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

• Label: 2.167.abv_bia
• Base field: $\F_{167}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{167}$
Dimension:	$2$
L-polynomial:	$( 1 - 25 x + 167 x^{2} )( 1 - 22 x + 167 x^{2} )$
	$1 - 47 x + 884 x^{2} - 7849 x^{3} + 27889 x^{4}$
Frobenius angles:	$\pm0.0816525061160$, $\pm0.175872025744$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
Cyclic group of points:	yes

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20878$	$765596260$	$21679273505032$	$604970520413802400$	$16871988296638429789978$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$121$	$27449$	$4654738$	$777800697$	$129892450391$	$21691970476346$	$3622557703449113$	$604967118129511633$	$101029508540859597886$	$16871927924958247498289$

Jacobians and polarizations

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

• $y^2=86 x^6+13 x^5+68 x^4+130 x^3+18 x^2+64 x+141$
• $y^2=114 x^6+158 x^5+53 x^4+23 x^3+120 x^2+50 x+146$
• $y^2=116 x^6+76 x^5+56 x^4+151 x^3+140 x^2+107 x+139$
• $y^2=139 x^6+55 x^5+164 x^4+118 x^3+x^2+74 x+29$
• $y^2=53 x^6+97 x^5+8 x^4+115 x^3+131 x^2+109 x+91$
• $y^2=59 x^6+9 x^5+42 x^4+107 x^3+20 x^2+71 x+23$
• $y^2=149 x^6+77 x^5+108 x^4+31 x^3+13 x^2+161 x+147$
• $y^2=26 x^6+120 x^5+124 x^4+132 x^3+20 x^2+64 x+46$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{167}$

The isogeny class factors as 1.167.az $\times$ 1.167.aw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.167.az : \(\Q(\sqrt{-43}) \).
• 1.167.aw : \(\Q(\sqrt{-46}) \).

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.ad_aii	$2$	(not in LMFDB)
2.167.d_aii	$2$	(not in LMFDB)
2.167.bv_bia	$2$	(not in LMFDB)