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

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

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$1$
L-polynomial:	$1 + 5 x + 9 x^{2}$
Frobenius angles:	$\pm0.813570501323$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-11}) \)
Galois group:	$C_2$
Jacobians:	$1$
Isomorphism classes:	1
Cyclic group of points:	yes

This isogeny class is simple and geometrically simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$15$	$75$	$720$	$6675$	$58575$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$15$	$75$	$720$	$6675$	$58575$	$532800$	$4780455$	$43047075$	$387441360$	$3486676875$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):

• $y^2+x y=x^3+2$

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 \(\Q(\sqrt{-11}) \).

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.ab
$\F_{3}$	1.3.b

Twists

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

Twist	Extension degree	Common base change
1.9.af	$2$	1.81.ah