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

• Label: 2.3.a_f
• Base field: $\F_{3}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$2$
L-polynomial:	$( 1 - x + 3 x^{2} )( 1 + x + 3 x^{2} )$
	$1 + 5 x^{2} + 9 x^{4}$
Frobenius angles:	$\pm0.406785250661$, $\pm0.593214749339$
Angle rank:	$1$ (numerical)
Jacobians:	$2$
Isomorphism classes:	4

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$15$	$225$	$720$	$5625$	$58575$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$20$	$28$	$68$	$244$	$710$	$2188$	$6788$	$19684$	$58100$

Jacobians and polarizations

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

• $y^2=2x^6+x^5+x^3+x+2$
• $y^2=x^6+2x^5+2x^3+2x+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ab $\times$ 1.3.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.3.ab : \(\Q(\sqrt{-11}) \).
• 1.3.b : \(\Q(\sqrt{-11}) \).

Endomorphism algebra over $\overline{\F}_{3}$

The base change of $A$ to $\F_{3^{2}}$ is 1.9.f^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.3.ac_h	$2$	2.9.k_br
2.3.c_h	$2$	2.9.k_br
2.3.a_af	$4$	2.81.ao_id