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

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

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$( 1 - 4 x^{2} + 9 x^{4} )^{2}$
	$1 - 8 x^{2} + 34 x^{4} - 72 x^{6} + 81 x^{8}$
Frobenius angles:	$\pm0.133860236401$, $\pm0.133860236401$, $\pm0.866139763599$, $\pm0.866139763599$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$1296$	$599076$	$49787136$	$3514829796$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$-6$	$28$	$90$	$244$	$906$	$2188$	$7194$	$19684$	$59994$

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 isogeny class factors as 2.3.a_ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}, \sqrt{-5})\)$)$

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

The base change of $A$ to $\F_{3^{2}}$ is 1.9.ae^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-5}) \)$)$

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.a_e_a_h	$3$	(not in LMFDB)
4.3.a_a_a_c	$4$	(not in LMFDB)
4.3.a_i_a_bi	$4$	(not in LMFDB)