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

• Label: 3.9.ap_dw_aou
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $2$
• 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:	$3$
L-polynomial:	$( 1 - 3 x )^{2}( 1 - 9 x + 37 x^{2} - 81 x^{3} + 81 x^{4} )$
	$1 - 15 x + 100 x^{2} - 384 x^{3} + 900 x^{4} - 1215 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.114191348093$, $\pm0.309392441858$
Angle rank:	$2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$116$	$387904$	$372652436$	$279639993600$	$204331003823936$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$57$	$703$	$6497$	$58600$	$529473$	$4778335$	$43045217$	$387451027$	$3486885852$

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 $\times$ 2.9.aj_bl 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 $\infty$.
• 2.9.aj_bl : 4.0.2725.1.

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.ad_ai_ci	$2$	(not in LMFDB)
3.9.d_ai_aci	$2$	(not in LMFDB)
3.9.p_dw_ou	$2$	(not in LMFDB)
3.9.ag_t_abz	$3$	(not in LMFDB)