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

• Label: 3.9.an_da_alc
• 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:	$( 1 - 3 x )^{2}( 1 - 7 x + 27 x^{2} - 63 x^{3} + 81 x^{4} )$
	$1 - 13 x + 78 x^{2} - 288 x^{3} + 702 x^{4} - 1053 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.154979380638$, $\pm0.408713257520$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$156$	$446784$	$377347932$	$275263622400$	$203809222009536$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$69$	$711$	$6393$	$58452$	$531189$	$4786779$	$43051857$	$387372969$	$3486534204$

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_bb 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_bb : 4.0.4901.1.

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_ag_bk	$2$	(not in LMFDB)
3.9.b_ag_abk	$2$	(not in LMFDB)
3.9.n_da_lc	$2$	(not in LMFDB)
3.9.ae_p_abt	$3$	(not in LMFDB)