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

• Label: 3.8.al_cm_aiq
• Base field: $\F_{2^{3}}$
• Dimension: $3$
• $p$-rank: $1$
• 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:	$( 1 - 3 x + 8 x^{2} )( 1 - 4 x + 8 x^{2} )^{2}$
	$1 - 11 x + 64 x^{2} - 224 x^{3} + 512 x^{4} - 704 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.250000000000$, $\pm0.322067999368$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$150$	$304200$	$165739950$	$74544210000$	$35605380303750$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$72$	$622$	$4432$	$33158$	$261144$	$2091038$	$16762784$	$134206966$	$1073792232$

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^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.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 1.8.ad : \(\Q(\sqrt{-23}) \).

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

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

• 1.4096.db : \(\Q(\sqrt{-23}) \).
• 1.4096.ey^2 : $\mathrm{M}_{2}(B)$, where $B$ is 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^2 $\times$ 1.64.h. The endomorphism algebra for each factor is:

  • 1.64.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 1.64.h : \(\Q(\sqrt{-23}) \).

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_q_abg	$2$	(not in LMFDB)
3.8.ad_i_a	$2$	(not in LMFDB)
3.8.d_i_a	$2$	(not in LMFDB)
3.8.f_q_bg	$2$	(not in LMFDB)
3.8.l_cm_iq	$2$	(not in LMFDB)
3.8.b_e_bo	$3$	(not in LMFDB)