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

• Label: 3.9.ao_do_ano
• 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:	$( 1 - 4 x + 9 x^{2} )( 1 - 5 x + 9 x^{2} )^{2}$
	$1 - 14 x + 92 x^{2} - 352 x^{3} + 828 x^{4} - 1134 x^{5} + 729 x^{6}$
Frobenius angles:	$\pm0.186429498677$, $\pm0.186429498677$, $\pm0.267720472801$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$150$	$472500$	$423842400$	$299413800000$	$210063674403750$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$70$	$794$	$6946$	$60236$	$533680$	$4783964$	$43035586$	$387367706$	$3486631750$

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^2 $\times$ 1.9.ae 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.af^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
• 1.9.ae : \(\Q(\sqrt{-5}) \).

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_m_ai	$2$	(not in LMFDB)
3.9.ae_c_bc	$2$	(not in LMFDB)
3.9.e_c_abc	$2$	(not in LMFDB)
3.9.g_m_i	$2$	(not in LMFDB)
3.9.o_do_no	$2$	(not in LMFDB)
3.9.b_f_ba	$3$	(not in LMFDB)