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

• Label: 3.4.ac_ab_g
• Base field: $\F_{2^{2}}$
• 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:	$\F_{2^{2}}$
Dimension:	$3$
L-polynomial:	$1 - 2 x - x^{2} + 6 x^{3} - 4 x^{4} - 32 x^{5} + 64 x^{6}$
Frobenius angles:	$\pm0.0754227771019$, $\pm0.367700328682$, $\pm0.839683286431$
Angle rank:	$3$ (numerical)
Number field:	6.0.37606811.1
Galois group:	$A_4\times C_2$
Jacobians:	$12$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$32$	$2816$	$279584$	$16772096$	$943782752$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$11$	$69$	$255$	$893$	$4103$	$16229$	$66527$	$263085$	$1048311$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which 4 are hyperelliptic):

• $a x^4+a x^3 y+a x^2 y z+x y^2 z+x z^3+y^4=0$
• $x^4+a x^3 y+x^3 z+a^2 x^2 y^2+a x^2 y z+x y^2 z+x z^3+y^4=0$
• $a x^4+a^2 x^3 y+a^2 x^2 y z+x y^2 z+x z^3+y^4=0$
• $x^4+a^2 x^3 y+x^3 z+a x^2 y^2+a^2 x^2 y z+x y^2 z+x z^3+y^4=0$
• $a^2 x^3 z+a x^2 y^2+x^2 y z+x^2 z^2+x z^3+y^3 z=0$
• $a x^3 z+a^2 x^2 y^2+x^2 y z+x^2 z^2+x z^3+y^3 z=0$
• $x^4+a x^2 y^2+x^2 y z+x^2 z^2+x y z^2+x z^3+y^3 z=0$
• $x^4+a^2 x^2 y^2+x^2 y z+x^2 z^2+x y z^2+x z^3+y^3 z=0$
• $y^2+(a x^3+a x^2) y=x^7+(a+1) x^2+a x$
• $y^2+(a x^3+a x^2) y=a x^7+(a+1) x^2+x$
• $y^2+((a+1) x^4+a x^3+a x^2) y=a x^3+a x^2+a x+1$
• $y^2+((a+1) x^4+a x^3+a x^2) y=x^8+x^2+x+a$

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 6.0.37606811.1.

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
3.4.c_ab_ag	$2$	3.16.ag_r_abc