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

• Label: 2.121.abk_vg
• 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:	$1 - 36 x + 552 x^{2} - 4356 x^{3} + 14641 x^{4}$
Frobenius angles:	$\pm0.0488289870000$, $\pm0.275561164275$
Angle rank:	$2$ (numerical)
Number field:	4.0.9947392.2
Galois group:	$D_{4}$
Jacobians:	$32$
Isomorphism classes:	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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10802$	$211567972$	$3138235160258$	$45950553622666000$	$672747108743598184802$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$86$	$14450$	$1771454$	$214362726$	$25937313326$	$3138424589522$	$379749776994086$	$45949729375898686$	$5559917312305349174$	$672749994967974036530$

Jacobians and polarizations

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

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

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

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 4.0.9947392.2.

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
2.121.bk_vg	$2$	(not in LMFDB)