## Invariants

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

Dimension: | $1$ |

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

Frobenius angles: | $\pm0.833333333333$ |

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 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 |

$A(\F_{q^r})$ | 7 | 7 | 28 | 91 | 217 | 784 | 2107 | 6643 | 19684 | 58807 |

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

$C(\F_{q^r})$ | 7 | 7 | 28 | 91 | 217 | 784 | 2107 | 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.