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

• Label: 3.9.ai_bn_afo
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x )^{2}( 1 - 2 x + 9 x^{2} )( 1 + 9 x^{2} )$
	$1 - 8 x + 39 x^{2} - 144 x^{3} + 351 x^{4} - 648 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.391826552031$, $\pm0.5$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$320$	$614400$	$382940480$	$267386880000$	$202540103585600$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$96$	$722$	$6204$	$58082$	$530784$	$4781618$	$43032444$	$387377858$	$3486670176$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains 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 $\times$ 1.9.ac $\times$ 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 : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.9.ac : \(\Q(\sqrt{-2}) \).
• 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^2 $\times$ 1.6561.abi. The endomorphism algebra for each factor is:

• 1.6561.agg^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.6561.abi : \(\Q(\sqrt{-2}) \).

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 $\times$ 1.81.o $\times$ 1.81.s. The endomorphism algebra for each factor is:

  • 1.81.as : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 1.81.o : \(\Q(\sqrt{-2}) \).
  • 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.ae_p_acu	$2$	(not in LMFDB)
3.9.e_p_cu	$2$	(not in LMFDB)
3.9.i_bn_fo	$2$	(not in LMFDB)
3.9.b_v_s	$3$	(not in LMFDB)