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SageMath

sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 435344p

sage: E.isogeny_class().curves

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

435344.p1 | 435344p1 | \([0, -1, 0, -31997, -5187599]\) | \(-2932006912/7750379\) | \(-9576857372316416\) | \([]\) | \(2322432\) | \(1.7541\) |
\(\Gamma_0(N)\)-optimal^{*} |

435344.p2 | 435344p2 | \([0, -1, 0, 278963, 117330641]\) | \(1942951190528/5944921619\) | \(-7345920300770245376\) | \([]\) | \(6967296\) | \(2.3034\) |
\(\Gamma_0(N)\)-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 435344p1.

## Rank

sage: E.rank()

The elliptic curves in class 435344p have rank \(1\).

## Complex multiplication

The elliptic curves in class 435344p do not have complex multiplication.## Modular form 435344.2.a.p

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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.