Abelian variety isogeny class 1.3.a over $\F_{3}$

• Label: 1.3.a
• Base field: $\F_{3}$
• Dimension: $1$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• 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, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

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 (of which all are hyperelliptic), and hence is principally polarizable:

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$16$	$28$	$64$	$244$

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