Abelian variety isogeny class 3.3.ad_ad_s over $\F_{3}$

• Label: 3.3.ad_ad_s
• Base field: $\F_{3}$
• Dimension: $3$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x + 3 x^{2} )( 1 - 3 x^{2} )^{2}$
	$1 - 3 x - 3 x^{2} + 18 x^{3} - 9 x^{4} - 27 x^{5} + 27 x^{6}$
Frobenius angles:	$0$, $0$, $\pm0.166666666667$, $1$, $1$
Angle rank:	$0$ (numerical)
Jacobians:	$0$

This isogeny class is not simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$112$	$18928$	$372736$	$15870844$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$-5$	$28$	$55$	$271$	$676$	$2269$	$6319$	$19684$	$57835$

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 $\F_{3}$

The isogeny class factors as 1.3.ad $\times$ 2.3.a_ag and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.3.ad : \(\Q(\sqrt{-3}) \).
• 2.3.a_ag : the quaternion algebra over \(\Q(\sqrt{3}) \) ramified at both real infinite places.

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

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 $\F_{3^{2}}$
  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ag^2 $\times$ 1.9.ad. 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 $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 2.27.a_acc. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 2.27.a_acc : the quaternion algebra over \(\Q(\sqrt{3}) \) ramified at both real infinite places.
• 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 is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.3.d_ad_as	$2$	3.9.ap_dv_aoo
3.3.ad_g_aj	$3$	(not in LMFDB)
3.3.a_ad_a	$3$	(not in LMFDB)
3.3.a_g_a	$3$	(not in LMFDB)
3.3.d_ad_as	$3$	(not in LMFDB)
3.3.d_g_j	$3$	(not in LMFDB)