# Properties

 Label 3.8.ag_bk_aea Base field $\F_{2^{3}}$ Dimension $3$ $p$-rank $0$ Ordinary no Supersingular no Simple yes Geometrically simple yes Primitive no Principally polarizable yes

The only example of a hyperspecial abelian variety of dimension greater than 1 in the LMFDB (see Chai-Oort [10.4310/PAMQ.2006.v2.n1.a2]).

## Invariants

 Base field: $\F_{2^{3}}$ Dimension: $3$ L-polynomial: $( 1 - 2 x + 8 x^{2} )^{3}$ $1 - 6x + 36x^{2} - 104x^{3} + 288x^{4} - 384x^{5} + 512x^{6}$ Frobenius angles: $\pm0.384973271919$, $\pm0.384973271919$, $\pm0.384973271919$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-7})$$ Galois group: $C_2$

This isogeny class is simple and geometrically simple.

## Newton polygon $p$-rank: $0$ Slopes: $[1/3, 1/3, 1/3, 2/3, 2/3, 2/3]$

## Point counts

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $343$ $456533$ $169112377$ $67967263441$ $34065789855713$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $101$ $633$ $4049$ $31713$ $260417$ $2102145$ $16801025$ $134225409$ $1073566721$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{3}}$
The endomorphism algebra of this simple isogeny class is the division algebra of dimension 9 over $$\Q(\sqrt{-7})$$ with the following ramification data at primes above $2$, and unramified at all archimedean places:
 $v$ ($2$,$$\pi + 1$$) ($2$,$$\pi$$) $\operatorname{inv}_v$ $1/3$ $2/3$
where $\pi$ is a root of $x^{2} - x + 2$.\n
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 $\F_{2}$ 3.2.a_a_ac

## Twists

Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
3.8.g_bk_ea$2$(not in LMFDB)
3.8.ag_i_i$7$(not in LMFDB)
3.8.i_bk_eq$7$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
3.8.g_bk_ea$2$(not in LMFDB)
3.8.ag_i_i$7$(not in LMFDB)
3.8.i_bk_eq$7$(not in LMFDB)
3.8.ai_bk_aeq$14$(not in LMFDB)
3.8.g_i_ai$14$(not in LMFDB)