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

• Label: 4.3.ai_bg_adk_gw
• 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:	$( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )^{2}$
	$1 - 8 x + 32 x^{2} - 88 x^{3} + 178 x^{4} - 264 x^{5} + 288 x^{6} - 216 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.0540867239847$, $\pm0.0540867239847$, $\pm0.445913276015$, $\pm0.445913276015$
Angle rank:	$1$ (numerical)

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$4624$	$391876$	$21381376$	$2428715524$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$10$	$20$	$26$	$156$	$730$	$2292$	$6426$	$19100$	$59050$

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 $\F_{3^{4}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 2.3.ae_i^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$

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

The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-2}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{2}}$

  The base change of $A$ to $\F_{3^{2}}$ is 2.9.a_ao^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$

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_a_a_ao	$2$	(not in LMFDB)
4.3.i_bg_dk_gw	$2$	(not in LMFDB)
4.3.e_i_i_h	$3$	(not in LMFDB)