Abelian variety isogeny class 2.9.af_n over $\F_{3^{2}}$

• Label: 2.9.af_n
• Base field: $\F_{3^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 5 x + 13 x^{2} - 45 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.0703393266913$, $\pm0.545465958288$
Angle rank:	$2$ (numerical)
Number field:	4.0.1525.1
Galois group:	$D_{4}$
Jacobians:	$5$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$45$	$6525$	$486405$	$41505525$	$3492738000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$83$	$665$	$6323$	$59150$	$532043$	$4779185$	$43043843$	$387474245$	$3486871598$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):

• $y^2=a x^6+a x^5+2 a x$
• $y^2=a x^6+2 a x^5+(a+1) x^4+(a+2) x^3+2 a x+1$
• $y^2=x^6+(2 a+1) x^5+(2 a+2) x^4+(2 a+1) x^3+(2 a+2) x^2+2 a x+a$
• $y^2=a x^6+(a+1) x^5+2 a x^4+x^3+2 x^2+(a+1) x+2 a+1$
• $y^2=a x^6+2 a x^5+2 x^4+(2 a+1) x^3+2 a x+2 a+2$

where $a$ is a root of the Conway polynomial.

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 4.0.1525.1.

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
2.9.f_n	$2$	2.81.b_aep