Show commands for:
SageMath

sage: E = EllipticCurve("425880.eb1")

sage: E.isogeny_class()

## Elliptic curves in class 425880eb

sage: E.isogeny_class().curves

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

425880.eb4 | 425880eb1 | [0, 0, 0, -266682, -53007019] | [2] | 2359296 | \(\Gamma_0(N)\)-optimal^{*} |

425880.eb3 | 425880eb2 | [0, 0, 0, -274287, -49823566] | [2, 2] | 4718592 | \(\Gamma_0(N)\)-optimal^{*} |

425880.eb2 | 425880eb3 | [0, 0, 0, -1034787, 351568334] | [2, 2] | 9437184 | \(\Gamma_0(N)\)-optimal^{*} |

425880.eb5 | 425880eb4 | [0, 0, 0, 364533, -247474474] | [2] | 9437184 | |

425880.eb1 | 425880eb5 | [0, 0, 0, -15940587, 24495983174] | [2] | 18874368 | \(\Gamma_0(N)\)-optimal^{*} |

425880.eb6 | 425880eb6 | [0, 0, 0, 1703013, 1896235094] | [2] | 18874368 |

^{*}optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 425880eb1.

## Rank

sage: E.rank()

The elliptic curves in class 425880eb have rank \(1\).

## Modular form 425880.2.a.eb

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

## Isogeny graph

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

The vertices are labelled with Cremona labels.