Abelian variety isogeny class 2.4.ab_a over $\F_{2^{2}}$

• Label: 2.4.ab_a
• Base field: $\F_{2^{2}}$
• Dimension: $2$
• $p$-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:	$2$
L-polynomial:	$1 - x - 4 x^{3} + 16 x^{4}$
Frobenius angles:	$\pm0.180745703069$, $\pm0.702084401492$
Angle rank:	$2$ (numerical)
Number field:	4.0.13068.1
Galois group:	$D_{4}$
Jacobians:	$2$
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:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12$	$264$	$3348$	$78672$	$1110972$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$16$	$52$	$304$	$1084$	$4120$	$16804$	$65440$	$261196$	$1048936$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2+x y=x^5+a x^3+x$
• $y^2+x y=x^5+(a+1) x^3+x$

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 4.0.13068.1.

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
2.4.b_a	$2$	2.16.ab_y