Show commands:
SageMath

sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 450840.c

sage: E.isogeny_class().curves

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

450840.c1 | 450840c2 | \([0, -1, 0, -456716, 114756516]\) | \(1705021456336/68471325\) | \(423099220917484800\) | \([2]\) | \(7077888\) | \(2.1485\) |
\(\Gamma_0(N)\)-optimal^{*} |

450840.c2 | 450840c1 | \([0, -1, 0, 12909, 6554916]\) | \(615962624/48481875\) | \(-18723753648990000\) | \([2]\) | \(3538944\) | \(1.8019\) |
\(\Gamma_0(N)\)-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.c1.

## Rank

sage: E.rank()

The elliptic curves in class 450840.c have rank \(1\).

## Complex multiplication

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

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.