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

• Label: 2.169.abs_bfg
• 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:	$1 - 44 x + 812 x^{2} - 7436 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.0810217339920$, $\pm0.242057894147$
Angle rank:	$2$ (numerical)
Number field:	4.0.22022400.1
Galois group:	$D_{4}$
Jacobians:	$24$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21894$	$806881476$	$23296599022086$	$665438513111666064$	$19005014539488235056774$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$126$	$28250$	$4826502$	$815757574$	$137858860086$	$23298086141786$	$3937376340009294$	$665416608316428094$	$112455406947982949118$	$19004963775040863169850$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.22022400.1.

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.bs_bfg	$2$	(not in LMFDB)