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

• Label: 3.9.am_cl_aii
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x )^{4}( 1 + 9 x^{2} )$
	$1 - 12 x + 63 x^{2} - 216 x^{3} + 567 x^{4} - 972 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $0$, $0$, $\pm0.5$
Angle rank:	$0$ (numerical)

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

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$160$	$409600$	$333592480$	$262144000000$	$202526270768800$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$64$	$622$	$6076$	$58078$	$529984$	$4774222$	$43007356$	$387341758$	$3486666304$

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.ag^2 $\times$ 1.9.a 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.ag^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.9.a : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{3^{8}}$ is 1.6561.agg^3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{4}}$
  The base change of $A$ to $\F_{3^{4}}$ is 1.81.as^2 $\times$ 1.81.s. The endomorphism algebra for each factor is:

  • 1.81.as^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 1.81.s : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

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.a_aj_a	$2$	(not in LMFDB)
3.9.m_cl_ii	$2$	(not in LMFDB)
3.9.ad_j_acc	$3$	(not in LMFDB)
3.9.g_bk_ee	$3$	(not in LMFDB)