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

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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.

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

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

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 $\F_{2^{3}}$.

SubfieldPrimitive 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)