## Invariants

Base field: | $\F_{2}$ |

Dimension: | $1$ |

L-polynomial: | $1 - 2 x + 2 x^{2}$ |

Frobenius angles: | $\pm0.250000000000$ |

Angle rank: | $0$ (numerical) |

Number field: | \(\Q(\sqrt{-1}) \) |

Galois group: | $C_2$ |

Jacobians: | 1 |

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is supersingular.

$p$-rank: | $0$ |

Slopes: | $[1/2, 1/2]$ |

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

$A(\F_{q^r})$ | 1 | 5 | 13 | 25 | 41 | 65 | 113 | 225 | 481 | 1025 |

$r$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

$C(\F_{q^r})$ | 1 | 5 | 13 | 25 | 41 | 65 | 113 | 225 | 481 | 1025 |

## Decomposition and endomorphism algebra

**Endomorphism algebra over $\F_{2}$**

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \). |

**Endomorphism algebra over $\overline{\F}_{2}$**

The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 1.16.i and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |

**Remainder of endomorphism lattice by field**

- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 1.4.a and its endomorphism algebra is \(\Q(\sqrt{-1}) \).

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.

Twist | Extension Degree | Common base change |

1.2.c | $2$ | 1.4.a |

1.2.a | $8$ | 1.256.abg |

# Additional information

This is the isogeny class of the Jacobian of a function field of class number 1. It also appears as a sporadic example in the classification of abelian varieties with one rational point.