Show commands for:
SageMath
sage: E = EllipticCurve("10780.k1")
sage: E.isogeny_class()
Elliptic curves in class 10780.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 10780.k1 | 10780f4 | [0, -1, 0, -347916, 79103816] | [2] | 62208 | |
| 10780.k2 | 10780f3 | [0, -1, 0, -21821, 1232330] | [2] | 31104 | |
| 10780.k3 | 10780f2 | [0, -1, 0, -4916, 76616] | [2] | 20736 | |
| 10780.k4 | 10780f1 | [0, -1, 0, -2221, -38730] | [2] | 10368 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10780.k have rank \(1\).
Modular form 10780.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
