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

• Label: 2.169.abw_bja
• 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:	$( 1 - 13 x )^{2}( 1 - 22 x + 169 x^{2} )$
	$1 - 48 x + 910 x^{2} - 8112 x^{3} + 28561 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.178912375022$
Angle rank:	$1$ (numerical)
Jacobians:	$20$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21312$	$802013184$	$23279325915456$	$665399220653260800$	$19004958209563460165952$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$122$	$28078$	$4822922$	$815709406$	$137858451482$	$23298084866446$	$3937376348274218$	$665416607901110206$	$112455406923550852538$	$19004963774387689319278$

Jacobians and polarizations

This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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.aw 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.aw : \(\Q(\sqrt{-3}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.169.ae_aja	$2$	(not in LMFDB)
2.169.e_aja	$2$	(not in LMFDB)
2.169.bw_bja	$2$	(not in LMFDB)
2.169.abb_oa	$3$	(not in LMFDB)
2.169.ad_aka	$3$	(not in LMFDB)