Abelian variety isogeny class 2.3.d_h over $\F_{3}$

• Label: 2.3.d_h
• Base field: $\F_{3}$
• 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:	$\F_{3}$
Dimension:	$2$
L-polynomial:	$1 + 3 x + 7 x^{2} + 9 x^{3} + 9 x^{4}$
Frobenius angles:	$\pm0.535169663346$, $\pm0.772732979144$
Angle rank:	$2$ (numerical)
Number field:	4.0.1525.1
Galois group:	$D_{4}$
Jacobians:	$1$
Isomorphism classes:	1

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$29$	$145$	$551$	$6525$	$56144$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$15$	$19$	$83$	$232$	$795$	$2149$	$6323$	$20197$	$58950$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=x^6+x^5+2 x^4+x^2+x+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3}$.

Endomorphism algebra over $\F_{3}$

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
2.3.ad_h	$2$	2.9.f_n