## Invariants

Base field: | $\F_{2}$ |

Dimension: | $2$ |

L-polynomial: | $1 - x^{2} + 4 x^{4}$ |

Frobenius angles: | $\pm0.209784688372$, $\pm0.790215311628$ |

Angle rank: | $1$ (numerical) |

Number field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |

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:

- $y^2+(x^2+x)y=x^5+x^3+x^2+x$

Point counts of the abelian variety

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

$A(\F_{q^r})$ | $4$ | $16$ | $76$ | $576$ | $964$ |

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

$C(\F_{q^r})$ | $3$ | $3$ | $9$ | $31$ | $33$ | $87$ | $129$ | $223$ | $513$ | $903$ |

## Decomposition and endomorphism algebra

**Endomorphism algebra over $\F_{2}$**

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\). |

**Endomorphism algebra over $\overline{\F}_{2}$**

The base change of $A$ to $\F_{2^{2}}$ is 1.4.ab^{ 2 } and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |

## Base change

This is a primitive isogeny class.