Abelian Variety Isogeny Class 1.27.i over $\F_{3^{3}}$

• Label: 1.27.i
• Base Field: $\F_{3^{3}}$
• Dimension: $1$
• Ordinary: No
• $p$-rank: $1$
• Principally polarizable: Yes
• Contains a Jacobian: Yes

Invariants

Base field:	$\F_{3^{3}}$
Dimension:	$1$
Weil polynomial:	$1 + 8 x + 27 x^{2}$
Frobenius angles:	$\pm0.779644248016$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-11}) \)
Galois group:	$C_2$
Jacobians:	4

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$	1	2	3	4	5	6	7	8	9	10
$A(\F_{q^r})$	36	720	19548	532800	14341716	387441360	10460380428	282428755200	7625603000196	205891109067600

Point counts of the curve

$r$	1	2	3	4	5	6	7	8	9	10
$C(\F_{q^r})$	36	720	19548	532800	14341716	387441360	10460380428	282428755200	7625603000196	205891109067600

Decomposition and endomorphism algebra

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

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-11}) \).

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

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.ab

Twists

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

Twist	Extension Degree	Common base change
1.27.ai	$2$	1.729.ak