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

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

Invariants

Base field:	$\F_{2^{3}}$
Dimension:	$3$
L-polynomial:	$( 1 + 8 x^{2} )( 1 - 5 x + 8 x^{2} )^{2}$
	$1 - 10 x + 49 x^{2} - 160 x^{3} + 392 x^{4} - 640 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.154919815756$, $\pm0.154919815756$, $\pm0.5$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$144$	$254016$	$132386832$	$68158589184$	$35780665616784$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$63$	$503$	$4063$	$33319$	$265167$	$2102743$	$16780991$	$134232839$	$1073787183$

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 $\F_{2^{6}}$.

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

The isogeny class factors as 1.8.af^2 $\times$ 1.8.a 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.a : \(\Q(\sqrt{-2}) \).

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

The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj^2 $\times$ 1.64.q. The endomorphism algebra for each factor is:

• 1.64.aj^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.64.q : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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.ab_b_ae
$\F_{2}$	3.2.c_h_i

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.8.a_ab_a	$2$	(not in LMFDB)
3.8.k_bx_ge	$2$	(not in LMFDB)
3.8.f_z_dc	$3$	(not in LMFDB)