## Invariants

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

Dimension: | $2$ |

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

Frobenius angles: | $\pm0.416666666667$, $\pm0.916666666667$ |

Angle rank: | $0$ (numerical) |

Number field: | \(\Q(\zeta_{12})\) |

Galois group: | $C_2^2$ |

Jacobians: | 1 |

This isogeny class is simple but not geometrically simple.

## Newton polygon

This isogeny class is supersingular.

$p$-rank: | $0$ |

Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

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

Point counts of the abelian variety

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

$A(\F_{q^r})$ | 13 | 13 | 169 | 169 | 793 | 4225 | 18577 | 74529 | 231361 | 1047553 |

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

$C(\F_{q^r})$ | 5 | 5 | 17 | 9 | 25 | 65 | 145 | 289 | 449 | 1025 |

## Decomposition and endomorphism algebra

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

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\). |

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey^{ 2 } and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |

**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.a_ae and its endomorphism algebra is \(\Q(\zeta_{12})\). - Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.e ^{ 2 }and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ - Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae ^{ 2 }and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a ^{ 2 }and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

## Base change

This is a primitive isogeny class.