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

• Label: 2.169.abs_bfh
• 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:	$( 1 - 25 x + 169 x^{2} )( 1 - 19 x + 169 x^{2} )$
	$1 - 44 x + 813 x^{2} - 7436 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.0885687144757$, $\pm0.239161554446$
Angle rank:	$2$ (numerical)
Jacobians:	$84$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21895$	$806940225$	$23297237074240$	$665442179281049625$	$19005029097392428768975$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$126$	$28252$	$4826634$	$815762068$	$137858965686$	$23298088032142$	$3937376366611254$	$665416608605924068$	$112455406950179714106$	$19004963775046849071052$

Jacobians and polarizations

This isogeny class contains the Jacobians of 84 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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

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.at 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.at : \(\Q(\sqrt{-35}) \).

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.ag_afh	$2$	(not in LMFDB)
2.169.g_afh	$2$	(not in LMFDB)
2.169.bs_bfh	$2$	(not in LMFDB)