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

• Label: 2.169.abw_bjb
• 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:	$1 - 48 x + 911 x^{2} - 8112 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.0457381254353$, $\pm0.172659114562$
Angle rank:	$2$ (numerical)
Number field:	4.0.359568.2
Galois group:	$D_{4}$
Jacobians:	$6$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21313$	$802072129$	$23280022209316$	$665403768037744713$	$19004979781804288431313$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$122$	$28080$	$4823066$	$815714980$	$137858607962$	$23298088431894$	$3937376418006314$	$665416609110550468$	$112455406942569159626$	$19004963774663884052880$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.359568.2.

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_bjb	$2$	(not in LMFDB)