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

• Label: 2.169.abs_bfe
• 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:	$1 - 44 x + 810 x^{2} - 7436 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.0647393693303$, $\pm0.247372633299$
Angle rank:	$2$ (numerical)
Number field:	4.0.330048.1
Galois group:	$D_{4}$
Jacobians:	$56$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21892$	$806763984$	$23295322932388$	$665431170999309312$	$19004985241716052615972$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$126$	$28246$	$4826238$	$815748574$	$137858647566$	$23298082285750$	$3937376283837486$	$665416607647046590$	$112455406941334247454$	$19004963774981224491286$

Jacobians and polarizations

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

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

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

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 4.0.330048.1.

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.bs_bfe	$2$	(not in LMFDB)