## Invariants

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

Dimension: | $2$ |

L-polynomial: | $1 + 3 x + 5 x^{2} + 6 x^{3} + 4 x^{4}$ |

Frobenius angles: | $\pm0.543118021706$, $\pm0.876451355039$ |

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 Jacobians of 1 curves, and hence is principally polarizable:

- $y^2+(x^3+x+1)y=x^5+x^4+x^3+x$

Point counts of the abelian variety

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

$A(\F_{q^r})$ | 19 | 19 | 76 | 171 | 1159 | 5776 | 11989 | 69939 | 261364 | 1113799 |

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

$C(\F_{q^r})$ | 6 | 6 | 9 | 10 | 36 | 87 | 90 | 274 | 513 | 1086 |

## 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 $\F_{2^{6}}$ is 1.64.l^{ 2 } and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |

**Remainder of endomorphism lattice by field**

- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 2.4.b_ad and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{5})\). - Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is the simple isogeny class 2.8.a_l and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{5})\).

## Base change

This is a primitive isogeny class.