## Invariants

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

Dimension: | $2$ |

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

Frobenius angles: | $\pm0.123548644961$, $\pm0.456881978294$ |

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:

- $y^2+(x^3+x+1)y=x^6+x^5+x^4+x^3+x^2+x+1$

Point counts of the abelian variety

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

$A(\F_{q^r})$ | $1$ | $19$ | $76$ | $171$ | $961$ |

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

$C(\F_{q^r})$ | $0$ | $6$ | $9$ | $10$ | $30$ | $87$ | $168$ | $274$ | $513$ | $1086$ |

## 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.

## Twists

# Additional information

This is the isogeny class of the Jacobian of a function field of class number 1.