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SageMath

sage: E = EllipticCurve("75.c1")

sage: E.isogeny_class()

sage: E.isogeny_class()

## Elliptic curves in class 75a

sage: E.isogeny_class().curves

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

75.c1 | 75a1 | [0, -1, 1, -8, -7] | 1 | 6 | \(\Gamma_0(N)\)-optimal |

75.c2 | 75a2 | [0, -1, 1, 42, 443] | 1 | 30 |

## Rank

sage: E.rank()

The elliptic curves in class 75a have rank \(0\).

## Modular form 75.2.a.c

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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

## Isogeny graph

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

The vertices are labelled with Cremona labels.