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

• Label: 3.2.ac_b_a
• Base field: $\F_{2}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• 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:	$1 - 2 x + x^{2} + 2 x^{4} - 8 x^{5} + 8 x^{6}$
Frobenius angles:	$\pm0.0693533547550$, $\pm0.339907131295$, $\pm0.770553776540$
Angle rank:	$2$ (numerical)
Number field:	6.0.2580992.1
Galois group:	$D_{6}$
Jacobians:	$1$
Cyclic group of points:	yes

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$44$	$386$	$5984$	$16742$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$3$	$7$	$23$	$11$	$51$	$127$	$255$	$619$	$1043$

Jacobians and polarizations

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

• $x^4+x^3 z+x y^3+x z^3+y^4=0$

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 6.0.2580992.1.

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.c_b_a	$2$	3.4.ac_f_am