Abelian variety isogeny class 3.4.af_r_abo over F22\F_{2^{2}}

• Label: 3.4.af_r_abo
• Base field: $\F_{2^{2}}$
• Dimension: $3$
• pp-rank: $2$
• Ordinary: no
• 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:	$( 1 + 4 x^{2} )( 1 - 5 x + 13 x^{2} - 20 x^{3} + 16 x^{4} )$
	$1 - 5 x + 17 x^{2} - 40 x^{3} + 68 x^{4} - 80 x^{5} + 64 x^{6}$
Frobenius angles:	$\pm0.140237960897$, $\pm0.387712212190$, $\pm0.5$
Angle rank:	$2$ (numerical)
Jacobians:	$0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$25$	$6875$	$313300$	$14911875$	$1048190625$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$26$	$75$	$226$	$1000$	$4271$	$16800$	$65986$	$262875$	$1049346$

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 $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2^{2}}$

The isogeny class factors as 1.4.a $\times$ 2.4.af_n and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.4.a : $\Q(\sqrt{-1})$.
• 2.4.af_n : 4.0.1025.1.

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 2.16.b_b. The endomorphism algebra for each factor is:

• 1.16.i : the quaternion algebra over $\Q$ ramified at $2$ and $\infty$.
• 2.16.b_b : 4.0.1025.1.

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
3.4.f_r_bo	$2$	3.16.j_z_bo
3.4.aj_bl_ado	$4$	(not in LMFDB)
3.4.ab_ad_m	$4$	(not in LMFDB)