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SageMath

sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

## Elliptic curves in class 414736.g

sage: E.isogeny_class().curves

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

414736.g1 | 414736g2 | \([0, 1, 0, -14109664, -19791928668]\) | \(1431644/49\) | \(10632485543889473395712\) | \([2]\) | \(23740416\) | \(2.9984\) | |

414736.g2 | 414736g1 | \([0, 1, 0, -2186004, 816925276]\) | \(21296/7\) | \(379731626567481192704\) | \([2]\) | \(11870208\) | \(2.6518\) |
\(\Gamma_0(N)\)-optimal^{*} |

^{*}optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 414736.g1.

## Rank

sage: E.rank()

The elliptic curves in class 414736.g have rank \(1\).

## Complex multiplication

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

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.