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

• Label: 2.169.abt_bgg
• Base field: $\F_{13^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{13^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 25 x + 169 x^{2} )( 1 - 20 x + 169 x^{2} )$
	$1 - 45 x + 838 x^{2} - 7605 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.0885687144757$, $\pm0.220639651288$
Angle rank:	$2$ (numerical)
Jacobians:	$32$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21750$	$805837500$	$23294178747000$	$665439475217400000$	$19005042530173093293750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$125$	$28213$	$4826000$	$815758753$	$137859063125$	$23298091147618$	$3937376414412125$	$665416608970146433$	$112455406947814508000$	$19004963774919168805573$

Jacobians and polarizations

This isogeny class contains the Jacobians of 32 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.169.az $\times$ 1.169.au and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.169.az : \(\Q(\sqrt{-51}) \).
• 1.169.au : \(\Q(\sqrt{-69}) \).

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.af_agg	$2$	(not in LMFDB)
2.169.f_agg	$2$	(not in LMFDB)
2.169.bt_bgg	$2$	(not in LMFDB)