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

• Label: 2.167.abw_bja
• 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:	$( 1 - 24 x + 167 x^{2} )^{2}$
	$1 - 48 x + 910 x^{2} - 8016 x^{3} + 27889 x^{4}$
Frobenius angles:	$\pm0.121023609245$, $\pm0.121023609245$
Angle rank:	$1$ (numerical)
Jacobians:	$33$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20736$	$764411904$	$21675207280896$	$604962784643383296$	$16871988646109807790336$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$120$	$27406$	$4653864$	$777790750$	$129892453080$	$21691973746222$	$3622557800122248$	$604967120056796734$	$101029508571146152248$	$16871927925339398065486$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 33 curves (of which all are hyperelliptic):

• $y^2=128 x^6+165 x^5+34 x^4+126 x^3+34 x^2+165 x+128$
• $y^2=100 x^6+9 x^4+9 x^2+100$
• $y^2=74 x^6+29 x^5+128 x^4+24 x^3+128 x^2+29 x+74$
• $y^2=153 x^6+110 x^4+110 x^2+153$
• $y^2=74 x^6+139 x^5+36 x^4+54 x^3+128 x^2+46 x+82$
• $y^2=100 x^6+81 x^5+6 x^4+25 x^3+72 x^2+141 x+122$
• $y^2=162 x^6+110 x^5+145 x^4+165 x^3+145 x^2+110 x+162$
• $y^2=12 x^6+136 x^5+112 x^4+158 x^3+112 x^2+136 x+12$
• $y^2=69 x^6+149 x^5+12 x^4+149 x^3+152 x^2+118 x+134$
• $y^2=15 x^6+152 x^5+78 x^4+114 x^3+39 x^2+38 x+148$
• $y^2=60 x^6+166 x^5+101 x^4+109 x^3+131 x^2+146 x+80$
• $y^2=66 x^6+12 x^4+12 x^2+66$
• $y^2=110 x^6+77 x^5+133 x^4+129 x^3+100 x^2+157 x+35$
• $y^2=126 x^6+44 x^5+84 x^4+73 x^3+124 x^2+108 x+77$
• $y^2=89 x^6+122 x^5+112 x^4+63 x^3+150 x^2+89 x+141$
• $y^2=80 x^6+63 x^5+18 x^4+12 x^3+147 x^2+152 x+30$
• $y^2=27 x^6+76 x^5+141 x^4+124 x^3+85 x^2+84 x+38$
• $y^2=25 x^6+50 x^4+50 x^2+25$
• $y^2=58 x^6+105 x^5+x^4+16 x^3+152 x^2+78 x+141$
• $y^2=114 x^6+161 x^5+147 x^4+152 x^3+29 x^2+45 x+29$
• and

• $y^2=143 x^6+148 x^5+62 x^4+2 x^3+62 x^2+148 x+143$
• $y^2=120 x^6+86 x^5+82 x^4+7 x^3+82 x^2+86 x+120$
• $y^2=166 x^6+145 x^5+159 x^4+31 x^3+165 x^2+103 x+60$
• $y^2=158 x^6+157 x^5+79 x^4+84 x^3+79 x^2+157 x+158$
• $y^2=136 x^6+65 x^5+141 x^4+135 x^3+9 x^2+11 x+95$
• $y^2=91 x^6+89 x^5+152 x^4+40 x^3+152 x^2+89 x+91$
• $y^2=159 x^6+x^5+49 x^4+22 x^3+49 x^2+x+159$
• $y^2=139 x^6+56 x^5+118 x^4+34 x^3+138 x^2+61 x+45$
• $y^2=123 x^6+61 x^4+61 x^2+123$
• $y^2=55 x^6+165 x^5+125 x^4+155 x^3+125 x^2+165 x+55$
• $y^2=33 x^6+91 x^5+100 x^4+63 x^3+100 x^2+91 x+33$
• $y^2=76 x^6+141 x^5+81 x^4+85 x^3+81 x^2+141 x+76$
• $y^2=5 x^6+150 x^4+150 x^2+5$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{167}$

The isogeny class factors as 1.167.ay^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-23}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.167.a_aji	$2$	(not in LMFDB)
2.167.bw_bja	$2$	(not in LMFDB)
2.167.y_pt	$3$	(not in LMFDB)