Abelian variety isogeny class 1.27.aj over F33\F_{3^{3}}

• Label: 1.27.aj
• Base field: $\F_{3^{3}}$
• Dimension: $1$
• pp-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{3}}$
Dimension:	$1$
L-polynomial:	$1 - 9 x + 27 x^{2}$
Frobenius angles:	$\pm0.166666666667$
Angle rank:	$0$ (numerical)
Number field:	$\Q(\sqrt{-3})$
Galois group:	$C_2$
Jacobians:	$1$

This isogeny class is simple and geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$19$	$703$	$19684$	$532171$	$14355469$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$19$	$703$	$19684$	$532171$	$14355469$	$387459856$	$10460530351$	$282430067923$	$7625597484988$	$205891117745743$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

Decomposition and endomorphism algebra

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

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

The endomorphism algebra of this simple isogeny class is $\Q(\sqrt{-3})$.

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

The base change of $A$ to $\F_{3^{18}}$ is the simple isogeny class 1.387420489.cggc and its endomorphism algebra is the quaternion algebra over $\Q$ ramified at $3$ and $\infty$.

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{3^{6}}$ is the simple isogeny class 1.729.abb and its endomorphism algebra is $\Q(\sqrt{-3})$.

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

  The base change of $A$ to $\F_{3^{9}}$ is the simple isogeny class 1.19683.a and its endomorphism algebra is $\Q(\sqrt{-3})$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
1.27.j	$2$	1.729.abb
1.27.a	$3$	(not in LMFDB)
1.27.j	$3$	(not in LMFDB)