## Invariants

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

Dimension: | $1$ |

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

Frobenius angles: | $\pm0.166666666667$ |

Angle rank: | $0$ (numerical) |

Number field: | \(\Q(\sqrt{-3}) \) |

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 Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Point counts of the abelian variety

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

$A(\F_{q^r})$ | $1$ | $7$ | $28$ | $91$ | $271$ |

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

$C(\F_{q^r})$ | $1$ | $7$ | $28$ | $91$ | $271$ | $784$ | $2269$ | $6643$ | $19684$ | $58807$ |

## Decomposition and endomorphism algebra

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

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |

**Endomorphism algebra over $\overline{\F}_{3}$**

The base change of $A$ to $\F_{3^{6}}$ is the simple isogeny class 1.729.cc and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$. |

**Remainder of endomorphism lattice by field**

- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 1.9.ad and its endomorphism algebra is \(\Q(\sqrt{-3}) \). - Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is the simple isogeny class 1.27.a and its endomorphism algebra is \(\Q(\sqrt{-3}) \).

## 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.d | $2$ | 1.9.ad |

1.3.a | $3$ | 1.27.a |

1.3.d | $3$ | 1.27.a |

# 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.