Abelian variety isogeny class 4.2.a_a_ac_a over $\F_{2}$

• Label: 4.2.a_a_ac_a
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2}$
Dimension:	$4$
L-polynomial:	$1 - 2 x^{3} - 4 x^{5} + 16 x^{8}$
Frobenius angles:	$\pm0.0640918585257$, $\pm0.393625989811$, $\pm0.651207033805$, $\pm0.823657974581$
Angle rank:	$4$ (numerical)
Number field:	8.0.5198736512.1
Galois group:	$C_2 \wr S_4$
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:	$0$
Slopes:	$[1/3, 1/3, 1/3, 1/2, 1/2, 2/3, 2/3, 2/3]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$11$	$253$	$1727$	$74129$	$537581$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$5$	$3$	$17$	$13$	$53$	$129$	$321$	$489$	$945$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2 + y=x^9 + x^7 + x^3 + 1$

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 8.0.5198736512.1.

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
4.2.a_a_c_a	$2$	4.4.a_a_ae_q