## Invariants

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

Dimension: | $2$ |

L-polynomial: | $1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4}$ |

Frobenius angles: | $\pm0.0540867239847$, $\pm0.445913276015$ |

Angle rank: | $1$ (numerical) |

Number field: | \(\Q(\zeta_{8})\) |

Galois group: | $C_2^2$ |

Jacobians: | 1 |

This isogeny class is simple but not geometrically simple.

## Newton polygon

This isogeny class is ordinary.

$p$-rank: | $2$ |

Slopes: | $[0, 0, 1, 1]$ |

## Point counts

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

- $y^2=2x^6+x^5+x^3+x^2+x+2$

Point counts of the abelian variety

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

$A(\F_{q^r})$ | $2$ | $68$ | $626$ | $4624$ | $49282$ |

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

$C(\F_{q^r})$ | $0$ | $10$ | $24$ | $54$ | $200$ | $730$ | $2240$ | $6494$ | $19392$ | $59050$ |

## Decomposition and endomorphism algebra

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

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\). |

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

The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao^{ 2 } and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |

**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 2.9.a_ao and its endomorphism algebra is \(\Q(\zeta_{8})\).

## Base change

This is a primitive isogeny class.