Abelian variety isogeny class 2.9.ak_br over $\F_{3^{2}}$

• Label: 2.9.ak_br
• Base field: $\F_{3^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 9 x^{2} )^{2}$
	$1 - 10 x + 43 x^{2} - 90 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.186429498677$, $\pm0.186429498677$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

This isogeny class is not simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$25$	$5625$	$547600$	$44555625$	$3543225625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$68$	$750$	$6788$	$60000$	$534158$	$4788000$	$43047428$	$387378750$	$3486569348$

Jacobians and polarizations

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

• $y^2=ax^6+2ax^4+ax^3+2ax^2+a$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3^{2}}$

The isogeny class factors as 1.9.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.

Subfield	Primitive Model
$\F_{3}$	2.3.a_af

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.9.a_ah	$2$	2.81.ao_id
2.9.k_br	$2$	2.81.ao_id
2.9.f_q	$3$	2.729.u_chy