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SageMath

sage: E = EllipticCurve("bz1")

sage: E.isogeny_class()

## Elliptic curves in class 414736bz

sage: E.isogeny_class().curves

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

414736.bz2 | 414736bz1 | \([0, -1, 0, 14299752, -97909629712]\) | \(4533086375/60669952\) | \(-4328015978908947381747712\) | \([2]\) | \(68124672\) | \(3.4080\) |
\(\Gamma_0(N)\)-optimal^{*} |

414736.bz1 | 414736bz2 | \([0, -1, 0, -251131288, -1433983312656]\) | \(24553362849625/1755162752\) | \(125208149764842438707904512\) | \([2]\) | \(136249344\) | \(3.7546\) |
\(\Gamma_0(N)\)-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 414736bz1.

## Rank

sage: E.rank()

The elliptic curves in class 414736bz have rank \(0\).

## Complex multiplication

The elliptic curves in class 414736bz do not have complex multiplication.## Modular form 414736.2.a.bz

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.