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

• Label: 3.4.ah_bb_aco
• Base field: $\F_{2^{2}}$
• Dimension: $3$
• $p$-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:	$( 1 - 2 x + 4 x^{2} )( 1 - 5 x + 13 x^{2} - 20 x^{3} + 16 x^{4} )$
	$1 - 7 x + 27 x^{2} - 66 x^{3} + 108 x^{4} - 112 x^{5} + 64 x^{6}$
Frobenius angles:	$\pm0.140237960897$, $\pm0.333333333333$, $\pm0.387712212190$
Angle rank:	$2$ (numerical)

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$15$	$5775$	$390420$	$18093075$	$1015466625$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$22$	$91$	$274$	$968$	$4015$	$16672$	$66754$	$263899$	$1048322$

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 $\F_{2^{6}}$.

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

The isogeny class factors as 1.4.ac $\times$ 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.ac : \(\Q(\sqrt{-3}) \).
• 2.4.af_n : 4.0.1025.1.

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

The base change of $A$ to $\F_{2^{6}}$ is 1.64.q $\times$ 2.64.k_cv. The endomorphism algebra for each factor is:

• 1.64.q : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.64.k_cv : 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.ad_h_ao	$2$	3.16.f_v_bk
3.4.d_h_o	$2$	3.16.f_v_bk
3.4.h_bb_co	$2$	3.16.f_v_bk
3.4.ab_ad_m	$3$	(not in LMFDB)