## Invariants

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

Dimension: | $2$ |

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

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

Frobenius angles: | $\pm0.250000000000$, $\pm0.5$ |

Angle rank: | $0$ (numerical) |

Jacobians: | 1 |

This isogeny class is not 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 Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

- $y^2+y=x^5+x^4+1$

Point counts of the abelian variety

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

$A(\F_{q^r})$ | $3$ | $45$ | $117$ | $225$ | $1353$ |

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

$C(\F_{q^r})$ | $1$ | $9$ | $13$ | $17$ | $41$ | $81$ | $113$ | $193$ | $481$ | $1089$ |

## Decomposition and endomorphism algebra

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

The isogeny class factors as 1.2.ac $\times$ 1.2.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |

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

The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg^{ 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 1.4.a $\times$ 1.4.e. The endomorphism algebra for each factor is: - 1.4.a : \(\Q(\sqrt{-1}) \).
- 1.4.e : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.i. The endomorphism algebra for each factor is:

## Base change

This is a primitive isogeny class.