# Properties

 Label 1.512.af Base Field $\F_{2^{9}}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{9}}$ Dimension: $1$ L-polynomial: $1 - 5 x + 512 x^{2}$ Frobenius angles: $\pm0.464759447268$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-7})$$ Galois group: $C_2$

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 a Jacobian, and hence is principally polarizable.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 508 263144 134225284 68719003024 35184365852108 18014398720839416 9223372041104766164 4722366482782680160800 2417851639226647529086108 1237940039285411746904576264

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 508 263144 134225284 68719003024 35184365852108 18014398720839416 9223372041104766164 4722366482782680160800 2417851639226647529086108 1237940039285411746904576264

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{9}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-7})$$.
All geometric endomorphisms are defined over $\F_{2^{9}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{9}}$.

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

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.512.f $2$ (not in LMFDB)