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

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

Invariants

Base field:	$\F_{2^{3}}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x + 8 x^{2} )( 1 - 5 x + 8 x^{2} )^{2}$
	$1 - 13 x + 79 x^{2} - 283 x^{3} + 632 x^{4} - 832 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.154919815756$, $\pm0.154919815756$, $\pm0.322067999368$
Angle rank:	$2$ (numerical)

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$96$	$225792$	$143999712$	$71713345536$	$35646361186656$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$54$	$548$	$4270$	$33196$	$263142$	$2100724$	$16791134$	$134254844$	$1073772054$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.8.af^2 $\times$ 1.8.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.8.af^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.8.ad : \(\Q(\sqrt{-23}) \).

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.ah_t_abl	$2$	(not in LMFDB)
3.8.ad_ab_bb	$2$	(not in LMFDB)
3.8.d_ab_abb	$2$	(not in LMFDB)
3.8.h_t_bl	$2$	(not in LMFDB)
3.8.n_db_kx	$2$	(not in LMFDB)
3.8.c_k_bd	$3$	(not in LMFDB)