Abelian variety isogeny class 2.169.abt_bgl over $\F_{13^{2}}$

• Label: 2.169.abt_bgl
• Base field: $\F_{13^{2}}$
• 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:	$\F_{13^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 45 x + 843 x^{2} - 7605 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.137315320176$, $\pm0.192643952110$
Angle rank:	$2$ (numerical)
Number field:	4.0.646525.1
Galois group:	$D_{4}$
Jacobians:	$22$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21755$	$806131525$	$23297441878595$	$665458801601185525$	$19005122092151551862000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$125$	$28223$	$4826675$	$815782443$	$137859640250$	$23298101855183$	$3937376567763575$	$665416610526784243$	$112455406953392137625$	$19004963774719400645198$

Jacobians and polarizations

This isogeny class contains the Jacobians of 22 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=(6a+10)x^6+3ax^5+(6a+4)x^4+(8a+11)x^3+(8a+5)x^2+(2a+3)x+9a+12$
• $y^2=x^6+(2a+8)x^5+x^4+(9a+7)x^3+(7a+10)x^2+(10a+11)x+4a+1$
• $y^2=(3a+2)x^6+3ax^5+(3a+12)x^4+(5a+4)x^3+(12a+11)x^2+(12a+4)x+a+2$
• $y^2=12ax^6+(8a+7)x^5+(3a+11)x^4+(10a+10)x^3+(11a+2)x^2+(a+3)x+11a+2$
• $y^2=(2a+4)x^6+(7a+2)x^5+3ax^4+(4a+3)x^3+(11a+3)x^2+(11a+7)x+9a+1$
• $y^2=(4a+10)x^6+(a+8)x^5+(9a+11)x^4+(10a+1)x^3+(a+8)x^2+(10a+9)x+10a+8$
• $y^2=(8a+7)x^6+(6a+8)x^5+6x^4+(10a+10)x^3+(10a+11)x^2+(2a+3)x+9a+12$
• $y^2=(7a+5)x^6+(8a+9)x^5+5ax^4+(3a+2)x^3+(6a+6)x^2+(10a+11)x+12$
• $y^2=(4a+9)x^6+(10a+12)x^5+9ax^4+7ax^3+(12a+3)x^2+(a+4)x+10a+10$
• $y^2=(9a+4)x^6+(5a+6)x^5+(4a+9)x^4+(11a+11)x^3+(5a+11)x^2+6x+10a+6$
• $y^2=(12a+7)x^6+(a+8)x^5+(10a+2)x^4+(10a+11)x^3+(7a+10)x^2+(5a+7)x+12a+7$
• $y^2=(9a+1)x^6+(5a+3)x^5+(12a+12)x^4+(5a+12)x^3+(6a+10)x^2+(7a+9)x+9$
• $y^2=(a+3)x^6+(10a+11)x^5+(5a+4)x^4+(2a+3)x^3+(11a+8)x^2+(a+4)x+10a+8$
• $y^2=(8a+7)x^6+(4a+10)x^5+(3a+5)x^4+(9a+4)x^3+(6a+7)x^2+(11a+12)x+5a+8$
• $y^2=(12a+8)x^6+(6a+6)x^5+(4a+4)x^4+8x^3+(10a+1)x^2+(7a+10)x+9a+11$
• $y^2=(3a+6)x^6+(10a+9)x^5+(4a+7)x^4+(7a+5)x^3+(9a+4)x^2+(8a+4)x+9a+5$
• $y^2=(6a+9)x^6+(a+8)x^5+(6a+1)x^4+(7a+12)x^3+(8a+12)x^2+(4a+9)x+9a+7$
• $y^2=(3a+10)x^6+(a+2)x^5+(9a+8)x^4+(a+6)x^3+(3a+3)x^2+(7a+5)x+11a+7$
• $y^2=(a+5)x^6+(7a+6)x^5+(8a+5)x^4+(5a+11)x^3+(3a+11)x^2+(2a+5)x+4a+10$
• $y^2=(3a+8)x^6+(5a+11)x^5+(9a+2)x^4+(5a+9)x^3+5x^2+(11a+6)x+2a+9$
• $y^2=(6a+9)x^6+(3a+6)x^5+(a+11)x^4+(2a+3)x^3+(4a+12)x^2+(4a+2)x+3a+1$
• $y^2=(10a+5)x^6+(11a+7)x^5+(9a+7)x^4+(9a+3)x^3+(6a+5)x^2+(5a+4)x+6a+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{2}}$.

Endomorphism algebra over $\F_{13^{2}}$

The endomorphism algebra of this simple isogeny class is 4.0.646525.1.

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
2.169.bt_bgl	$2$	(not in LMFDB)