Show commands for:
SageMath

sage: E = EllipticCurve("493680.cp1")

sage: E.isogeny_class()

## Elliptic curves in class 493680.cp

sage: E.isogeny_class().curves

LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|

493680.cp1 | 493680cp2 | [0, -1, 0, -22767400, 41797155952] | [u'2'] | 30965760 | \(\Gamma_0(N)\)-optimal^{*} |

493680.cp2 | 493680cp1 | [0, -1, 0, -1703720, 377535600] | [u'2'] | 15482880 | \(\Gamma_0(N)\)-optimal^{*} |

^{*}optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 493680.cp2.

## Rank

sage: E.rank()

The elliptic curves in class 493680.cp have rank \(0\).

## Modular form 493680.2.a.cp

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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.