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

• Label: 3.9.an_df_amc
• 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:	$( 1 - 5 x + 9 x^{2} )( 1 - 4 x + 9 x^{2} )^{2}$
	$1 - 13 x + 83 x^{2} - 314 x^{3} + 747 x^{4} - 1053 x^{5} + 729 x^{6}$
Frobenius angles:	$\pm0.186429498677$, $\pm0.267720472801$, $\pm0.267720472801$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$180$	$529200$	$443316240$	$301432320000$	$209220243606900$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$79$	$828$	$6991$	$59997$	$531844$	$4777413$	$43023391$	$387377532$	$3486801679$

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

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.af_l_ak	$2$	(not in LMFDB)
3.9.ad_d_ba	$2$	(not in LMFDB)
3.9.d_d_aba	$2$	(not in LMFDB)
3.9.f_l_k	$2$	(not in LMFDB)
3.9.n_df_mc	$2$	(not in LMFDB)
3.9.ab_ae_bl	$3$	(not in LMFDB)