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

• Label: 3.9.ap_dy_apf
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• 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:	$( 1 - 5 x + 9 x^{2} )^{3}$
	$1 - 15 x + 102 x^{2} - 395 x^{3} + 918 x^{4} - 1215 x^{5} + 729 x^{6}$
Frobenius angles:	$\pm0.186429498677$, $\pm0.186429498677$, $\pm0.186429498677$
Angle rank:	$1$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$125$	$421875$	$405224000$	$297408796875$	$210910505328125$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$61$	$760$	$6901$	$60475$	$535516$	$4790515$	$43047781$	$387357880$	$3486461821$

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.af^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-11}) \)$)$

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
3.9.af_c_bj	$2$	(not in LMFDB)
3.9.f_c_abj	$2$	(not in LMFDB)
3.9.p_dy_pf	$2$	(not in LMFDB)
3.9.a_a_k	$3$	(not in LMFDB)