Abelian variety isogeny class 2.167.abu_bgz over $\F_{167}$

• Label: 2.167.abu_bgz
• Base field: $\F_{167}$
• 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:	$\F_{167}$
Dimension:	$2$
L-polynomial:	$1 - 46 x + 857 x^{2} - 7682 x^{3} + 27889 x^{4}$
Frobenius angles:	$\pm0.0558069158019$, $\pm0.207406191075$
Angle rank:	$2$ (numerical)
Number field:	4.0.2875968.2
Galois group:	$D_{4}$
Jacobians:	$8$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21019$	$766668025$	$21682109682052$	$604971287016069625$	$16871961020557309209379$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$122$	$27488$	$4655348$	$777801684$	$129892240402$	$21691963677926$	$3622557564190750$	$604967115954496036$	$101029508514287983916$	$16871927924723441268368$

Jacobians and polarizations

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

• $y^2=86 x^6+9 x^5+9 x^4+79 x^3+46 x^2+107 x+33$
• $y^2=16 x^6+89 x^5+141 x^4+78 x^3+158 x^2+162 x+43$
• $y^2=8 x^6+131 x^5+4 x^4+131 x^3+121 x^2+118 x+163$
• $y^2=67 x^6+7 x^5+163 x^4+33 x^3+151 x^2+92 x+153$
• $y^2=51 x^6+45 x^5+61 x^4+14 x^3+139 x^2+69 x+101$
• $y^2=81 x^6+3 x^5+59 x^4+120 x^3+150 x^2+109 x+76$
• $y^2=145 x^6+163 x^5+63 x^4+89 x^3+31 x^2+83 x+3$
• $y^2=35 x^6+148 x^5+41 x^4+123 x^3+155 x^2+117 x+151$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{167}$.

Endomorphism algebra over $\F_{167}$

The endomorphism algebra of this simple isogeny class is 4.0.2875968.2.

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
2.167.bu_bgz	$2$	(not in LMFDB)