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SageMath

sage: E = EllipticCurve("s1")

sage: E.isogeny_class()

## Elliptic curves in class 424830.s

sage: E.isogeny_class().curves

LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|

424830.s1 | 424830s2 | \([1, 1, 0, -10875981078, -436571405239668]\) | \(-12242088317612041/600\) | \(-6973082191981377389400\) | \([]\) | \(349780032\) | \(4.1151\) | |

424830.s2 | 424830s1 | \([1, 1, 0, -133092453, -609944793093]\) | \(-22434194041/843750\) | \(-9805896832473811953843750\) | \([]\) | \(116593344\) | \(3.5658\) |
\(\Gamma_0(N)\)-optimal^{*} |

^{*}optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 424830.s1.

## Rank

sage: E.rank()

The elliptic curves in class 424830.s have rank \(1\).

## Complex multiplication

The elliptic curves in class 424830.s do not have complex multiplication.## Modular form 424830.2.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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

## Isogeny graph

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

The vertices are labelled with LMFDB labels.