## Invariants

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

Dimension: | $1$ |

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

Frobenius angles: | $\pm0.250000000000$ |

Angle rank: | $0$ (numerical) |

Number field: | \(\Q(\sqrt{-1}) \) |

Galois group: | $C_2$ |

Jacobians: | $1$ |

Isomorphism classes: | 1 |

This isogeny class is simple and geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

## Newton polygon

This isogeny class is supersingular.

$p$-rank: | $0$ |

Slopes: | $[1/2, 1/2]$ |

## Point counts

Point counts of the abelian variety

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

$A(\F_{q^r})$ | $1$ | $5$ | $13$ | $25$ | $41$ |

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

$C(\F_{q^r})$ | $1$ | $5$ | $13$ | $25$ | $41$ | $65$ | $113$ | $225$ | $481$ | $1025$ |

## Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

## Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{4}}$.

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

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \). |

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

The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 1.16.i and its endomorphism algebra 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 the simple isogeny class 1.4.a and its endomorphism algebra is \(\Q(\sqrt{-1}) \).

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.

Twist | Extension degree | Common base change |
---|---|---|

1.2.c | $2$ | 1.4.a |

1.2.a | $8$ | 1.256.abg |

# Additional information

This is the isogeny class of the Jacobian of a function field of class number 1. It also appears as a sporadic example in the classification of abelian varieties with one rational point.