## Invariants

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

Dimension: | $2$ |

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

Frobenius angles: | $\pm0.250000000000$, $\pm0.750000000000$ |

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 Jacobians of 1 curves, and hence is principally polarizable:

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

Point counts of the abelian variety

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

$A(\F_{q^r})$ | 5 | 25 | 65 | 625 | 1025 | 4225 | 16385 | 50625 | 262145 | 1050625 |

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

$C(\F_{q^r})$ | 3 | 5 | 9 | 33 | 33 | 65 | 129 | 193 | 513 | 1025 |

## Decomposition and endomorphism algebra

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

The isogeny class factors as 1.2.ac $\times$ 1.2.c 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 $\F_{2^{4}}$ is 1.16.i^{ 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 ^{ 2 }and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

## Base change

This is a primitive isogeny class.