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

• Label: 3.8.al_cl_ain
• 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:	$( 1 - 5 x + 8 x^{2} )( 1 - 3 x + 8 x^{2} )^{2}$
	$1 - 11 x + 63 x^{2} - 221 x^{3} + 504 x^{4} - 704 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.154919815756$, $\pm0.322067999368$, $\pm0.322067999368$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$144$	$290304$	$158172912$	$72267116544$	$35217016925904$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$70$	$598$	$4302$	$32798$	$261142$	$2095910$	$16787102$	$134269294$	$1073832550$

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

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.af_p_abj	$2$	(not in LMFDB)
3.8.ab_d_bd	$2$	(not in LMFDB)
3.8.b_d_abd	$2$	(not in LMFDB)
3.8.f_p_bj	$2$	(not in LMFDB)
3.8.l_cl_in	$2$	(not in LMFDB)
3.8.ac_ag_br	$3$	(not in LMFDB)