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

• Label: 3.8.ak_cf_aho
• Base field: $\F_{2^{3}}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• 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:	$( 1 - 4 x + 8 x^{2} )( 1 - 3 x + 8 x^{2} )^{2}$
	$1 - 10 x + 57 x^{2} - 196 x^{3} + 456 x^{4} - 640 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.322067999368$, $\pm0.322067999368$
Angle rank:	$1$ (numerical)

This isogeny class is not simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$180$	$336960$	$169693380$	$73679673600$	$35196767460900$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$79$	$635$	$4383$	$32779$	$260143$	$2091067$	$16772927$	$134245355$	$1073842639$

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 $\F_{2^{12}}$.

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

The isogeny class factors as 1.8.ae $\times$ 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.ae : \(\Q(\sqrt{-1}) \).
• 1.8.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-23}) \)$)$

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.db^2 $\times$ 1.4096.ey. The endomorphism algebra for each factor is:

• 1.4096.db^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-23}) \)$)$
• 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.h^2. The endomorphism algebra for each factor is:

  • 1.64.a : \(\Q(\sqrt{-1}) \).
  • 1.64.h^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.ae_p_abc	$2$	(not in LMFDB)
3.8.ac_j_e	$2$	(not in LMFDB)
3.8.c_j_ae	$2$	(not in LMFDB)
3.8.e_p_bc	$2$	(not in LMFDB)
3.8.k_cf_ho	$2$	(not in LMFDB)
3.8.ab_ad_bs	$3$	(not in LMFDB)