## Invariants

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

Dimension: | $2$ |

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

$1 + 4x + 8x^{2} + 8x^{3} + 4x^{4}$ | |

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

Angle rank: | $0$ (numerical) |

Jacobians: | 0 |

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 is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

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

$A(\F_{q^r})$ | $25$ | $25$ | $25$ | $625$ | $625$ |

Point counts of the (virtual) curve

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

$C(\F_{q^r})$ | $7$ | $5$ | $1$ | $33$ | $17$ | $65$ | $161$ | $193$ | $577$ | $1025$ |

## Decomposition and endomorphism algebra

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

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

**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.