Abelian variety isogeny class 3.9.as_ff_auu over $\F_{3^{2}}$

• Label: 3.9.as_ff_auu
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x )^{6}$
	$1 - 18 x + 135 x^{2} - 540 x^{3} + 1215 x^{4} - 1458 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $0$, $0$, $0$, $0$
Angle rank:	$0$ (numerical)

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$64$	$262144$	$308915776$	$262144000000$	$200859416110144$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-8$	$28$	$568$	$6076$	$57592$	$527068$	$4769848$	$43007356$	$387302392$	$3486430108$

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^3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is 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
3.9.ag_aj_ee	$2$	(not in LMFDB)
3.9.g_aj_aee	$2$	(not in LMFDB)
3.9.s_ff_uu	$2$	(not in LMFDB)
3.9.aj_bb_acc	$3$	(not in LMFDB)
3.9.a_a_acc	$3$	(not in LMFDB)
3.9.j_cc_hh	$3$	(not in LMFDB)