Abelian variety isogeny class 3.4.ab_a_ae over $\F_{2^{2}}$

• Label: 3.4.ab_a_ae
• Base field: $\F_{2^{2}}$
• Dimension: $3$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$3$
L-polynomial:	$1 - x - 4 x^{3} - 16 x^{5} + 64 x^{6}$
Frobenius angles:	$\pm0.0879689769532$, $\pm0.473922960988$, $\pm0.791945394104$
Angle rank:	$3$ (numerical)
Number field:	6.0.1539727.2
Galois group:	$D_{6}$
Jacobians:	$12$
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:	$1$
Slopes:	$[0, 1/2, 1/2, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$44$	$3784$	$219956$	$15658192$	$973513244$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$16$	$52$	$240$	$924$	$4312$	$16580$	$65184$	$263212$	$1049896$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which 8 are hyperelliptic):

• $a x^4+x^3 y+x^2 y^2+x^2 y z+x z^3+y^4=0$
• $a^2 x^4+x^3 y+x^2 y^2+x^2 y z+x z^3+y^4=0$
• $a x^4+x^3 y+x^2 y^2+x y^2 z+x z^3+y^4=0$
• $a^2 x^4+x^3 y+x^2 y^2+x y^2 z+x z^3+y^4=0$
• $y^2+a x^3 y=x^7+x^2+x+a+1$
• $y^2+a x^3 y=x^7+x^2+a x+a+1$
• $y^2+a x^3 y=x^7+(a+1) x^2+x+a+1$
• $y^2+a x^3 y=x^7+(a+1) x^2+(a+1) x+a+1$
• $y^2+a x^2 y=x^7+x+1$
• $y^2+a x^2 y=a x^7+a x+1$
• $y^2+((a+1) x^4+a x^2+a) y=x$
• $y^2+((a+1) x^4+a x^2+a) y=a x+1$

where $a$ is a root of the Conway polynomial.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{2^{2}}$

The endomorphism algebra of this simple isogeny class is 6.0.1539727.2.

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
3.4.b_a_e	$2$	3.16.ab_ai_dc