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SageMath

sage: E = EllipticCurve("400752.cv1")

sage: E.isogeny_class()

## Elliptic curves in class 400752.cv

sage: E.isogeny_class().curves

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

400752.cv1 | 400752cv2 | [0, 0, 0, -21571275, 38262222394] | [2] | 19660800 | \(\Gamma_0(N)\)-optimal^{*} |

400752.cv2 | 400752cv1 | [0, 0, 0, -401115, 1421909962] | [2] | 9830400 | \(\Gamma_0(N)\)-optimal^{*} |

^{*}optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 400752.cv2.

## Rank

sage: E.rank()

The elliptic curves in class 400752.cv have rank \(1\).

## Modular form 400752.2.a.cv

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.