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

• Label: 3.9.an_cz_akw
• 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:	$( 1 - 3 x )^{2}( 1 - 7 x + 26 x^{2} - 63 x^{3} + 81 x^{4} )$
	$1 - 13 x + 77 x^{2} - 282 x^{3} + 693 x^{4} - 1053 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.122441590128$, $\pm0.422937410221$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$152$	$432896$	$366619136$	$271512371200$	$203209984268632$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$67$	$690$	$6303$	$58277$	$531136$	$4786877$	$43051391$	$387386178$	$3486667507$

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 $\F_{3^{2}}$.

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

The isogeny class factors as 1.9.ag $\times$ 2.9.ah_ba 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.ah_ba : 4.0.49708.1.

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_ah_be	$2$	(not in LMFDB)
3.9.b_ah_abe	$2$	(not in LMFDB)
3.9.n_cz_kw	$2$	(not in LMFDB)
3.9.ae_o_abw	$3$	(not in LMFDB)