# Properties

 Label 1.8.af Base field $\F_{2^{3}}$ Dimension $1$ $p$-rank $1$ Ordinary yes Supersingular no Simple yes Geometrically simple yes Primitive no Principally polarizable yes Contains a Jacobian yes

## Invariants

 Base field: $\F_{2^{3}}$ Dimension: $1$ L-polynomial: $1 - 5 x + 8 x^{2}$ Frobenius angles: $\pm0.154919815756$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-7})$$ Galois group: $C_2$ Jacobians: 1

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $1$ Slopes: $[0, 1]$

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $56$ $508$ $4144$ $33044$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $56$ $508$ $4144$ $33044$ $263144$ $2099948$ $16783200$ $134225284$ $1073731736$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{3}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-7})$$.
All geometric endomorphisms are defined over $\F_{2^{3}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{3}}$.

 Subfield Primitive Model $\F_{2}$ 1.2.b

## Twists

Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
1.8.f$2$1.64.aj