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SageMath

sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 6400.x

sage: E.isogeny_class().curves

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

6400.x1 | 6400f2 | \([0, -1, 0, -333, -1963]\) | \(8000\) | \(512000000\) | \([2]\) | \(2304\) | \(0.40075\) | \(-8\) | |

6400.x2 | 6400f1 | \([0, -1, 0, -83, 287]\) | \(8000\) | \(8000000\) | \([2]\) | \(1152\) | \(0.054173\) | \(\Gamma_0(N)\)-optimal | \(-8\) |

## Rank

sage: E.rank()

The elliptic curves in class 6400.x have rank \(1\).

## Complex multiplication

Each elliptic curve in class 6400.x has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-2}) \).## Modular form 6400.2.a.x

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.