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SageMath

sage: E = EllipticCurve("cp1")

sage: E.isogeny_class()

## Elliptic curves in class 20160.cp

sage: E.isogeny_class().curves

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

20160.cp1 | 20160ee3 | [0, 0, 0, -215148, 38410832] | [2] | 98304 | |

20160.cp2 | 20160ee2 | [0, 0, 0, -13548, 590672] | [2, 2] | 49152 | |

20160.cp3 | 20160ee1 | [0, 0, 0, -2028, -22192] | [2] | 24576 | \(\Gamma_0(N)\)-optimal |

20160.cp4 | 20160ee4 | [0, 0, 0, 3732, 1993808] | [2] | 98304 |

## Rank

sage: E.rank()

The elliptic curves in class 20160.cp have rank \(0\).

## Complex multiplication

The elliptic curves in class 20160.cp do not have complex multiplication.## Modular form 20160.2.a.cp

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.