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

• Label: 2.9.ai_be
• Base field: $\F_{3^{2}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 3 x )^{2}( 1 - 2 x + 9 x^{2} )$
	$1 - 8 x + 30 x^{2} - 72 x^{3} + 81 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.391826552031$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

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

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$32$	$6144$	$524576$	$41779200$	$3429976352$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$78$	$722$	$6366$	$58082$	$529326$	$4781618$	$43045566$	$387377858$	$3486552078$

Jacobians and polarizations

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

• $y^2=ax^5+2ax^4+ax^3+2ax^2+ax$

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.ag $\times$ 1.9.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.9.ag : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.9.ac : \(\Q(\sqrt{-2}) \).

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
2.9.ae_g	$2$	2.81.ae_adm
2.9.e_g	$2$	2.81.ae_adm
2.9.i_be	$2$	2.81.ae_adm
2.9.b_m	$3$	2.729.ai_abnm