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

• Label: 3.9.am_cu_akg
• 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:	$( 1 - 2 x + 9 x^{2} )( 1 - 5 x + 9 x^{2} )^{2}$
	$1 - 12 x + 72 x^{2} - 266 x^{3} + 648 x^{4} - 972 x^{5} + 729 x^{6}$
Frobenius angles:	$\pm0.186429498677$, $\pm0.186429498677$, $\pm0.391826552031$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$200$	$540000$	$424937600$	$290859120000$	$207519638405000$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$82$	$796$	$6754$	$59518$	$533500$	$4791022$	$43059394$	$387375484$	$3486455122$

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.ac 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.ac : \(\Q(\sqrt{-2}) \).

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.ai_bg_adq	$2$	(not in LMFDB)
3.9.ac_c_o	$2$	(not in LMFDB)
3.9.c_c_ao	$2$	(not in LMFDB)
3.9.i_bg_dq	$2$	(not in LMFDB)
3.9.m_cu_kg	$2$	(not in LMFDB)
3.9.d_p_cg	$3$	(not in LMFDB)