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SageMath
sage: E = EllipticCurve("6069.b1")
sage: E.isogeny_class()
Elliptic curves in class 6069.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 6069.b1 | 6069b5 | [1, 1, 1, -226582, -41607616] | [2] | 18432 | |
| 6069.b2 | 6069b3 | [1, 1, 1, -14167, -654004] | [2, 2] | 9216 | |
| 6069.b3 | 6069b4 | [1, 1, 1, -11277, 453444] | [2] | 9216 | |
| 6069.b4 | 6069b6 | [1, 1, 1, -9832, -1056292] | [2] | 18432 | |
| 6069.b5 | 6069b2 | [1, 1, 1, -1162, -3754] | [2, 2] | 4608 | |
| 6069.b6 | 6069b1 | [1, 1, 1, 283, -286] | [2] | 2304 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6069.b have rank \(0\).
Modular form 6069.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
