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SageMath

sage: E = EllipticCurve("h1")

sage: E.isogeny_class()

## Elliptic curves in class 800h

sage: E.isogeny_class().curves

LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality | CM discriminant |
---|---|---|---|---|---|---|

800.e1 | 800h1 | [0, 0, 0, -5, 0] | [2] | 32 | \(\Gamma_0(N)\)-optimal | -4 |

800.e2 | 800h2 | [0, 0, 0, 20, 0] | [2] | 64 | -4 |

## Rank

sage: E.rank()

The elliptic curves in class 800h have rank \(1\).

## Complex multiplication

Each elliptic curve in class 800h has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).## Modular form 800.2.a.h

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 Cremona 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 Cremona labels.