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

• Label: 3.9.ap_dv_aoo
• Base field: $\F_{3^{2}}$
• Dimension: $3$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x )^{4}( 1 - 3 x + 9 x^{2} )$
	$1 - 15 x + 99 x^{2} - 378 x^{3} + 891 x^{4} - 1215 x^{5} + 729 x^{6}$
Frobenius angles:	$0$, $0$, $0$, $0$, $\pm0.333333333333$
Angle rank:	$0$ (numerical)

This isogeny class is not simple, not primitive, not ordinary, and supersingular. It is principally polarizable.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$112$	$372736$	$358269184$	$272097280000$	$201692843439472$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$55$	$676$	$6319$	$57835$	$527068$	$4772035$	$43027039$	$387381124$	$3486607255$

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 $\F_{3^{12}}$.

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

The isogeny class factors as 1.9.ag^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.ag^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.9.ad : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec^3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

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^2 $\times$ 1.81.j. The endomorphism algebra for each factor is:

  • 1.81.as^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 1.81.j : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{3^{6}}$
  The base change of $A$ to $\F_{3^{6}}$ is 1.729.acc^2 $\times$ 1.729.cc. The endomorphism algebra for each factor is:

  • 1.729.acc^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 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_ad_s
$\F_{3}$	3.3.d_ad_as

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.9.aj_bb_acc	$2$	(not in LMFDB)
3.9.ad_aj_cc	$2$	(not in LMFDB)
3.9.d_aj_acc	$2$	(not in LMFDB)
3.9.j_bb_cc	$2$	(not in LMFDB)
3.9.p_dv_oo	$2$	(not in LMFDB)
3.9.ag_aj_ee	$3$	(not in LMFDB)
3.9.ag_s_acc	$3$	(not in LMFDB)
3.9.d_aj_acc	$3$	(not in LMFDB)
3.9.d_s_bb	$3$	(not in LMFDB)
3.9.m_cu_kk	$3$	(not in LMFDB)