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

• Label: 2.81.abc_nl
• Base field: $\F_{3^{4}}$
• 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:	$\F_{3^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 17 x + 81 x^{2} )( 1 - 11 x + 81 x^{2} )$
	$1 - 28 x + 349 x^{2} - 2268 x^{3} + 6561 x^{4}$
Frobenius angles:	$\pm0.106600758076$, $\pm0.290722850198$
Angle rank:	$2$ (numerical)
Jacobians:	$56$
Isomorphism classes:	160

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4615$	$42490305$	$282727157440$	$1853383296595545$	$12157767880071565375$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$6476$	$532002$	$43055156$	$3486813774$	$282429249758$	$22876789630014$	$1853020221100196$	$150094636342860162$	$12157665472560576476$

Jacobians and polarizations

This isogeny class contains the Jacobians of 56 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.81.ar $\times$ 1.81.al and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.81.ar : \(\Q(\sqrt{-35}) \).
• 1.81.al : \(\Q(\sqrt{-203}) \).

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.ag_az	$2$	(not in LMFDB)
2.81.g_az	$2$	(not in LMFDB)
2.81.bc_nl	$2$	(not in LMFDB)