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

• Label: 2.9.ae_n
• 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:	$( 1 - 5 x + 9 x^{2} )( 1 + x + 9 x^{2} )$
	$1 - 4 x + 13 x^{2} - 36 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.186429498677$, $\pm0.553300379038$
Angle rank:	$2$ (numerical)
Jacobians:	$14$
Isomorphism classes:	40

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$55$	$7425$	$520960$	$42953625$	$3536439775$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$92$	$714$	$6548$	$59886$	$533582$	$4781454$	$43044068$	$387438906$	$3486664652$

Jacobians and polarizations

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

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

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.af $\times$ 1.9.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.9.af : \(\Q(\sqrt{-11}) \).
• 1.9.b : \(\Q(\sqrt{-35}) \).

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.ag_x	$2$	2.81.k_br
2.9.e_n	$2$	2.81.k_br
2.9.g_x	$2$	2.81.k_br