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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 127296.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127296.x1 | 127296cd4 | \([0, 0, 0, -628716, 191873360]\) | \(143820170742457/5826444\) | \(1113450787897344\) | \([2]\) | \(786432\) | \(1.9695\) | |
| 127296.x2 | 127296cd3 | \([0, 0, 0, -190956, -29596336]\) | \(4029546653497/351790452\) | \(67228202305585152\) | \([2]\) | \(786432\) | \(1.9695\) | |
| 127296.x3 | 127296cd2 | \([0, 0, 0, -41196, 2691920]\) | \(40459583737/7033104\) | \(1344047104917504\) | \([2, 2]\) | \(393216\) | \(1.6229\) | |
| 127296.x4 | 127296cd1 | \([0, 0, 0, 4884, 240464]\) | \(67419143/169728\) | \(-32435525910528\) | \([2]\) | \(196608\) | \(1.2763\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 127296.x have rank \(1\).
Complex multiplication
The elliptic curves in class 127296.x do not have complex multiplication.Modular form 127296.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
