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SageMath

sage: E = EllipticCurve("gj1")

sage: E.isogeny_class()

## Elliptic curves in class 476850gj

sage: E.isogeny_class().curves

LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|

476850.gj2 | 476850gj1 | [1, 1, 1, 191312, 28264781] | [2] | 7077888 |
\(\Gamma_0(N)\)-optimal^{*} |

476850.gj1 | 476850gj2 | [1, 1, 1, -1036938, 259175781] | [2] | 14155776 |
\(\Gamma_0(N)\)-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 476850gj1.

## Rank

sage: E.rank()

The elliptic curves in class 476850gj have rank \(0\).

## Complex multiplication

The elliptic curves in class 476850gj do not have complex multiplication.## Modular form 476850.2.a.gj

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.