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

• Label: 2.9.ag_ba
• Base field: $\F_{3^{2}}$
• 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:	$\F_{3^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 4 x + 9 x^{2} )( 1 - 2 x + 9 x^{2} )$
	$1 - 6 x + 26 x^{2} - 54 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.267720472801$, $\pm0.391826552031$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
Isomorphism classes:	24

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$48$	$8064$	$600624$	$43868160$	$3472262448$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$98$	$820$	$6686$	$58804$	$530306$	$4781956$	$43046846$	$387406180$	$3486732578$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{3^{2}}$

The isogeny class factors as 1.9.ae $\times$ 1.9.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.9.ae : \(\Q(\sqrt{-5}) \).
• 1.9.ac : \(\Q(\sqrt{-2}) \).

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_k	$2$	2.81.q_hi
2.9.c_k	$2$	2.81.q_hi
2.9.g_ba	$2$	2.81.q_hi