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SageMath

sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 485100.d

sage: E.isogeny_class().curves

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

485100.d1 | 485100d2 | \([0, 0, 0, -5692575, 5218487750]\) | \(59466754384/121275\) | \(41605145297100000000\) | \([2]\) | \(17694720\) | \(2.6511\) |
\(\Gamma_0(N)\)-optimal^{*} |

485100.d2 | 485100d1 | \([0, 0, 0, -235200, 137671625]\) | \(-67108864/343035\) | \(-7355195329308750000\) | \([2]\) | \(8847360\) | \(2.3045\) |
\(\Gamma_0(N)\)-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 485100.d1.

## Rank

sage: E.rank()

The elliptic curves in class 485100.d have rank \(2\).

## Complex multiplication

The elliptic curves in class 485100.d do not have complex multiplication.## Modular form 485100.2.a.d

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.