Abelian variety isogeny class 2.121.abk_vk over $\F_{11^{2}}$

• Label: 2.121.abk_vk
• Base field: $\F_{11^{2}}$
• 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:	$\F_{11^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 36 x + 556 x^{2} - 4356 x^{3} + 14641 x^{4}$
Frobenius angles:	$\pm0.0881234832712$, $\pm0.264384292094$
Angle rank:	$2$ (numerical)
Number field:	4.0.15264000.3
Galois group:	$D_{4}$
Jacobians:	$32$
Isomorphism classes:	64

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10806$	$211689540$	$3139002100566$	$45953098779453840$	$672752879319587625366$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$86$	$14458$	$1771886$	$214374598$	$25937535806$	$3138427747258$	$379749812808326$	$45949729719963838$	$5559917315503102166$	$672749995003518030298$

Jacobians and polarizations

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

• $y^2=(8a+1)x^6+(7a+6)x^5+(8a+6)x^4+(a+5)x^3+(a+8)x^2+(2a+5)x+10a$
• $y^2=(2a+4)x^6+2ax^5+5ax^4+(9a+9)x^3+(4a+7)x^2+(6a+9)x+5a+4$
• $y^2=(5a+7)x^6+(3a+6)x^5+(a+3)x^4+6x^3+(8a+5)x^2+(9a+1)x+7a+5$
• $y^2=(9a+9)x^6+(9a+9)x^5+(6a+10)x^4+(6a+5)x^3+(4a+1)x^2+(7a+10)x+10a$
• $y^2=(7a+9)x^6+(9a+8)x^5+(6a+9)x^4+4x^3+(8a+9)x^2+(10a+10)x+4a+1$
• $y^2=(6a+2)x^6+ax^5+(10a+9)x^4+(5a+5)x^3+(5a+3)x^2+(8a+2)x+4$
• $y^2=(9a+4)x^6+(9a+3)x^5+2x^4+(7a+4)x^3+10ax^2+(3a+7)x+10$
• $y^2=(2a+1)x^6+(6a+9)x^5+(3a+8)x^4+(10a+6)x^3+(2a+6)x^2+(7a+10)x+5a+2$
• $y^2=(4a+4)x^6+(6a+1)x^5+(4a+6)x^4+(8a+3)x^3+(8a+2)x^2+5x+2a+5$
• $y^2=(8a+1)x^6+(8a+5)x^5+(7a+5)x^4+(8a+2)x^3+(8a+9)x^2+7ax+9a+2$
• $y^2=(3a+10)x^6+(5a+7)x^5+(9a+6)x^4+ax^3+(3a+8)x^2+2ax+10a+3$
• $y^2=(a+8)x^6+(5a+4)x^5+(a+5)x^4+(6a+10)x^3+(2a+9)x^2+(7a+5)x+5a+8$
• $y^2=(a+8)x^6+(9a+2)x^5+(5a+10)x^4+(10a+10)x^3+(5a+10)x^2+(3a+7)x+2a+9$
• $y^2=(4a+8)x^6+(8a+2)x^5+(7a+7)x^4+(7a+10)x^3+(8a+7)x^2+(10a+3)x+4a+5$
• $y^2=6ax^6+4ax^5+5ax^4+(4a+7)x^3+(a+3)x^2+(6a+3)x+3a+10$
• $y^2=(4a+4)x^6+3x^5+(8a+6)x^4+(5a+7)x^3+6ax^2+(8a+8)x+3a+4$
• $y^2=(7a+9)x^6+(6a+3)x^5+(2a+1)x^4+(10a+5)x^3+(7a+1)x^2+(2a+3)x+6a+4$
• $y^2=(2a+2)x^6+(4a+9)x^5+(5a+2)x^4+(4a+9)x^3+9ax^2+(2a+10)x+10a+10$
• $y^2=(5a+10)x^6+(2a+4)x^5+(6a+7)x^4+ax^3+(8a+2)x^2+(4a+10)x+8a+5$
• $y^2=(8a+5)x^6+(4a+9)x^5+(5a+2)x^4+(9a+1)x^3+(2a+5)x^2+9x+3a$
• and

• $y^2=(3a+6)x^6+(6a+2)x^5+(8a+3)x^4+(6a+5)x^3+(10a+5)x^2+(8a+4)x+10a+4$
• $y^2=(2a+5)x^6+(5a+8)x^5+(2a+10)x^4+(9a+8)x^3+(5a+2)x^2+(a+10)x+4a+6$
• $y^2=(4a+8)x^6+(5a+3)x^5+(a+3)x^3+(10a+3)x^2+(6a+3)x+4$
• $y^2=(6a+7)x^6+(a+2)x^5+(2a+3)x^4+(7a+8)x^3+9x^2+(3a+8)x+8a+7$
• $y^2=(6a+2)x^6+(7a+6)x^5+(6a+2)x^4+(5a+7)x^3+(8a+5)x^2+3a+2$
• $y^2=(4a+2)x^6+(7a+4)x^5+(3a+2)x^4+(a+1)x^3+9ax^2+(3a+7)x+9a+2$
• $y^2=x^6+6ax^5+(9a+5)x^4+(5a+8)x^3+(2a+5)x^2+(6a+4)x+7a+10$
• $y^2=(a+8)x^6+(10a+1)x^5+(3a+10)x^4+(6a+7)x^3+(8a+3)x^2+(7a+1)x+9a+9$
• $y^2=2x^6+(8a+9)x^5+(a+6)x^4+(a+8)x^3+(6a+8)x^2+(9a+9)x+10a+7$
• $y^2=(9a+1)x^6+(3a+1)x^5+(a+3)x^4+(10a+6)x^3+(3a+2)x^2+2ax+9a+2$
• $y^2=9x^6+(9a+6)x^5+(a+8)x^4+(4a+7)x^3+(6a+4)x^2+x+8a+3$
• $y^2=(4a+6)x^6+(3a+4)x^5+(a+4)x^4+8ax^3+(7a+1)x^2+2x+9a+10$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{2}}$.

Endomorphism algebra over $\F_{11^{2}}$

The endomorphism algebra of this simple isogeny class is 4.0.15264000.3.

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
2.121.bk_vk	$2$	(not in LMFDB)