Abelian variety isogeny class 2.197.abx_bma over $\F_{197}$

• Label: 2.197.abx_bma
• Base field: $\F_{197}$
• 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:	$\F_{197}$
Dimension:	$2$
L-polynomial:	$( 1 - 27 x + 197 x^{2} )( 1 - 22 x + 197 x^{2} )$
	$1 - 49 x + 988 x^{2} - 9653 x^{3} + 38809 x^{4}$
Frobenius angles:	$\pm0.0882242067266$, $\pm0.213320986704$
Angle rank:	$2$ (numerical)
Jacobians:	$60$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$30096$	$1489752000$	$58441245375744$	$2268505705356000000$	$88036654027586402728656$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$149$	$38385$	$7644002$	$1506173393$	$296710146049$	$58451739466230$	$11514990558930637$	$2268453123943144513$	$446885265410734782074$	$88036397287347313708425$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 60 curves (of which all are hyperelliptic):

• $y^2=15 x^6+44 x^5+143 x^4+34 x^3+167 x^2+46 x+170$
• $y^2=81 x^6+48 x^5+12 x^4+137 x^3+38 x^2+16 x+139$
• $y^2=183 x^6+74 x^5+79 x^4+37 x^3+59 x^2+187 x+20$
• $y^2=159 x^6+174 x^5+59 x^4+86 x^3+183 x^2+127 x+54$
• $y^2=128 x^6+142 x^5+88 x^4+107 x^3+50 x^2+127 x+113$
• $y^2=76 x^6+42 x^5+51 x^4+114 x^3+188 x^2+32 x+165$
• $y^2=164 x^6+156 x^5+162 x^4+192 x^3+85 x^2+53 x+84$
• $y^2=87 x^6+31 x^5+164 x^4+61 x^3+62 x^2+98 x+108$
• $y^2=95 x^6+5 x^5+29 x^4+145 x^3+29 x^2+142 x+140$
• $y^2=71 x^6+69 x^5+27 x^4+141 x^3+185 x^2+59 x+82$
• $y^2=78 x^6+6 x^5+88 x^4+127 x^3+153 x^2+107 x+82$
• $y^2=16 x^6+64 x^5+2 x^4+49 x^3+147 x^2+114 x+38$
• $y^2=156 x^6+116 x^5+190 x^4+60 x^3+120 x^2+165 x+106$
• $y^2=146 x^6+40 x^5+58 x^4+43 x^3+28 x^2+187 x+113$
• $y^2=26 x^6+111 x^5+142 x^4+25 x^3+103 x^2+78 x+188$
• $y^2=16 x^6+69 x^5+8 x^4+121 x^3+39 x^2+79 x+48$
• $y^2=83 x^6+128 x^5+143 x^4+2 x^3+54 x^2+16 x+21$
• $y^2=32 x^6+183 x^5+73 x^4+73 x^3+44 x^2+117 x+75$
• $y^2=104 x^6+91 x^5+51 x^4+34 x^3+104 x^2+79 x+120$
• $y^2=74 x^6+127 x^5+96 x^4+6 x^3+169 x^2+176 x+171$
• and

• $y^2=14 x^6+163 x^5+114 x^4+56 x^3+6 x^2+107 x+43$
• $y^2=147 x^6+30 x^5+35 x^4+106 x^3+49 x^2+109 x+90$
• $y^2=192 x^6+184 x^5+130 x^4+8 x^3+75 x^2+23 x+90$
• $y^2=193 x^5+69 x^4+34 x^3+57 x^2+12 x+56$
• $y^2=57 x^6+120 x^5+9 x^4+59 x^3+111 x^2+157 x+76$
• $y^2=145 x^6+16 x^5+146 x^4+137 x^3+144 x^2+33 x+196$
• $y^2=171 x^6+82 x^5+151 x^4+132 x^3+105 x^2+107 x+124$
• $y^2=83 x^6+54 x^5+77 x^4+84 x^3+64 x^2+109 x+6$
• $y^2=167 x^6+16 x^5+173 x^4+89 x^3+6 x^2+4 x+89$
• $y^2=73 x^6+156 x^5+152 x^4+161 x^3+44 x^2+34 x+74$
• $y^2=121 x^6+54 x^5+161 x^4+152 x^3+64 x^2+108 x+131$
• $y^2=19 x^6+13 x^5+114 x^4+186 x^3+111 x^2+117 x+129$
• $y^2=75 x^6+153 x^5+173 x^4+12 x^3+25 x^2+76 x+176$
• $y^2=191 x^6+178 x^5+54 x^4+62 x^3+23 x^2+102 x+67$
• $y^2=17 x^6+64 x^5+137 x^4+126 x^3+34 x^2+21 x+163$
• $y^2=58 x^6+72 x^5+37 x^4+153 x^3+92 x^2+174 x+118$
• $y^2=87 x^6+51 x^5+42 x^4+119 x^3+177 x^2+196 x+30$
• $y^2=94 x^6+176 x^5+141 x^4+82 x^2+99 x+67$
• $y^2=3 x^6+146 x^5+9 x^4+90 x^3+85 x^2+13 x+115$
• $y^2=98 x^6+183 x^5+107 x^4+66 x^3+25 x^2+16 x+30$
• $y^2=56 x^6+17 x^5+77 x^4+156 x^3+129 x^2+124 x+181$
• $y^2=80 x^6+34 x^5+84 x^4+173 x^3+122 x^2+151 x+104$
• $y^2=138 x^6+86 x^5+27 x^4+19 x^3+58 x^2+161 x+12$
• $y^2=71 x^6+129 x^5+85 x^4+58 x^3+134 x^2+144 x+38$
• $y^2=31 x^6+42 x^5+71 x^4+79 x^3+145 x^2+50 x+5$
• $y^2=139 x^6+133 x^5+186 x^4+182 x^3+96 x^2+89 x+63$
• $y^2=114 x^6+44 x^5+191 x^4+158 x^3+124 x^2+8 x+152$
• $y^2=140 x^6+151 x^5+73 x^4+93 x^3+109 x^2+93 x+126$
• $y^2=96 x^6+37 x^5+170 x^4+164 x^3+130 x^2+62 x+79$
• $y^2=67 x^6+111 x^5+181 x^4+174 x^3+41 x^2+101 x+66$
• $y^2=52 x^6+105 x^5+63 x^4+98 x^3+174 x^2+140 x+176$
• $y^2=174 x^6+83 x^5+79 x^4+109 x^3+x^2+173 x+94$
• $y^2=156 x^6+52 x^5+64 x^4+112 x^3+123 x^2+128 x+141$
• $y^2=167 x^6+54 x^5+3 x^4+40 x^3+24 x^2+64 x+81$
• $y^2=65 x^6+33 x^5+109 x^4+23 x^3+169 x^2+118 x+184$
• $y^2=67 x^6+17 x^5+76 x^4+108 x^3+56 x^2+24 x+7$
• $y^2=105 x^6+139 x^5+164 x^4+116 x^3+144 x^2+78 x+98$
• $y^2=176 x^6+116 x^5+63 x^4+106 x^3+57 x^2+196 x+40$
• $y^2=91 x^6+44 x^5+43 x^4+34 x^3+54 x^2+174 x+40$
• $y^2=73 x^6+16 x^5+12 x^4+106 x^3+108 x^2+63 x+94$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{197}$.

Endomorphism algebra over $\F_{197}$

The isogeny class factors as 1.197.abb $\times$ 1.197.aw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.197.abb : \(\Q(\sqrt{-59}) \).
• 1.197.aw : \(\Q(\sqrt{-19}) \).

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
2.197.af_ahs	$2$	(not in LMFDB)
2.197.f_ahs	$2$	(not in LMFDB)
2.197.bx_bma	$2$	(not in LMFDB)