061294921875000\) | \([2]\) | \(53084160\) | \(3.2761\) | |
296450.gc3 | 296450gc3 | \([1, 0, 0, -13343338, -154794266208]\) | \(-80677568161/3131816380\) | \(-10199072248770563403437500\) | \([2]\) | \(79626240\) | \(3.4789\) | |
296450.gc1 | 296450gc4 | \([1, 0, 0, -521755088, -4562215726958]\) | \(4823468134087681/30382271150\) | \(98942894774880611442968750\) | \([2]\) | \(159252480\) | \(3.8254\) |
Rank
sage: E.rank()
The elliptic curves in class 296450gc have rank \(2\).
Complex multiplication
The elliptic curves in class 296450gc do not have complex multiplication.Modular form 296450.2.a.gc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.