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SageMath

sage: E = EllipticCurve("kg1")

sage: E.isogeny_class()

## Elliptic curves in class 486720kg

sage: E.isogeny_class().curves

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

486720.kg2 | 486720kg1 | \([0, 0, 0, 387348, 257558704]\) | \(6967871/35100\) | \(-32376856513373798400\) | \([2]\) | \(12386304\) | \(2.4259\) |
\(\Gamma_0(N)\)-optimal^{*} |

486720.kg1 | 486720kg2 | \([0, 0, 0, -4479852, 3269382064]\) | \(10779215329/1232010\) | \(1136427663619420323840\) | \([2]\) | \(24772608\) | \(2.7725\) |
\(\Gamma_0(N)\)-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 486720kg1.

## Rank

sage: E.rank()

The elliptic curves in class 486720kg have rank \(1\).

## Complex multiplication

The elliptic curves in class 486720kg do not have complex multiplication.## Modular form 486720.2.a.kg

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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.