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

• Label: 3.8.ap_dv_aob
• Base field: $\F_{2^{3}}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• 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:	$( 1 - 5 x + 8 x^{2} )^{3}$
	$1 - 15 x + 99 x^{2} - 365 x^{3} + 792 x^{4} - 960 x^{5} + 512 x^{6}$
Frobenius angles:	$\pm0.154919815756$, $\pm0.154919815756$, $\pm0.154919815756$
Angle rank:	$1$ (numerical)

This isogeny class is not simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$64$	$175616$	$131096512$	$71163817984$	$36080939749184$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$38$	$498$	$4238$	$33594$	$265142$	$2105538$	$16795166$	$134240394$	$1073711558$

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^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

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.a_a_af
$\F_{2}$	3.2.d_j_n

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.8.af_ab_bt	$2$	(not in LMFDB)
3.8.f_ab_abt	$2$	(not in LMFDB)
3.8.p_dv_ob	$2$	(not in LMFDB)
3.8.a_a_af	$3$	(not in LMFDB)