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

• Label: 3.9.al_cf_ahq
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $1$
• 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:	$( 1 - 3 x )^{2}( 1 - 5 x + 9 x^{2} )( 1 + 9 x^{2} )$
	$1 - 11 x + 57 x^{2} - 198 x^{3} + 513 x^{4} - 891 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.186429498677$, $\pm0.5$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$200$	$480000$	$365175200$	$273408000000$	$205849605005000$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$75$	$686$	$6351$	$59039$	$532800$	$4781111$	$43020831$	$387360254$	$3486676875$

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 $\times$ 1.9.af $\times$ 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.af : \(\Q(\sqrt{-11}) \).
• 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.ej. 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.ej : \(\Q(\sqrt{-11}) \).

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.ah $\times$ 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.ah : \(\Q(\sqrt{-11}) \).
  • 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.ab_ad_as	$2$	(not in LMFDB)
3.9.b_ad_s	$2$	(not in LMFDB)
3.9.l_cf_hq	$2$	(not in LMFDB)
3.9.ac_m_abk	$3$	(not in LMFDB)