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SageMath

sage: E = EllipticCurve("eb1")

sage: E.isogeny_class()

## Elliptic curves in class 468270.eb

sage: E.isogeny_class().curves

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

468270.eb1 | 468270eb2 | [1, -1, 1, -180530087, -933578487201] | [2] | 103219200 |
\(\Gamma_0(N)\)-optimal^{*} |

468270.eb2 | 468270eb1 | [1, -1, 1, -11168807, -14895159969] | [2] | 51609600 |
\(\Gamma_0(N)\)-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 468270.eb1.

## Rank

sage: E.rank()

The elliptic curves in class 468270.eb have rank \(0\).

## Complex multiplication

The elliptic curves in class 468270.eb do not have complex multiplication.## Modular form 468270.2.a.eb

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.