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SageMath

sage: E = EllipticCurve("s1")

sage: E.isogeny_class()

## Elliptic curves in class 450840.s

sage: E.isogeny_class().curves

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

450840.s1 | 450840s2 | \([0, -1, 0, -179897678420, -29368756943723100]\) | \(21208997008348807455199568/61790625\) | \(1875870465580144800000\) | \([2]\) | \(1027768320\) | \(4.6786\) |
\(\Gamma_0(N)\)-optimal^{*} |

450840.s2 | 450840s1 | \([0, -1, 0, -11243600295, -458884411110600]\) | \(-82847542748407455193088/141410419921875\) | \(-268312982609347534218750000\) | \([2]\) | \(513884160\) | \(4.3321\) |
\(\Gamma_0(N)\)-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 450840.s1.

## Rank

sage: E.rank()

The elliptic curves in class 450840.s have rank \(0\).

## Complex multiplication

The elliptic curves in class 450840.s do not have complex multiplication.## Modular form 450840.2.a.s

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.