Abelian variety isogeny class 2.2.ab_ab over $\F_{2}$

• Label: 2.2.ab_ab
• Base field: $\F_{2}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: no
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$2$
L-polynomial:	$1 - x - x^{2} - 2 x^{3} + 4 x^{4}$
Frobenius angles:	$\pm0.0516399385854$, $\pm0.718306605252$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-7})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	1

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$7$	$16$	$259$	$751$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$2$	$-1$	$18$	$22$	$47$	$142$	$226$	$503$	$1082$

Jacobians and polarizations

This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{3}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-7})\).

Endomorphism algebra over $\overline{\F}_{2}$

The base change of $A$ to $\F_{2^{3}}$ is 1.8.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.2.b_ab	$2$	2.4.ad_f
2.2.c_f	$3$	2.8.ak_bp
2.2.ac_f	$6$	2.64.as_ib