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SageMath

sage: E = EllipticCurve("s1")

sage: E.isogeny_class()

## Elliptic curves in class 414400s

sage: E.isogeny_class().curves

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

414400.s2 | 414400s1 | \([0, 1, 0, 1567, 1055263]\) | \(415292/469567\) | \(-480836608000000\) | \([2]\) | \(1769472\) | \(1.4962\) |
\(\Gamma_0(N)\)-optimal^{*} |

414400.s1 | 414400s2 | \([0, 1, 0, -146433, 21035263]\) | \(169556172914/4353013\) | \(8914970624000000\) | \([2]\) | \(3538944\) | \(1.8428\) |
\(\Gamma_0(N)\)-optimal^{*} |

^{*}optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 414400s1.

## Rank

sage: E.rank()

The elliptic curves in class 414400s have rank \(0\).

## Complex multiplication

The elliptic curves in class 414400s do not have complex multiplication.## Modular form 414400.2.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.