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

• Label: 1.27.k
• 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:	$1 + 10 x + 27 x^{2}$
Frobenius angles:	$\pm0.912260171954$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}) \)
Galois group:	$C_2$
Jacobians:	1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$	1	2	3	4	5	6	7	8	9	10
$A(\F_{q^r})$	38	684	19874	530784	14350358	387423756	10460281394	282430166400	7625593124678	205891158689964

Point counts of the curve

$r$	1	2	3	4	5	6	7	8	9	10
$C(\F_{q^r})$	38	684	19874	530784	14350358	387423756	10460281394	282430166400	7625593124678	205891158689964

Decomposition and endomorphism algebra

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

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

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.ac

Twists

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

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