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

• Label: 2.9.af_s
• Base field: $\F_{3^{2}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 - 5 x + 9 x^{2} )( 1 + 9 x^{2} )$
	$1 - 5 x + 18 x^{2} - 45 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.186429498677$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$2$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$50$	$7500$	$540200$	$42720000$	$3514951250$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$93$	$740$	$6513$	$59525$	$534258$	$4785485$	$43033953$	$387399620$	$3486794973$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{4}}$.

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

The isogeny class factors as 1.9.af $\times$ 1.9.a 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.a : \(\Q(\sqrt{-1}) \).

Endomorphism algebra over $\overline{\F}_{3^{2}}$

The base change of $A$ to $\F_{3^{4}}$ is 1.81.ah $\times$ 1.81.s. The endomorphism algebra for each factor is:

• 1.81.ah : \(\Q(\sqrt{-11}) \).
• 1.81.s : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

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
2.9.f_s	$2$	2.81.l_bk
2.9.al_bw	$4$	(not in LMFDB)
2.9.ab_am	$4$	(not in LMFDB)