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

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

Invariants

Base field:	$\F_{3}$
Dimension:	$2$
L-polynomial:	$( 1 - x + 3 x^{2} )^{2}$
	$1 - 2 x + 7 x^{2} - 6 x^{3} + 9 x^{4}$
Frobenius angles:	$\pm0.406785250661$, $\pm0.406785250661$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Cyclic group of points:	no
Non-cyclic primes:	$3$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9$	$225$	$1296$	$5625$	$45369$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$20$	$44$	$68$	$182$	$710$	$2354$	$6788$	$19412$	$58100$

Jacobians and polarizations

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

• $y^2=2 x^6+x^4+2 x^3+x^2+2$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ab^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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
2.3.a_f	$2$	2.9.k_br
2.3.c_h	$2$	2.9.k_br
2.3.b_ac	$3$	2.27.q_eo