Abelian variety isogeny class 3.2.a_ab_b over $\F_{2}$

• Label: 3.2.a_ab_b
• Base field: $\F_{2}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2}$
Dimension:	$3$
L-polynomial:	$1 - x^{2} + x^{3} - 2 x^{4} + 8 x^{6}$
Frobenius angles:	$\pm0.136803573082$, $\pm0.483868732217$, $\pm0.909340018530$
Angle rank:	$3$ (numerical)
Number field:	6.0.12663175.1
Galois group:	$S_4\times C_2$
Jacobians:	$1$
Isomorphism classes:	1

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7$	$35$	$931$	$2275$	$40957$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$3$	$12$	$7$	$38$	$96$	$150$	$279$	$489$	$1168$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):

• $x^4+x^3 y+x^3 z+x z^3+y^3 z=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 6.0.12663175.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
3.2.a_ab_ab	$2$	3.4.ac_ad_t