## Invariants

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

Dimension: | $2$ |

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

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

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 | 6 | 7 | 8 | 9 | 10 |

$A(\F_{q^r})$ | 1 | 25 | 169 | 625 | 1681 | 4225 | 12769 | 50625 | 231361 | 1050625 |

Point counts of the (virtual) curve

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

$C(\F_{q^r})$ | -1 | 5 | 17 | 33 | 49 | 65 | 97 | 193 | 449 | 1025 |

## Decomposition and endomorphism algebra

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

The isogeny class factors as 1.2.ac^{ 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.