Abelian variety isogeny class 2.81.abb_mr over $\F_{3^{4}}$

• Label: 2.81.abb_mr
• Base field: $\F_{3^{4}}$
• 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:	$\F_{3^{4}}$
Dimension:	$2$
L-polynomial:	$1 - 27 x + 329 x^{2} - 2187 x^{3} + 6561 x^{4}$
Frobenius angles:	$\pm0.0820625813014$, $\pm0.321046276823$
Angle rank:	$2$ (numerical)
Number field:	4.0.18177085.2
Galois group:	$D_{4}$
Jacobians:	$36$
Isomorphism classes:	36

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4677$	$42584085$	$282644322525$	$1853084316026565$	$12157417305869843712$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$55$	$6491$	$531847$	$43048211$	$3486713230$	$282428476523$	$22876787850895$	$1853020247846051$	$150094636553409967$	$12157665470456223326$

Jacobians and polarizations

This isogeny class contains the Jacobians of 36 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{3^{4}}$

The endomorphism algebra of this simple isogeny class is 4.0.18177085.2.

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
2.81.bb_mr	$2$	(not in LMFDB)