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

• Label: 2.4.ab_e
• Base field: $\F_{2^{2}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$2$
L-polynomial:	$1 - x + 4 x^{2} - 4 x^{3} + 16 x^{4}$
Frobenius angles:	$\pm0.278773483733$, $\pm0.627659203843$
Angle rank:	$2$ (numerical)
Number field:	4.0.2312.1
Galois group:	$D_{4}$
Jacobians:	$1$

This isogeny class is simple and geometrically simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$416$	$3952$	$74048$	$1132816$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$24$	$64$	$288$	$1104$	$3936$	$16048$	$65664$	$261712$	$1048864$

Jacobians and polarizations

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

• $y^2+xy=x^5+x$

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 4.0.2312.1.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

Subfield	Primitive Model
$\F_{2}$	2.2.ab_a
$\F_{2}$	2.2.b_a

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.4.b_e	$2$	2.16.h_bo