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

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

Invariants

Base field:	$\F_{2^{3}}$
Dimension:	$3$
L-polynomial:	$1 - 8 x + 36 x^{2} - 120 x^{3} + 288 x^{4} - 512 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.0435981566527$, $\pm0.329312442367$, $\pm0.527830414776$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{7})\)
Galois group:	$C_6$

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$197$	$290969$	$131937401$	$66073531489$	$34877605586657$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$73$	$505$	$3937$	$32481$	$261313$	$2092161$	$16770561$	$134249473$	$1073810433$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

• $y^2+ax^4y=x^8+(a+1)x$
• $y^2+ax^4y=x^8+(a^2+1)x$
• $y^2+ax^4y=x^8+(a^2+a)x$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{21}}$.

Endomorphism algebra over $\F_{2^{3}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{7})\).

Endomorphism algebra over $\overline{\F}_{2^{3}}$

The base change of $A$ to $\F_{2^{21}}$ is the simple isogeny class 3.2097152.ahka_bfyoxc_adesazpwa and its endomorphism algebra 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 $,\( \pi \))	($ 2 $,\( \pi + 1 \))
$\operatorname{inv}_v$	$1/3$	$2/3$

where $\pi$ is a root of $x^{2} - x + 2$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.8.i_bk_eq	$2$	(not in LMFDB)
3.8.g_i_ai	$7$	(not in LMFDB)
3.8.g_bk_ea	$7$	(not in LMFDB)