Abelian variety isogeny class 3.4.aj_bn_adv over $\F_{2^{2}}$

• Label: 3.4.aj_bn_adv
• Base field: $\F_{2^{2}}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x + 4 x^{2} )^{3}$
	$1 - 9 x + 39 x^{2} - 99 x^{3} + 156 x^{4} - 144 x^{5} + 64 x^{6}$
Frobenius angles:	$\pm0.230053456163$, $\pm0.230053456163$, $\pm0.230053456163$
Angle rank:	$1$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$4096$	$405224$	$23887872$	$1266723368$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$14$	$92$	$350$	$1196$	$4238$	$16124$	$64190$	$259148$	$1044974$

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^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

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
3.4.ad_d_d	$2$	3.16.ad_bz_adt
3.4.d_d_ad	$2$	3.16.ad_bz_adt
3.4.j_bn_dv	$2$	3.16.ad_bz_adt
3.4.a_a_j	$3$	(not in LMFDB)