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

• Label: 3.3.ad_i_aq
• 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:	$1 - 3 x + 8 x^{2} - 16 x^{3} + 24 x^{4} - 27 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.144152045695$, $\pm0.430919798031$, $\pm0.579935378442$
Angle rank:	$3$ (numerical)
Number field:	6.0.15522536.1
Galois group:	$S_4\times C_2$
Jacobians:	$1$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14$	$1484$	$18494$	$439264$	$15630944$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$17$	$25$	$65$	$266$	$797$	$2283$	$6801$	$19807$	$58472$

Jacobians and polarizations

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

• $y^2=2x^8+2x^4+2x^3+x^2+x$

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 6.0.15522536.1.

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.d_i_q	$2$	3.9.h_q_u