Abelian Variety Isogeny Class 1.3.a over $\F_{3}$

• Label: 1.3.a
• Base Field: $\F_{3}$
• Dimension: $1$
• Ordinary: No
• $p$-rank: $0$
• Principally polarizable: Yes
• Contains a Jacobian: Yes

Invariants

Base field:	$\F_{3}$
Dimension:	$1$
L-polynomial:	$1 + 3 x^{2}$
Frobenius angles:	$\pm0.5$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
Jacobians:	2

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

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

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$	1	2	3	4	5	6	7	8	9	10
$A(\F_{q^r})$	4	16	28	64	244	784	2188	6400	19684	59536

Point counts of the curve

$r$	1	2	3	4	5	6	7	8	9	10
$C(\F_{q^r})$	4	16	28	64	244	784	2188	6400	19684	59536

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$

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

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

The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 1.9.g and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

All geometric endomorphisms are defined over $\F_{3^{2}}$.

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
1.3.ad	$3$	1.27.a
1.3.d	$3$	1.27.a
1.3.d	$6$	1.729.cc