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

• Label: 2.169.abu_bhh
• 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:	$1 - 46 x + 865 x^{2} - 7774 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.111746331041$, $\pm0.188213296655$
Angle rank:	$2$ (numerical)
Number field:	4.0.1074752.1
Galois group:	$D_{4}$
Jacobians:	$14$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21607$	$804795929$	$23291875918108$	$665442128117176697$	$19005083179182936672247$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$124$	$28176$	$4825522$	$815762004$	$137859357984$	$23298098917350$	$3937376551030144$	$665416610695718820$	$112455406961026648066$	$19004963774880925783936$

Jacobians and polarizations

This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.1074752.1.

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.bu_bhh	$2$	(not in LMFDB)