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

• Label: 4.3.ah_z_acn_fa
• 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:	$( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$
	$1 - 7 x + 25 x^{2} - 65 x^{3} + 130 x^{4} - 195 x^{5} + 225 x^{6} - 189 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.0540867239847$, $\pm0.0975263560046$, $\pm0.445913276015$, $\pm0.527857038681$
Angle rank:	$3$ (numerical)
Isomorphism classes:	1

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$5508$	$343674$	$22847184$	$3077562336$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$11$	$15$	$31$	$212$	$791$	$2237$	$6495$	$19851$	$59846$

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 $\times$ 2.3.ad_f and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 2.3.ae_i : \(\Q(\zeta_{8})\).
• 2.3.ad_f : 4.0.2197.1.

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

The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao^2 $\times$ 2.81.ax_kf. The endomorphism algebra for each factor is:

• 1.81.ao^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
• 2.81.ax_kf : 4.0.2197.1.

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 $\times$ 2.9.b_al. The endomorphism algebra for each factor is:

  • 2.9.a_ao : \(\Q(\zeta_{8})\).
  • 2.9.b_al : 4.0.2197.1.

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.ab_b_b_ao	$2$	(not in LMFDB)
4.3.b_b_ab_ao	$2$	(not in LMFDB)
4.3.h_z_cn_fa	$2$	(not in LMFDB)