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

• Label: 4.3.a_ae_a_h
• 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:	$1 - 4 x^{2} + 7 x^{4} - 36 x^{6} + 81 x^{8}$
Frobenius angles:	$\pm0.0328064302661$, $\pm0.300526903067$, $\pm0.699473096933$, $\pm0.967193569734$
Angle rank:	$1$ (numerical)
Number field:	8.0.3317760000.3
Galois group:	$C_2^3$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$49$	$2401$	$470596$	$39955041$	$3500716849$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$2$	$28$	$78$	$244$	$554$	$2188$	$6246$	$19684$	$59522$

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 $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3}$

The endomorphism algebra of this simple isogeny class is 8.0.3317760000.3.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.abs^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-5}) \)$)$

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{3^{2}}$ is 2.9.ae_h^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{-5})\)$)$

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

  The base change of $A$ to $\F_{3^{3}}$ is 2.27.a_abs^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{2}, \sqrt{-5})\)$)$

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_i_a_bi	$3$	(not in LMFDB)
4.3.a_e_a_h	$4$	(not in LMFDB)
4.3.ac_c_i_ar	$8$	(not in LMFDB)