Abelian variety isogeny class 1.9.g over $\F_{3^{2}}$

• Label: 1.9.g
• Base field: $\F_{3^{2}}$
• Dimension: $1$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$1$
L-polynomial:	$( 1 + 3 x )^{2}$
	$1 + 6 x + 9 x^{2}$
Frobenius angles:	$1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q\)
Galois group:	Trivial
Jacobians:	$1$

This isogeny class is simple and geometrically simple, not 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

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$64$	$784$	$6400$	$59536$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$64$	$784$	$6400$	$59536$	$529984$	$4787344$	$43033600$	$387459856$	$3486666304$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{3^{2}}$

The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.

Subfield	Primitive Model
$\F_{3}$	1.3.a

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
1.9.ag	$2$	1.81.as
1.9.ad	$3$	1.729.cc
1.9.a	$4$	(not in LMFDB)