Show commands for:
SageMath
sage: E = EllipticCurve("36414.cg1")
sage: E.isogeny_class()
Elliptic curves in class 36414cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 36414.cg5 | 36414cf1 | [1, -1, 1, -1355, -38505] | [2] | 36864 | \(\Gamma_0(N)\)-optimal |
| 36414.cg4 | 36414cf2 | [1, -1, 1, -27365, -1734357] | [2] | 73728 | |
| 36414.cg6 | 36414cf3 | [1, -1, 1, 11650, 809421] | [2] | 110592 | |
| 36414.cg3 | 36414cf4 | [1, -1, 1, -92390, 8882925] | [2] | 221184 | |
| 36414.cg2 | 36414cf5 | [1, -1, 1, -443525, 114129789] | [2] | 331776 | |
| 36414.cg1 | 36414cf6 | [1, -1, 1, -7102085, 7286730621] | [2] | 663552 |
Rank
sage: E.rank()
The elliptic curves in class 36414cf have rank \(1\).
Modular form 36414.2.a.cg
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
