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

• Label: 2.167.abt_bfz
• 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:	$1 - 45 x + 831 x^{2} - 7515 x^{3} + 27889 x^{4}$
Frobenius angles:	$\pm0.0488944454644$, $\pm0.228665706648$
Angle rank:	$2$ (numerical)
Number field:	4.0.6194725.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21161$	$767742241$	$21685045372319$	$604973500995383821$	$16871944875756536434736$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$123$	$27527$	$4655979$	$777804531$	$129892116108$	$21691959611123$	$3622557491917809$	$604967115102230323$	$101029508508975796053$	$16871927924758703773022$

Jacobians and polarizations

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

• $y^2=127 x^6+140 x^5+104 x^4+128 x^3+66 x^2+107 x+25$
• $y^2=17 x^6+95 x^5+24 x^4+136 x^3+138 x^2+103 x+56$
• $y^2=60 x^6+45 x^5+111 x^4+84 x^3+143 x^2+18 x+131$
• $y^2=143 x^6+51 x^5+10 x^4+136 x^3+101 x^2+74 x+146$
• $y^2=142 x^6+104 x^5+121 x^4+112 x^3+144 x^2+14 x+70$
• $y^2=5 x^6+165 x^5+30 x^4+45 x^2+163 x+64$
• $y^2=166 x^6+125 x^5+37 x^4+132 x^3+95 x^2+25 x+63$
• $y^2=21 x^6+82 x^5+114 x^4+110 x^3+44 x^2+161 x+80$

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 4.0.6194725.1.

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.bt_bfz	$2$	(not in LMFDB)