Abelian variety isogeny class 2.3.ac_f over $\F_{3}$

• Label: 2.3.ac_f
• Base field: $\F_{3}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$2$
L-polynomial:	$1 - 2 x + 5 x^{2} - 6 x^{3} + 9 x^{4}$
Frobenius angles:	$\pm0.254551732336$, $\pm0.538152604671$
Angle rank:	$2$ (numerical)
Number field:	4.0.4672.2
Galois group:	$D_{4}$
Jacobians:	$1$
Isomorphism classes:	1
Cyclic group of points:	yes

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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7$	$161$	$868$	$6601$	$69167$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$16$	$32$	$84$	$282$	$766$	$2046$	$6308$	$19760$	$59296$

Jacobians and polarizations

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

• $y^2=2 x^6+x^5+2 x^3+2 x^2+x+2$

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 4.0.4672.2.

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
2.3.c_f	$2$	2.9.g_t