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

• Label: 2.9.ag_bb
• Base field: $\F_{3^{2}}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 3 x + 9 x^{2} )^{2}$
	$1 - 6 x + 27 x^{2} - 54 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.333333333333$, $\pm0.333333333333$
Angle rank:	$0$ (numerical)
Jacobians:	$1$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$49$	$8281$	$614656$	$44129449$	$3458263249$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$100$	$838$	$6724$	$58564$	$528526$	$4778596$	$43059844$	$387499222$	$3486902500$

Jacobians and polarizations

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

• $y^2=ax^6+2ax+2a$

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^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

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.ag_p
$\F_{3}$	2.3.a_ad
$\F_{3}$	2.3.g_p

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.9.a_j	$2$	2.81.s_jj
2.9.g_bb	$2$	2.81.s_jj
2.9.d_a	$3$	2.729.ee_gmg
2.9.m_cc	$3$	2.729.ee_gmg