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

• Label: 2.4.ag_r
• 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: no

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 3 x + 4 x^{2} )^{2}$
	$1 - 6 x + 17 x^{2} - 24 x^{3} + 16 x^{4}$
Frobenius angles:	$\pm0.230053456163$, $\pm0.230053456163$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$256$	$5476$	$82944$	$1170724$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$15$	$83$	$319$	$1139$	$4191$	$16211$	$64639$	$260147$	$1046175$

Jacobians and polarizations

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

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.ad^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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_ad

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.4.a_ab	$2$	2.16.ac_bh
2.4.g_r	$2$	2.16.ac_bh
2.4.d_f	$3$	2.64.s_ib