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

• Label: 4.2.ae_h_ah_h
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• 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:	$( 1 - x + 2 x^{2} )( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$
	$1 - 4 x + 7 x^{2} - 7 x^{3} + 7 x^{4} - 14 x^{5} + 28 x^{6} - 32 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.0992589862044$, $\pm0.186455299510$, $\pm0.384973271919$, $\pm0.757883870938$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Isomorphism classes:	3

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$232$	$4214$	$130384$	$766942$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$3$	$8$	$27$	$24$	$78$	$181$	$283$	$575$	$958$

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 $\F_{2^{7}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ab $\times$ 3.2.ad_c_b 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.ab : \(\Q(\sqrt{-7}) \).
• 3.2.ad_c_b : \(\Q(\zeta_{7})\).

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

The base change of $A$ to $\F_{2^{7}}$ is 1.128.n^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-7}) \)$)$

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
4.2.ac_b_ad_j	$2$	4.4.ac_h_ah_v
4.2.c_b_d_j	$2$	4.4.ac_h_ah_v
4.2.e_h_h_h	$2$	4.4.ac_h_ah_v