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

• Label: 3.9.am_cv_akm
• 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:	$( 1 - 4 x + 9 x^{2} )( 1 - 8 x + 32 x^{2} - 72 x^{3} + 81 x^{4} )$
	$1 - 12 x + 73 x^{2} - 272 x^{3} + 657 x^{4} - 972 x^{5} + 729 x^{6}$
Frobenius angles:	$\pm0.141826552031$, $\pm0.267720472801$, $\pm0.358173447969$
Angle rank:	$2$ (numerical)
Isomorphism classes:	22

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$204$	$554064$	$434556108$	$292368491520$	$206409655841964$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$84$	$814$	$6788$	$59198$	$530964$	$4783406$	$43058812$	$387464926$	$3486846804$

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 $\F_{3^{8}}$.

Endomorphism algebra over $\F_{3^{2}}$

The isogeny class factors as 1.9.ae $\times$ 2.9.ai_bg 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.ae : \(\Q(\sqrt{-5}) \).
• 2.9.ai_bg : \(\Q(\zeta_{8})\).

Endomorphism algebra over $\overline{\F}_{3^{2}}$

The base change of $A$ to $\F_{3^{8}}$ is 1.6561.bi^2 $\times$ 1.6561.gc. The endomorphism algebra for each factor is:

• 1.6561.bi^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
• 1.6561.gc : \(\Q(\sqrt{-5}) \).

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{4}}$
  The base change of $A$ to $\F_{3^{4}}$ is 1.81.c $\times$ 2.81.a_bi. The endomorphism algebra for each factor is:

  • 1.81.c : \(\Q(\sqrt{-5}) \).
  • 2.81.a_bi : \(\Q(\zeta_{8})\).

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.ae_j_aq	$2$	(not in LMFDB)
3.9.e_j_q	$2$	(not in LMFDB)
3.9.m_cv_km	$2$	(not in LMFDB)