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

• Label: 4.3.a_ae_a_w
• 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:	$( 1 - 2 x^{2} + 9 x^{4} )^{2}$
	$1 - 4 x^{2} + 22 x^{4} - 36 x^{6} + 81 x^{8}$
Frobenius angles:	$\pm0.195913276015$, $\pm0.195913276015$, $\pm0.804086723985$, $\pm0.804086723985$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$64$	$4096$	$602176$	$84934656$	$3430210624$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$2$	$28$	$138$	$244$	$914$	$2188$	$6426$	$19684$	$57122$

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_ac^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$

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

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

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_c_a_af	$3$	(not in LMFDB)
4.3.ai_bk_aea_ig	$4$	(not in LMFDB)
4.3.ae_i_ae_ac	$4$	(not in LMFDB)