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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 53550.cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53550.cs1 | 53550dt6 | \([1, -1, 1, -3086685905, 66007278247097]\) | \(285531136548675601769470657/17941034271597192\) | \(204359593499911765125000\) | \([2]\) | \(31457280\) | \(3.9332\) | |
| 53550.cs2 | 53550dt4 | \([1, -1, 1, -193284905, 1027278589097]\) | \(70108386184777836280897/552468975892674624\) | \(6292966928527496889000000\) | \([2, 2]\) | \(15728640\) | \(3.5866\) | |
| 53550.cs3 | 53550dt5 | \([1, -1, 1, -65835905, 2361669619097]\) | \(-2770540998624539614657/209924951154647363208\) | \(-2391176396745905121541125000\) | \([2]\) | \(31457280\) | \(3.9332\) | |
| 53550.cs4 | 53550dt2 | \([1, -1, 1, -20412905, -8916178903]\) | \(82582985847542515777/44772582831427584\) | \(509987701314229824000000\) | \([2, 2]\) | \(7864320\) | \(3.2400\) | |
| 53550.cs5 | 53550dt1 | \([1, -1, 1, -15804905, -24150226903]\) | \(38331145780597164097/55468445663232\) | \(631820263882752000000\) | \([2]\) | \(3932160\) | \(2.8934\) | \(\Gamma_0(N)\)-optimal |
| 53550.cs6 | 53550dt3 | \([1, -1, 1, 78731095, -70187170903]\) | \(4738217997934888496063/2928751705237796928\) | \(-33360312392474280633000000\) | \([2]\) | \(15728640\) | \(3.5866\) |
Rank
sage: E.rank()
The elliptic curves in class 53550.cs have rank \(0\).
Complex multiplication
The elliptic curves in class 53550.cs do not have complex multiplication.Modular form 53550.2.a.cs
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.
