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

• Label: 2.27.ao_dw
• Base field: $\F_{3^{3}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{3}}$
Dimension:	$2$
L-polynomial:	$1 - 14 x + 100 x^{2} - 378 x^{3} + 729 x^{4}$
Frobenius angles:	$\pm0.182412883256$, $\pm0.330788140050$
Angle rank:	$2$ (numerical)
Number field:	4.0.366912.2
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$438$	$535236$	$393809742$	$283351797456$	$205945173103758$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$734$	$20006$	$533174$	$14352674$	$387418814$	$10460373098$	$282430143518$	$7625600615150$	$205891123955054$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

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

The endomorphism algebra of this simple isogeny class is 4.0.366912.2.

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
2.27.o_dw	$2$	2.729.e_bhq