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

• Label: 2.9.ae_k
• 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:	$1 - 4 x + 10 x^{2} - 36 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.135555600465$, $\pm0.578465250809$
Angle rank:	$2$ (numerical)
Number field:	4.0.7488.1
Galois group:	$D_{4}$
Jacobians:	$14$
Isomorphism classes:	18

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$52$	$6864$	$495508$	$42611712$	$3529300372$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$86$	$678$	$6494$	$59766$	$532790$	$4782966$	$43064510$	$387482214$	$3486744086$

Jacobians and polarizations

This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.7488.1.

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.e_k	$2$	2.81.e_aba