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SageMath

sage: E = EllipticCurve("486720.dv1")

sage: E.isogeny_class()

## Elliptic curves in class 486720.dv

sage: E.isogeny_class().curves

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

486720.dv1 | 486720dv5 | [0, 0, 0, -877558188, 10006033356592] | [2] | 132120576 | \(\Gamma_0(N)\)-optimal^{*} |

486720.dv2 | 486720dv4 | [0, 0, 0, -82257708, -286926126032] | [2] | 66060288 | |

486720.dv3 | 486720dv3 | [0, 0, 0, -55001388, 155422142512] | [2, 2] | 66060288 | \(\Gamma_0(N)\)-optimal^{*} |

486720.dv4 | 486720dv6 | [0, 0, 0, -11196588, 396190845232] | [2] | 132120576 | |

486720.dv5 | 486720dv2 | [0, 0, 0, -6329388, -2255668688] | [2, 2] | 33030144 | \(\Gamma_0(N)\)-optimal^{*} |

486720.dv6 | 486720dv1 | [0, 0, 0, 1458132, -271408592] | [2] | 16515072 | \(\Gamma_0(N)\)-optimal^{*} |

^{*}optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 486720.dv6.

## Rank

sage: E.rank()

The elliptic curves in class 486720.dv have rank \(0\).

## Modular form 486720.2.a.dv

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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.