Abelian variety isogeny class 3.8.ag_bk_aea over $\F_{2^{3}}$

• 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:	$( 1 - 2 x + 8 x^{2} )^{3}$
	$1 - 6 x + 36 x^{2} - 104 x^{3} + 288 x^{4} - 384 x^{5} + 512 x^{6}$
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, not primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

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

Point counts of the (virtual) curve

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

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{3}}$.

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 $x^{2} - x + 2$.

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.

Twist	Extension degree	Common 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)