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SageMath
E = EllipticCurve("ni1")
E.isogeny_class()
Elliptic curves in class 479808.ni
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
479808.ni1 | 479808ni5 | \([0, 0, 0, -387193879884, 92734389270274192]\) | \(285531136548675601769470657/17941034271597192\) | \(403369601585106552706100625408\) | \([2]\) | \(2264924160\) | \(5.1411\) | \(\Gamma_0(N)\)-optimal* |
479808.ni2 | 479808ni3 | \([0, 0, 0, -24245658444, 1443184565779600]\) | \(70108386184777836280897/552468975892674624\) | \(12421200880640252920624186392576\) | \([2, 2]\) | \(1132462080\) | \(4.7945\) | \(\Gamma_0(N)\)-optimal* |
479808.ni3 | 479808ni6 | \([0, 0, 0, -8258455884, 3317952650942608]\) | \(-2770540998624539614657/209924951154647363208\) | \(-4719758216173604166756801227784192\) | \([2]\) | \(2264924160\) | \(5.1411\) | |
479808.ni4 | 479808ni2 | \([0, 0, 0, -2560594764, -12533759058800]\) | \(82582985847542515777/44772582831427584\) | \(1006625294018868876956575727616\) | \([2, 2]\) | \(566231040\) | \(4.4480\) | \(\Gamma_0(N)\)-optimal* |
479808.ni5 | 479808ni1 | \([0, 0, 0, -1982567244, -33934881170288]\) | \(38331145780597164097/55468445663232\) | \(1247101169810717006383546368\) | \([2]\) | \(283115520\) | \(4.1014\) | \(\Gamma_0(N)\)-optimal* |
479808.ni6 | 479808ni4 | \([0, 0, 0, 9876028596, -98580268761968]\) | \(4738217997934888496063/2928751705237796928\) | \(-65847341385090655872686827241472\) | \([2]\) | \(1132462080\) | \(4.7945\) |
Rank
sage: E.rank()
The elliptic curves in class 479808.ni have rank \(1\).
Complex multiplication
The elliptic curves in class 479808.ni do not have complex multiplication.Modular form 479808.2.a.ni
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.