# Properties

 Label 1.512.bg Base Field $\F_{2^{9}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{9}}$ Dimension: $1$ L-polynomial: $1 + 32 x + 512 x^{2}$ Frobenius angles: $\pm0.750000000000$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-1})$$ Galois group: $C_2$

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 a Jacobian, and hence is principally polarizable.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 545 262145 134201345 68720001025 35184363700225 18014398509481985 9223372041149743105 4722366482732206260225 2417851639231457372667905 1237940039285380274899124225

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 545 262145 134201345 68720001025 35184363700225 18014398509481985 9223372041149743105 4722366482732206260225 2417851639231457372667905 1237940039285380274899124225

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{9}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-1})$$.
Endomorphism algebra over $\overline{\F}_{2^{9}}$
 The base change of $A$ to $\F_{2^{36}}$ is the simple isogeny class 1.68719476736.bdvoy and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{36}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{18}}$  The base change of $A$ to $\F_{2^{18}}$ is the simple isogeny class 1.262144.a and its endomorphism algebra is $$\Q(\sqrt{-1})$$.

## 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.c

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.512.abg $2$ (not in LMFDB) 1.512.a $8$ (not in LMFDB)