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

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

Invariants

Base field:	$\F_{3}$
Dimension:	$3$
L-polynomial:	$1 - 9 x^{3} + 27 x^{6}$
Frobenius angles:	$\pm0.0555555555556$, $\pm0.611111111111$, $\pm0.722222222222$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\zeta_{9})\)
Galois group:	$C_6$
Jacobians:	$0$
Isomorphism classes:	5

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$19$	$703$	$6859$	$532171$	$14355469$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$10$	$1$	$82$	$244$	$649$	$2188$	$6562$	$19684$	$59050$

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 $\F_{3^{18}}$.

Endomorphism algebra over $\F_{3}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{9})\).

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

The base change of $A$ to $\F_{3^{18}}$ is 1.387420489.cggc^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 the simple isogeny class 3.9.a_a_abb and its endomorphism algebra is \(\Q(\zeta_{9})\).

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

  The base change of $A$ to $\F_{3^{3}}$ is 1.27.aj^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{3^{6}}$

  The base change of $A$ to $\F_{3^{6}}$ is 1.729.abb^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{3^{9}}$

  The base change of $A$ to $\F_{3^{9}}$ is 1.19683.a^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$

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.a_a_j	$2$	3.9.a_a_abb
3.3.aj_bk_add	$9$	(not in LMFDB)
3.3.ag_s_abk	$9$	(not in LMFDB)