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

• Label: 2.81.abf_pi
• 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:	$1 - 31 x + 398 x^{2} - 2511 x^{3} + 6561 x^{4}$
Frobenius angles:	$\pm0.0704000604239$, $\pm0.231694314237$
Angle rank:	$2$ (numerical)
Number field:	4.0.646204.2
Galois group:	$D_{4}$
Jacobians:	$8$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4418$	$41979836$	$282264888992$	$1853212286251456$	$12157842620564660098$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$51$	$6397$	$531132$	$43051185$	$3486835211$	$282429639442$	$22876788665243$	$1853020128692129$	$150094634865530892$	$12157665459433474477$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.646204.2.

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.bf_pi	$2$	(not in LMFDB)