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

• Label: 2.169.abv_bib
• 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:	$1 - 47 x + 885 x^{2} - 7943 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.0403592669200$, $\pm0.196341187291$
Angle rank:	$2$ (numerical)
Number field:	4.0.1078245.4
Galois group:	$D_{4}$
Jacobians:	$8$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21457$	$803285709$	$23284251363121$	$665412276240794805$	$19004983497851678762752$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$123$	$28123$	$4823943$	$815725411$	$137858634918$	$23298086312611$	$3937376354719287$	$665416607961932131$	$112455406927343082867$	$19004963774524665642478$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.1078245.4.

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.bv_bib	$2$	(not in LMFDB)