Abelian variety isogeny class 4.3.a_ad_a_q over $\F_{3}$

• Label: 4.3.a_ad_a_q
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$1 - 3 x^{2} + 16 x^{4} - 27 x^{6} + 81 x^{8}$
Frobenius angles:	$\pm0.148854628963$, $\pm0.264917483542$, $\pm0.735082516458$, $\pm0.851145371037$
Angle rank:	$2$ (numerical)
Number field:	8.0.3613452544.3
Galois group:	$D_4\times C_2$
Jacobians:	$0$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$68$	$4624$	$558416$	$75759616$	$3500901188$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$4$	$28$	$128$	$244$	$802$	$2188$	$6528$	$19684$	$59524$

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 $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3}$

The endomorphism algebra of this simple isogeny class is 8.0.3613452544.3.

Endomorphism algebra over $\overline{\F}_{3}$

The base change of $A$ to $\F_{3^{2}}$ is 2.9.ad_q^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.3757.1$)$

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
4.3.ac_f_ak_bc	$4$	(not in LMFDB)
4.3.a_d_a_q	$4$	(not in LMFDB)
4.3.c_f_k_bc	$4$	(not in LMFDB)