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

• Label: 2.27.ao_dz
• 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:	$( 1 - 7 x + 27 x^{2} )^{2}$
	$1 - 14 x + 103 x^{2} - 378 x^{3} + 729 x^{4}$
Frobenius angles:	$\pm0.264757707515$, $\pm0.264757707515$
Angle rank:	$1$ (numerical)
Jacobians:	$12$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$441$	$540225$	$396328464$	$283955765625$	$206005480057881$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$740$	$20132$	$534308$	$14356874$	$387398870$	$10459986782$	$282427555268$	$7625593509884$	$205891157761700$

Jacobians and polarizations

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

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

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.ah^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-59}) \)$)$

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
2.27.a_f	$2$	2.729.k_cfb
2.27.o_dz	$2$	2.729.k_cfb
2.27.h_w	$3$	(not in LMFDB)