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

• Label: 3.9.an_de_alx
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x + 9 x^{2} )( 1 - 5 x + 9 x^{2} )^{2}$
	$1 - 13 x + 82 x^{2} - 309 x^{3} + 738 x^{4} - 1053 x^{5} + 729 x^{6}$
Frobenius angles:	$\pm0.186429498677$, $\pm0.186429498677$, $\pm0.333333333333$
Angle rank:	$1$ (numerical)

This isogeny class is not simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$175$	$511875$	$429318400$	$295983016875$	$208366469329375$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$77$	$804$	$6869$	$59757$	$532700$	$4785813$	$43053989$	$387418116$	$3486628397$

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 $\F_{3^{6}}$.

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

The isogeny class factors as 1.9.af^2 $\times$ 1.9.ad 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.af^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
• 1.9.ad : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{3^{2}}$

The base change of $A$ to $\F_{3^{6}}$ is 1.729.k^2 $\times$ 1.729.cc. The endomorphism algebra for each factor is:

• 1.729.k^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
• 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.

Subfield	Primitive Model
$\F_{3}$	3.3.ad_ac_p
$\F_{3}$	3.3.d_ac_ap

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.9.ah_w_abz	$2$	(not in LMFDB)
3.9.ad_c_v	$2$	(not in LMFDB)
3.9.d_c_av	$2$	(not in LMFDB)
3.9.h_w_bz	$2$	(not in LMFDB)
3.9.n_de_lx	$2$	(not in LMFDB)
3.9.ae_ai_da	$3$	(not in LMFDB)
3.9.c_k_bq	$3$	(not in LMFDB)
3.9.l_cd_he	$3$	(not in LMFDB)