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

• Label: 3.9.ag_bc_aee
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• 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:	$( 1 + 9 x^{2} )( 1 - 6 x + 19 x^{2} - 54 x^{3} + 81 x^{4} )$
	$1 - 6 x + 28 x^{2} - 108 x^{3} + 252 x^{4} - 486 x^{5} + 729 x^{6}$
Frobenius angles:	$\pm0.0763052093420$, $\pm0.490896535327$, $\pm0.5$
Angle rank:	$2$ (numerical)
Cyclic group of points:	yes

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$410$	$660100$	$369695360$	$265011667200$	$205044537441050$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$102$	$694$	$6146$	$58804$	$534144$	$4784308$	$43025282$	$387432262$	$3487102662$

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 $\F_{3^{4}}$.

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

The isogeny class factors as 1.9.a $\times$ 2.9.ag_t 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.a : \(\Q(\sqrt{-1}) \).
• 2.9.ag_t : \(\Q(\sqrt{-9 -2 \sqrt{2}})\).

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

The base change of $A$ to $\F_{3^{4}}$ is 1.81.s $\times$ 2.81.c_aev. The endomorphism algebra for each factor is:

• 1.81.s : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 2.81.c_aev : \(\Q(\sqrt{-9 -2 \sqrt{2}})\).

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.g_bc_ee	$2$	(not in LMFDB)
3.9.am_cm_aio	$4$	(not in LMFDB)
3.9.a_ai_ag	$4$	(not in LMFDB)