## Invariants

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

Dimension: | $1$ |

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

$1 - 4x + 4x^{2}$ | |

Frobenius angles: | $0$, $0$ |

Angle rank: | $0$ (numerical) |

Number field: | \(\Q\) |

Galois group: | Trivial |

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

$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|

$A(\F_{q^r})$ | $1$ | $9$ | $49$ | $225$ | $961$ |

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

$C(\F_{q^r})$ | $1$ | $9$ | $49$ | $225$ | $961$ | $3969$ | $16129$ | $65025$ | $261121$ | $1046529$ |

## Decomposition and endomorphism algebra

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

The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |

## Base change

This is a primitive isogeny class.

## Twists

# 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.