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

• Label: 2.9.ah_ba
• 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:	$1 - 7 x + 26 x^{2} - 63 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.122441590128$, $\pm0.422937410221$
Angle rank:	$2$ (numerical)
Number field:	4.0.49708.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$38$	$6764$	$542336$	$42423808$	$3469878838$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$85$	$744$	$6465$	$58763$	$532594$	$4791251$	$43064513$	$387425544$	$3486785605$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.49708.1.

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.h_ba	$2$	2.81.d_abs