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

• Label: 2.169.abt_bgb
• 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:	$1 - 45 x + 833 x^{2} - 7605 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.0337370003399$, $\pm0.236533266559$
Angle rank:	$2$ (numerical)
Number field:	4.0.58525.1
Galois group:	$D_{4}$
Jacobians:	$15$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21745$	$805543525$	$23290915741105$	$665420067397059525$	$19004961417376725898000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$125$	$28203$	$4825325$	$815734963$	$137858474750$	$23298079780203$	$3937376234206925$	$665416606564969123$	$112455406920191412125$	$19004963774632781217198$

Jacobians and polarizations

This isogeny class contains the Jacobians of 15 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.58525.1.

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_bgb	$2$	(not in LMFDB)