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SageMath

sage: E = EllipticCurve("q1")

sage: E.isogeny_class()

## Elliptic curves in class 405600.q

sage: E.isogeny_class().curves

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

405600.q1 | 405600q2 | \([0, -1, 0, -2435008, -1192138988]\) | \(18821096/3645\) | \(309227201716680000000\) | \([2]\) | \(20127744\) | \(2.6487\) |
\(\Gamma_0(N)\)-optimal^{*} |

405600.q2 | 405600q1 | \([0, -1, 0, 311242, -110116488]\) | \(314432/675\) | \(-7158037076775000000\) | \([2]\) | \(10063872\) | \(2.3022\) |
\(\Gamma_0(N)\)-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 405600.q1.

## Rank

sage: E.rank()

The elliptic curves in class 405600.q have rank \(1\).

## Complex multiplication

The elliptic curves in class 405600.q do not have complex multiplication.## Modular form 405600.2.a.q

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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.