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SageMath

sage: E = EllipticCurve("h1")

sage: E.isogeny_class()

## Elliptic curves in class 435344.h

sage: E.isogeny_class().curves

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

435344.h1 | 435344h2 | \([0, 1, 0, -82528, -9152220]\) | \(12576878500/1127\) | \(5570369272832\) | \([2]\) | \(1228800\) | \(1.4857\) |
\(\Gamma_0(N)\)-optimal^{*} |

435344.h2 | 435344h1 | \([0, 1, 0, -4788, -165476]\) | \(-9826000/3703\) | \(-4575660474112\) | \([2]\) | \(614400\) | \(1.1391\) |
\(\Gamma_0(N)\)-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 435344.h1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

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.