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

• Label: 4.3.a_ac_a_ad
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$1 - 2 x^{2} - 3 x^{4} - 18 x^{6} + 81 x^{8}$
Frobenius angles:	$\pm0.0513492633383$, $\pm0.355435987323$, $\pm0.644564012677$, $\pm0.948650736662$
Angle rank:	$2$ (numerical)
Number field:	8.0.138169810944.4
Galois group:	$D_4\times C_2$
Jacobians:	$0$
Cyclic group of points:	yes

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$59$	$3481$	$476012$	$34117281$	$3483239699$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$6$	$28$	$62$	$244$	$570$	$2188$	$6758$	$19684$	$58926$

Jacobians and polarizations

This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.

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 8.0.138169810944.4.

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

The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 4.9.ae_ac_ay_jj and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{-2}, \sqrt{-11})\) with the following ramification data at primes above $3$, and unramified at all archimedean places:

$v$	($ 3 $,\( \frac{1}{3} \pi^{3} + 2 \pi^{2} + \frac{2}{3} \pi + 1 \))	($ 3 $,\( \frac{1}{3} \pi^{3} + \frac{8}{3} \pi \))	($ 3 $,\( \frac{5}{3} \pi^{3} + \pi^{2} + \frac{4}{3} \pi + 1 \))	($ 3 $,\( \frac{1}{3} \pi^{3} + \frac{8}{3} \pi + 2 \))
$\operatorname{inv}_v$	$0$	$1/2$	$1/2$	$0$

where $\pi$ is a root of $x^{4} - 2x^{3} + 11x^{2} - 10x + 3$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
4.3.a_c_a_ad	$4$	(not in LMFDB)