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

• Label: 2.9.c_ac
• 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:	$1 + 2 x - 2 x^{2} + 18 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.296321154275$, $\pm0.880565730135$
Angle rank:	$2$ (numerical)
Number field:	4.0.49392.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	16

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$100$	$6000$	$588100$	$43872000$	$3479930500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$74$	$804$	$6686$	$58932$	$531242$	$4774908$	$43054526$	$387401916$	$3486998474$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.49392.1.

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.ac_ac	$2$	2.81.ai_dq