Abelian variety isogeny class 2.27.am_cw over $\F_{3^{3}}$

• Label: 2.27.am_cw
• Base field: $\F_{3^{3}}$
• 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:	$\F_{3^{3}}$
Dimension:	$2$
L-polynomial:	$( 1 - 10 x + 27 x^{2} )( 1 - 2 x + 27 x^{2} )$
	$1 - 12 x + 74 x^{2} - 324 x^{3} + 729 x^{4}$
Frobenius angles:	$\pm0.0877398280459$, $\pm0.438356648427$
Angle rank:	$2$ (numerical)
Jacobians:	$48$
Isomorphism classes:	204

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$468$	$533520$	$386721972$	$281527833600$	$205780798043028$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$734$	$19648$	$529742$	$14341216$	$387439406$	$10460624848$	$282430143518$	$7625596404016$	$205891148425214$

Jacobians and polarizations

This isogeny class contains the Jacobians of 48 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.27.ak $\times$ 1.27.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.27.ak : \(\Q(\sqrt{-2}) \).
• 1.27.ac : \(\Q(\sqrt{-26}) \).

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
2.27.ai_bi	$2$	2.729.e_abgk
2.27.i_bi	$2$	2.729.e_abgk
2.27.m_cw	$2$	2.729.e_abgk