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

• Label: 2.9.al_bw
• 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: no

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 3 x )^{2}( 1 - 5 x + 9 x^{2} )$
	$1 - 11 x + 48 x^{2} - 99 x^{3} + 81 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.186429498677$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20$	$4800$	$500240$	$42720000$	$3486022100$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$57$	$686$	$6513$	$59039$	$531342$	$4781111$	$43033953$	$387360254$	$3486558777$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

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.af 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.af : \(\Q(\sqrt{-11}) \).

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
2.9.ab_am	$2$	2.81.az_lc
2.9.b_am	$2$	2.81.az_lc
2.9.l_bw	$2$	2.81.az_lc
2.9.ac_d	$3$	2.729.abs_bji