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SageMath

sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 435344.d

sage: E.isogeny_class().curves

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

435344.d1 | 435344d2 | \([0, 1, 0, -63600, 2653444]\) | \(5756278756/2705927\) | \(13374456624069632\) | \([2]\) | \(4423680\) | \(1.7886\) |
\(\Gamma_0(N)\)-optimal^{*} |

435344.d2 | 435344d1 | \([0, 1, 0, 14140, 321244]\) | \(253012016/181447\) | \(-224207363231488\) | \([2]\) | \(2211840\) | \(1.4420\) |
\(\Gamma_0(N)\)-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 435344.d1.

## Rank

sage: E.rank()

The elliptic curves in class 435344.d have rank \(1\).

## Complex multiplication

The elliptic curves in class 435344.d do not have complex multiplication.## Modular form 435344.2.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.