## Invariants

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

Dimension: | $2$ |

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

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

Frobenius angles: | $0$, $0$, $1$, $1$ |

Angle rank: | $0$ (numerical) |

Number field: | \(\Q(\sqrt{2}) \) |

Galois group: | $C_2$ |

Jacobians: | 0 |

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 is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

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

$A(\F_{q^r})$ | $1$ | $1$ | $49$ | $81$ | $961$ |

Point counts of the (virtual) curve

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

$C(\F_{q^r})$ | $3$ | $-3$ | $9$ | $1$ | $33$ | $33$ | $129$ | $193$ | $513$ | $897$ |

## Decomposition and endomorphism algebra

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

The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q(\sqrt{2}) \) ramified at both real infinite places. |

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

The base change of $A$ to $\F_{2^{2}}$ is 1.4.ae^{ 2 } and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |

## Base change

This is a primitive isogeny class.

## Twists

# Additional information

This isogeny class appears as a sporadic example in the classification of abelian varieties with one rational point.