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SageMath

sage: E = EllipticCurve("409101.s1")

sage: E.isogeny_class()

## Elliptic curves in class 409101s

sage: E.isogeny_class().curves

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

409101.s2 | 409101s1 | [1, 0, 0, -429976, -171169657] | [2] | 12533760 | \(\Gamma_0(N)\)-optimal^{*} |

409101.s1 | 409101s2 | [1, 0, 0, -7930161, -8593877412] | [2] | 25067520 | \(\Gamma_0(N)\)-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 409101s1.

## Rank

sage: E.rank()

The elliptic curves in class 409101s have rank \(0\).

## Modular form 409101.2.a.s

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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.