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

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

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - x + 4 x^{2} )^{2}$
	$1 - 2 x + 9 x^{2} - 8 x^{3} + 16 x^{4}$
Frobenius angles:	$\pm0.419569376745$, $\pm0.419569376745$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

This isogeny class is not simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$576$	$5776$	$57600$	$929296$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$31$	$87$	$223$	$903$	$4111$	$16887$	$65983$	$260583$	$1045231$

Jacobians and polarizations

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

• $y^2+(x^2+x)y=x^5+x^3+x^2+x$

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.4.ab^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

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.a_ab

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.4.a_h	$2$	2.16.o_dd
2.4.c_j	$2$	2.16.o_dd
2.4.b_ad	$3$	2.64.w_jp