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

• Label: 3.2.ab_ab_f
• 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:	$1 - x - x^{2} + 5 x^{3} - 2 x^{4} - 4 x^{5} + 8 x^{6}$
Frobenius angles:	$\pm0.188077943068$, $\pm0.335703956083$, $\pm0.922415663513$
Angle rank:	$3$ (numerical)
Number field:	6.0.1142512.1
Galois group:	$S_4\times C_2$
Jacobians:	$1$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$36$	$1962$	$5904$	$43926$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$2$	$20$	$22$	$42$	$62$	$128$	$286$	$506$	$1042$

Jacobians and polarizations

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

• $x^3 y+x^2 y z+x y^3+x z^3+y^4=0$

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 6.0.1142512.1.

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.b_ab_af	$2$	3.4.ad_h_an