# Properties

 Label 1.256.abg Base Field $\F_{2^{8}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{8}}$ Dimension: $1$ L-polynomial: $( 1 - 16 x )^{2}$ Frobenius angles: $0$, $0$ Angle rank: $0$ (numerical) Number field: $$\Q$$ Galois group: Trivial Jacobians: 1

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is supersingular.

 $p$-rank: $0$ Slopes: $[1/2, 1/2]$

## Point counts

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 225 65025 16769025 4294836225 1099509530625 281474943156225 72057593501057025 18446744065119617025 4722366482732206260225 1208925819612430151450625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 225 65025 16769025 4294836225 1099509530625 281474943156225 72057593501057025 18446744065119617025 4722366482732206260225 1208925819612430151450625

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
 The endomorphism algebra of this simple isogeny class is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{8}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{8}}$.

 Subfield Primitive Model $\F_{2}$ 1.2.ac $\F_{2}$ 1.2.a $\F_{2}$ 1.2.c

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 1.256.bg $2$ (not in LMFDB) 1.256.q $3$ (not in LMFDB) 1.256.a $4$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.256.bg $2$ (not in LMFDB) 1.256.q $3$ (not in LMFDB) 1.256.a $4$ (not in LMFDB) 1.256.aq $6$ (not in LMFDB)