# Properties

 Label 1.256.az Base Field $\F_{2^{8}}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{8}}$ Dimension: $1$ L-polynomial: $1 - 25 x + 256 x^{2}$ Frobenius angles: $\pm0.214582404850$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-399})$$ Galois group: $C_2$ Jacobians: 16

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 16 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 232 65424 16780792 4295085600 1099513670152 281474997484464 72057594034425112 18446744068303886400 4722366482735400306472 1208925819612656902321104

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 232 65424 16780792 4295085600 1099513670152 281474997484464 72057594034425112 18446744068303886400 4722366482735400306472 1208925819612656902321104

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-399})$$.
All geometric endomorphisms are defined over $\F_{2^{8}}$.

## 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 1.256.z $2$ (not in LMFDB)