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

• Label: 1.27.ai
• 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:	$1 - 8 x + 27 x^{2}$
Frobenius angles:	$\pm0.220355751984$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-11}) \)
Galois group:	$C_2$
Jacobians:	$4$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20$	$720$	$19820$	$532800$	$14356100$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$720$	$19820$	$532800$	$14356100$	$387441360$	$10460325980$	$282428755200$	$7625591969780$	$205891109067600$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are 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 \(\Q(\sqrt{-11}) \).

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.b

Twists

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

Twist	Extension degree	Common base change
1.27.i	$2$	1.729.ak