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

• Label: 2.169.abs_bfa
• Base field: $\F_{13^{2}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 - 13 x )^{2}( 1 - 18 x + 169 x^{2} )$
	$1 - 44 x + 806 x^{2} - 7436 x^{3} + 28561 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.256594102998$
Angle rank:	$1$ (numerical)
Jacobians:	$12$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21888$	$806529024$	$23292770811264$	$665416447679692800$	$19004925918315703563648$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$126$	$28238$	$4825710$	$815730526$	$137858217246$	$23298074272046$	$3937376159570574$	$665416605942610366$	$112455406918791204030$	$19004963774661792286478$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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.aba $\times$ 1.169.as 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.aba : the quaternion algebra over \(\Q\) ramified at $13$ and $\infty$.
• 1.169.as : \(\Q(\sqrt{-22}) \).

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.ai_afa	$2$	(not in LMFDB)
2.169.i_afa	$2$	(not in LMFDB)
2.169.bs_bfa	$2$	(not in LMFDB)