## Invariants

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

Dimension: | $2$ |

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

$1 - 2 x + 7 x^{2} - 6 x^{3} + 9 x^{4}$ | |

Frobenius angles: | $\pm0.406785250661$, $\pm0.406785250661$ |

Angle rank: | $1$ (numerical) |

Jacobians: | 1 |

This isogeny class is not 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:

- $y^2=2x^6+x^4+2x^3+x^2+2$

Point counts of the abelian variety

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

$A(\F_{q^r})$ | $9$ | $225$ | $1296$ | $5625$ | $45369$ |

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

$C(\F_{q^r})$ | $2$ | $20$ | $44$ | $68$ | $182$ | $710$ | $2354$ | $6788$ | $19412$ | $58100$ |

## Decomposition and endomorphism algebra

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

The isogeny class factors as 1.3.ab^{ 2 } and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |

## Base change

This is a primitive isogeny class.