Show commands for:
SageMath
sage: E = EllipticCurve("j1")
sage: E.isogeny_class()
Elliptic curves in class 66654.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 66654.j1 | 66654w4 | [1, -1, 0, -6398883, 6231834121] | [2] | 1441792 | |
| 66654.j2 | 66654w6 | [1, -1, 0, -4351653, -3459419429] | [2] | 2883584 | |
| 66654.j3 | 66654w3 | [1, -1, 0, -495243, 47599825] | [2, 2] | 1441792 | |
| 66654.j4 | 66654w2 | [1, -1, 0, -400023, 97399885] | [2, 2] | 720896 | |
| 66654.j5 | 66654w1 | [1, -1, 0, -19143, 2256061] | [2] | 360448 | \(\Gamma_0(N)\)-optimal |
| 66654.j6 | 66654w5 | [1, -1, 0, 1837647, 366272599] | [2] | 2883584 |
Rank
sage: E.rank()
The elliptic curves in class 66654.j have rank \(0\).
Complex multiplication
The elliptic curves in class 66654.j do not have complex multiplication.Modular form 66654.2.a.j
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
