Abelian variety isogeny class 4.2.b_b_ad_ag over $\F_{2}$

• Label: 4.2.b_b_ad_ag
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $3$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$4$
L-polynomial:	$( 1 + x + 2 x^{2} )( 1 - x^{2} - 2 x^{3} - 2 x^{4} + 8 x^{6} )$
	$1 + x + x^{2} - 3 x^{3} - 6 x^{4} - 6 x^{5} + 4 x^{6} + 8 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.0599223455698$, $\pm0.532599854765$, $\pm0.615026728081$, $\pm0.842522200335$
Angle rank:	$3$ (numerical)
Jacobians:	$0$
Isomorphism classes:	52

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

Newton polygon

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$256$	$1168$	$34816$	$1077296$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$6$	$-2$	$6$	$34$	$78$	$74$	$286$	$538$	$1006$

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 $\F_{2}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.b $\times$ 3.2.a_ab_ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.2.b : \(\Q(\sqrt{-7}) \).
• 3.2.a_ab_ac : 6.0.399424.1.

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
4.2.ab_b_ab_ac	$2$	4.4.b_af_ab_y
4.2.ab_b_d_ag	$2$	4.4.b_af_ab_y
4.2.b_b_b_ac	$2$	4.4.b_af_ab_y