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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 56550.u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 56550.u1 | 56550v6 | \([1, 0, 1, -5830976, -5419994902]\) | \(1403225087856519288817/286925652\) | \(4483213312500\) | \([2]\) | \(1179648\) | \(2.2561\) | |
| 56550.u2 | 56550v4 | \([1, 0, 1, -364476, -84690902]\) | \(342695799974030257/154911513744\) | \(2420492402250000\) | \([2, 2]\) | \(589824\) | \(1.9096\) | |
| 56550.u3 | 56550v5 | \([1, 0, 1, -305976, -112770902]\) | \(-202751340503592817/234115321265748\) | \(-3658051894777312500\) | \([2]\) | \(1179648\) | \(2.2561\) | |
| 56550.u4 | 56550v3 | \([1, 0, 1, -200476, 33933098]\) | \(57027947016536497/1135497163632\) | \(17742143181750000\) | \([2]\) | \(589824\) | \(1.9096\) | |
| 56550.u5 | 56550v2 | \([1, 0, 1, -26476, -866902]\) | \(131352161359537/55341621504\) | \(864712836000000\) | \([2, 2]\) | \(294912\) | \(1.5630\) | |
| 56550.u6 | 56550v1 | \([1, 0, 1, 5524, -98902]\) | \(1193377118543/963575808\) | \(-15055872000000\) | \([2]\) | \(147456\) | \(1.2164\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 56550.u have rank \(1\).
Complex multiplication
The elliptic curves in class 56550.u do not have complex multiplication.Modular form 56550.2.a.u
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
