## Invariants

Base field: | $\F_{3}$ |

Dimension: | $1$ |

L-polynomial: | $1 - 2 x + 3 x^{2}$ |

Frobenius angles: | $\pm0.304086723985$ |

Angle rank: | $1$ (numerical) |

Number field: | \(\Q(\sqrt{-2}) \) |

Galois group: | $C_2$ |

Jacobians: | 1 |

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

$p$-rank: | $1$ |

Slopes: | $[0, 1]$ |

## Point counts

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Point counts of the abelian variety

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

$A(\F_{q^r})$ | $2$ | $12$ | $38$ | $96$ | $242$ |

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

$C(\F_{q^r})$ | $2$ | $12$ | $38$ | $96$ | $242$ | $684$ | $2102$ | $6528$ | $19874$ | $59532$ |

## Decomposition and endomorphism algebra

**Endomorphism algebra over $\F_{3}$**

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \). |

## 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 |
---|---|---|

1.3.c | $2$ | 1.9.c |