Abelian variety isogeny class 3.3.ae_j_ar over $\F_{3}$

• Label: 3.3.ae_j_ar
• Base field: $\F_{3}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$3$
L-polynomial:	$1 - 4 x + 9 x^{2} - 17 x^{3} + 27 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.0653366913680$, $\pm0.328985474983$, $\pm0.609104440316$
Angle rank:	$3$ (numerical)
Number field:	6.0.10338167.1
Galois group:	$S_4\times C_2$
Jacobians:	$1$
Isomorphism classes:	1

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7$	$847$	$14308$	$502271$	$14422352$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$12$	$21$	$76$	$245$	$633$	$2058$	$6788$	$20055$	$59107$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=2x^8+2x^7+x^6+2x^5+x^4+x^3+2x^2+x+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3}$.

Endomorphism algebra over $\F_{3}$

The endomorphism algebra of this simple isogeny class is 6.0.10338167.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
3.3.e_j_r	$2$	3.9.c_ab_abl