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SageMath

sage: E = EllipticCurve("df1")

sage: E.isogeny_class()

## Elliptic curves in class 443760.df

sage: E.isogeny_class().curves

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

443760.df1 | 443760df3 | [0, 1, 0, -24806800, 47223347348] | [2] | 34062336 |
\(\Gamma_0(N)\)-optimal^{*} |

443760.df2 | 443760df2 | [0, 1, 0, -2618800, -409851052] | [2, 2] | 17031168 |
\(\Gamma_0(N)\)-optimal^{*} |

443760.df3 | 443760df1 | [0, 1, 0, -2027120, -1110163500] | [2] | 8515584 |
\(\Gamma_0(N)\)-optimal^{*} |

443760.df4 | 443760df4 | [0, 1, 0, 10102320, -3213585900] | [4] | 34062336 |

^{*}optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 443760.df1.

## Rank

sage: E.rank()

The elliptic curves in class 443760.df have rank \(1\).

## Complex multiplication

The elliptic curves in class 443760.df do not have complex multiplication.## Modular form 443760.2.a.df

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

## Isogeny graph

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

The vertices are labelled with LMFDB labels.