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SageMath

sage: E = EllipticCurve("bi1")

sage: E.isogeny_class()

## Elliptic curves in class 405600.bi

sage: E.isogeny_class().curves

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

405600.bi1 | 405600bi2 | \([0, -1, 0, -5578408, -5067697688]\) | \(497169541448/190125\) | \(7341576489000000000\) | \([2]\) | \(15482880\) | \(2.5859\) |
\(\Gamma_0(N)\)-optimal^{*} |

405600.bi2 | 405600bi1 | \([0, -1, 0, -297158, -103322688]\) | \(-601211584/609375\) | \(-2941336734375000000\) | \([2]\) | \(7741440\) | \(2.2393\) |
\(\Gamma_0(N)\)-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.bi1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

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.