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

• Label: 2.169.abw_bjc
• 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:	$1 - 48 x + 912 x^{2} - 8112 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.0676966056373$, $\pm0.164966217013$
Angle rank:	$2$ (numerical)
Number field:	4.0.319744.3
Galois group:	$D_{4}$
Jacobians:	$12$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21314$	$802131076$	$23280718508642$	$665408312165921296$	$19005001287878055651074$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$122$	$28082$	$4823210$	$815720550$	$137858763962$	$23298091966802$	$3937376486364842$	$665416610271001534$	$112455406960115183930$	$19004963774901398348402$

Jacobians and polarizations

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

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

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 4.0.319744.3.

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.bw_bjc	$2$	(not in LMFDB)