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

• Label: 3.9.an_db_ali
• 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:	$( 1 - 3 x )^{2}( 1 - 5 x + 9 x^{2} )( 1 - 2 x + 9 x^{2} )$
	$1 - 13 x + 79 x^{2} - 294 x^{3} + 711 x^{4} - 1053 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.186429498677$, $\pm0.391826552031$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$160$	$460800$	$388186240$	$278876160000$	$204169342352800$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$71$	$732$	$6479$	$58557$	$530684$	$4784133$	$43045919$	$387356988$	$3486444551$

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 $\times$ 1.9.af $\times$ 1.9.ac 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.ac : \(\Q(\sqrt{-2}) \).

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.aj_bj_ady	$2$	(not in LMFDB)
3.9.ad_ab_g	$2$	(not in LMFDB)
3.9.ab_af_bq	$2$	(not in LMFDB)
3.9.b_af_abq	$2$	(not in LMFDB)
3.9.d_ab_ag	$2$	(not in LMFDB)
3.9.j_bj_dy	$2$	(not in LMFDB)
3.9.n_db_li	$2$	(not in LMFDB)
3.9.ae_q_abq	$3$	(not in LMFDB)