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

• Label: 2.9.af_y
• 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:	$( 1 - 3 x + 9 x^{2} )( 1 - 2 x + 9 x^{2} )$
	$1 - 5 x + 24 x^{2} - 45 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.333333333333$, $\pm0.391826552031$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$56$	$8736$	$608384$	$43365504$	$3444208376$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$105$	$830$	$6609$	$58325$	$529326$	$4783805$	$43065249$	$387456590$	$3486729225$

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 $\F_{3^{6}}$.

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

The isogeny class factors as 1.9.ad $\times$ 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.ad : \(\Q(\sqrt{-3}) \).
• 1.9.ac : \(\Q(\sqrt{-2}) \).

Endomorphism algebra over $\overline{\F}_{3^{2}}$

The base change of $A$ to $\F_{3^{6}}$ is 1.729.bu $\times$ 1.729.cc. The endomorphism algebra for each factor is:

• 1.729.bu : \(\Q(\sqrt{-2}) \).
• 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

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_m	$2$	2.81.x_lc
2.9.b_m	$2$	2.81.x_lc
2.9.f_y	$2$	2.81.x_lc
2.9.e_g	$3$	2.729.dw_fvq