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

• Label: 2.81.abc_np
• 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:	$1 - 28 x + 353 x^{2} - 2268 x^{3} + 6561 x^{4}$
Frobenius angles:	$\pm0.142096013077$, $\pm0.273278171236$
Angle rank:	$2$ (numerical)
Number field:	4.0.280225.1
Galois group:	$D_{4}$
Jacobians:	$32$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4619$	$42545609$	$282906359600$	$1853681593837129$	$12158086157032608859$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$6484$	$532338$	$43062084$	$3486905054$	$282430035838$	$22876792809134$	$1853020195344644$	$150094635694716498$	$12157665465428043604$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

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

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

where $a$ is a root of the Conway polynomial.

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 4.0.280225.1.

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.bc_np	$2$	(not in LMFDB)