Abelian variety isogeny class 1.27.ak over $\F_{3^{3}}$

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

Invariants

Base field:	$\F_{3^{3}}$
Dimension:	$1$
L-polynomial:	$1 - 10 x + 27 x^{2}$
Frobenius angles:	$\pm0.0877398280459$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}) \)
Galois group:	$C_2$
Jacobians:	$1$
Isomorphism classes:	1

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18$	$684$	$19494$	$530784$	$14347458$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$684$	$19494$	$530784$	$14347458$	$387423756$	$10460425014$	$282430166400$	$7625601845298$	$205891158689964$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

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

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

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{3}}$.

Subfield	Primitive Model
$\F_{3}$	1.3.c

Twists

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

Twist	Extension degree	Common base change
1.27.k	$2$	1.729.abu