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

• Label: 3.4.ah_bc_acq
• Base field: $\F_{2^{2}}$
• Dimension: $3$
• $p$-rank: $1$
• 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:	$( 1 - 3 x + 4 x^{2} )( 1 - 2 x + 4 x^{2} )^{2}$
	$1 - 7 x + 28 x^{2} - 68 x^{3} + 112 x^{4} - 112 x^{5} + 64 x^{6}$
Frobenius angles:	$\pm0.230053456163$, $\pm0.333333333333$, $\pm0.333333333333$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18$	$7056$	$485514$	$21464352$	$1066905018$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$24$	$106$	$320$	$1018$	$3888$	$16042$	$65600$	$263194$	$1049424$

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.ad $\times$ 1.4.ac^2 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.ad : \(\Q(\sqrt{-7}) \).
• 1.4.ac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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

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

• 1.64.j : \(\Q(\sqrt{-7}) \).
• 1.64.q^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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_i_am	$2$	3.16.h_ce_ia
3.4.ab_e_e	$2$	3.16.h_ce_ia
3.4.b_e_ae	$2$	3.16.h_ce_ia
3.4.d_i_m	$2$	3.16.h_ce_ia
3.4.h_bc_cq	$2$	3.16.h_ce_ia
3.4.ab_ac_q	$3$	(not in LMFDB)
3.4.f_e_ai	$3$	(not in LMFDB)