## Invariants

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

Dimension: | $2$ |

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

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 | 6 | 7 | 8 | 9 | 10 |

$A(\F_{q^r})$ | 1 | 1 | 49 | 81 | 961 | 2401 | 16129 | 50625 | 261121 | 923521 |

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.