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

• Label: 2.167.abw_biz
• 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:	$( 1 - 25 x + 167 x^{2} )( 1 - 23 x + 167 x^{2} )$
	$1 - 48 x + 909 x^{2} - 8016 x^{3} + 27889 x^{4}$
Frobenius angles:	$\pm0.0816525061160$, $\pm0.150776270497$
Angle rank:	$2$ (numerical)
Jacobians:	$9$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20735$	$764354305$	$21674535362480$	$604958445574622425$	$16871968320451579081175$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$120$	$27404$	$4653720$	$777785172$	$129892296600$	$21691970201486$	$3622557731985480$	$604967118921767908$	$101029508554764086280$	$16871927925140570573564$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):

• $y^2=23 x^6+165 x^5+41 x^4+29 x^3+41 x^2+165 x+23$
• $y^2=13 x^6+91 x^5+77 x^4+17 x^3+157 x^2+164 x+159$
• $y^2=125 x^6+127 x^5+91 x^4+141 x^3+148 x^2+81 x+15$
• $y^2=133 x^6+126 x^5+13 x^4+155 x^2+54 x+63$
• $y^2=164 x^6+137 x^5+42 x^4+36 x^3+76 x^2+54 x+5$
• $y^2=90 x^6+93 x^5+16 x^4+12 x^3+127 x^2+122 x+55$
• $y^2=32 x^6+34 x^5+7 x^4+49 x^3+162 x^2+123 x+38$
• $y^2=75 x^6+60 x^5+135 x^4+25 x^3+135 x^2+60 x+75$
• $y^2=25 x^6+150 x^5+38 x^4+154 x^3+97 x^2+107 x+36$

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.ax 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.ax : \(\Q(\sqrt{-139}) \).

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.ac_ajh	$2$	(not in LMFDB)
2.167.c_ajh	$2$	(not in LMFDB)
2.167.bw_biz	$2$	(not in LMFDB)