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

• Label: 3.9.ao_dl_amy
• 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:	$( 1 - 3 x )^{2}( 1 - 8 x + 32 x^{2} - 72 x^{3} + 81 x^{4} )$
	$1 - 14 x + 89 x^{2} - 336 x^{3} + 801 x^{4} - 1134 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.141826552031$, $\pm0.358173447969$
Angle rank:	$1$ (numerical)

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$136$	$422144$	$379534792$	$278446182400$	$203895946508936$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$64$	$716$	$6468$	$58476$	$529984$	$4783068$	$43057532$	$387436604$	$3486666304$

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 $\times$ 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.ag : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 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.agg $\times$ 1.6561.bi^2. The endomorphism algebra for each factor is:

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

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 $\times$ 2.81.a_bi. The endomorphism algebra for each factor is:

  • 1.81.as : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 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.ac_ah_bw	$2$	(not in LMFDB)
3.9.c_ah_abw	$2$	(not in LMFDB)
3.9.o_dl_my	$2$	(not in LMFDB)
3.9.af_r_abw	$3$	(not in LMFDB)