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

• Label: 2.167.abu_bgx
• 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:	$1 - 46 x + 855 x^{2} - 7682 x^{3} + 27889 x^{4}$
Frobenius angles:	$\pm0.0116366186044$, $\pm0.214984126313$
Angle rank:	$2$ (numerical)
Number field:	4.0.14912.2
Galois group:	$D_{4}$
Jacobians:	$3$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21017$	$766553041$	$21680822113136$	$604963446088901561$	$16871926902961694800177$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$122$	$27484$	$4655072$	$777791604$	$129891977742$	$21691958216014$	$3622557468289490$	$604967114480502948$	$101029508493801990176$	$16871927924454950153004$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=117 x^6+116 x^5+73 x^4+24 x^3+82 x^2+165 x+79$
• $y^2=83 x^6+34 x^5+74 x^4+6 x^3+68 x^2+58 x+31$
• $y^2=32 x^6+165 x^5+44 x^4+120 x^3+60 x^2+24 x+11$

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 4.0.14912.2.

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_bgx	$2$	(not in LMFDB)