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

• Label: 2.169.abu_bhe
• 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:	$1 - 46 x + 862 x^{2} - 7774 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.0773526067278$, $\pm0.205567582789$
Angle rank:	$2$ (numerical)
Number field:	4.0.239600.1
Galois group:	$D_{4}$
Jacobians:	$36$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21604$	$804619376$	$23289874383556$	$665429866510601984$	$19005030100615640363524$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$124$	$28170$	$4825108$	$815746974$	$137858972964$	$23298091209066$	$3937376425915756$	$665416609062508734$	$112455406945315935532$	$19004963774818114765450$

Jacobians and polarizations

This isogeny class contains the Jacobians of 36 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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

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 4.0.239600.1.

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_bhe	$2$	(not in LMFDB)