Abelian variety isogeny class 4.3.ac_d_aj_s over $\F_{3}$

• Label: 4.3.ac_d_aj_s
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$( 1 - 2 x + 3 x^{2} )( 1 - 9 x^{3} + 27 x^{6} )$
	$1 - 2 x + 3 x^{2} - 9 x^{3} + 18 x^{4} - 27 x^{5} + 27 x^{6} - 54 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.0555555555556$, $\pm0.304086723985$, $\pm0.611111111111$, $\pm0.722222222222$
Angle rank:	$1$ (numerical)
Isomorphism classes:	53

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$38$	$8436$	$260642$	$51088416$	$3474023498$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$12$	$11$	$96$	$242$	$603$	$2102$	$6528$	$19874$	$59532$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{18}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ac $\times$ 3.3.a_a_aj 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.ac : \(\Q(\sqrt{-2}) \).
• 3.3.a_a_aj : \(\Q(\zeta_{9})\).

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

The base change of $A$ to $\F_{3^{18}}$ is 1.387420489.evq $\times$ 1.387420489.cggc^3. The endomorphism algebra for each factor is:

• 1.387420489.evq : \(\Q(\sqrt{-2}) \).
• 1.387420489.cggc^3 : $\mathrm{M}_{3}(B)$, 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.c $\times$ 3.9.a_a_abb. The endomorphism algebra for each factor is:

  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 3.9.a_a_abb : \(\Q(\zeta_{9})\).
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.aj^3 $\times$ 1.27.k. The endomorphism algebra for each factor is:

  • 1.27.aj^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 1.27.k : \(\Q(\sqrt{-2}) \).
• Endomorphism algebra over $\F_{3^{6}}$
  The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.abb^3. The endomorphism algebra for each factor is:

  • 1.729.abu : \(\Q(\sqrt{-2}) \).
  • 1.729.abb^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
• Endomorphism algebra over $\F_{3^{9}}$
  The base change of $A$ to $\F_{3^{9}}$ is 1.19683.a^3 $\times$ 1.19683.hi. The endomorphism algebra for each factor is:

  • 1.19683.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 1.19683.hi : \(\Q(\sqrt{-2}) \).

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
4.3.ac_d_j_as	$2$	(not in LMFDB)
4.3.c_d_aj_as	$2$	(not in LMFDB)
4.3.c_d_j_s	$2$	(not in LMFDB)