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

• Label: 2.197.aby_bnb
• 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:	$( 1 - 27 x + 197 x^{2} )( 1 - 23 x + 197 x^{2} )$
	$1 - 50 x + 1015 x^{2} - 9850 x^{3} + 38809 x^{4}$
Frobenius angles:	$\pm0.0882242067266$, $\pm0.194339516027$
Angle rank:	$2$ (numerical)
Jacobians:	$72$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$29925$	$1488020625$	$58434153926400$	$2268490456054265625$	$88036659801248398057125$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$148$	$38340$	$7643074$	$1506163268$	$296710165508$	$58451742974070$	$11514990645526964$	$2268453125248368388$	$446885265421874979418$	$88036397287305461777700$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 72 curves (of which all are hyperelliptic):

• $y^2=171 x^6+16 x^5+106 x^4+177 x^3+27 x^2+159 x+52$
• $y^2=73 x^6+30 x^5+141 x^4+166 x^3+128 x^2+28 x+109$
• $y^2=88 x^6+83 x^5+161 x^4+151 x^3+188 x^2+116 x+26$
• $y^2=58 x^6+125 x^5+137 x^4+140 x^3+19 x^2+155 x+102$
• $y^2=49 x^6+94 x^5+90 x^4+58 x^3+133 x^2+128 x+79$
• $y^2=177 x^6+33 x^5+110 x^4+42 x^3+80 x^2+107 x+82$
• $y^2=81 x^6+191 x^5+47 x^4+147 x^3+151 x^2+85 x+35$
• $y^2=111 x^6+122 x^5+123 x^4+14 x^3+55 x^2+47 x+79$
• $y^2=12 x^6+28 x^5+39 x^4+4 x^3+142 x^2+7 x+70$
• $y^2=12 x^6+150 x^5+131 x^4+85 x^3+178 x^2+193 x+71$
• $y^2=128 x^6+161 x^5+42 x^4+65 x^3+63 x^2+116 x+38$
• $y^2=52 x^6+13 x^5+182 x^4+127 x^3+42 x^2+31 x+165$
• $y^2=195 x^6+45 x^5+160 x^4+30 x^3+37 x^2+45 x+2$
• $y^2=19 x^6+90 x^5+67 x^4+169 x^3+153 x^2+101 x+196$
• $y^2=189 x^6+64 x^5+177 x^4+188 x^3+27 x^2+93 x+12$
• $y^2=80 x^6+83 x^5+187 x^4+93 x^3+62 x^2+85 x+12$
• $y^2=82 x^6+52 x^5+56 x^4+189 x^3+78 x^2+124 x+5$
• $y^2=181 x^6+38 x^5+65 x^4+9 x^3+155 x^2+117 x+121$
• $y^2=73 x^6+15 x^5+102 x^4+118 x^3+194 x^2+7 x+89$
• $y^2=184 x^6+194 x^5+9 x^4+38 x^3+157 x^2+140 x+141$
• and

• $y^2=30 x^6+184 x^5+91 x^4+179 x^3+92 x^2+128 x+20$
• $y^2=184 x^6+170 x^5+55 x^4+184 x^3+55 x^2+170 x+184$
• $y^2=130 x^6+158 x^5+96 x^4+137 x^3+129 x^2+50 x+167$
• $y^2=178 x^6+142 x^5+27 x^4+59 x^3+115 x^2+81 x+155$
• $y^2=96 x^6+38 x^5+116 x^4+120 x^3+80 x^2+6 x+118$
• $y^2=76 x^6+3 x^5+96 x^4+143 x^3+190 x^2+189 x+168$
• $y^2=21 x^6+46 x^5+141 x^4+145 x^3+81 x^2+177 x+12$
• $y^2=159 x^6+181 x^5+164 x^4+115 x^3+41 x^2+196 x+141$
• $y^2=71 x^6+158 x^5+190 x^4+85 x^3+155 x^2+172 x+167$
• $y^2=105 x^6+152 x^5+97 x^4+108 x^3+19 x^2+3 x+20$
• $y^2=92 x^6+95 x^5+168 x^4+124 x^3+136 x^2+165 x+39$
• $y^2=100 x^6+151 x^5+51 x^4+27 x^3+93 x^2+12 x+93$
• $y^2=178 x^6+38 x^5+52 x^4+89 x^3+192 x^2+168 x+21$
• $y^2=146 x^6+128 x^5+66 x^4+27 x^3+2 x^2+52 x+153$
• $y^2=114 x^6+40 x^5+175 x^4+17 x^3+165 x^2+195 x+126$
• $y^2=169 x^6+114 x^5+41 x^4+92 x^3+164 x^2+51 x+178$
• $y^2=74 x^6+110 x^5+49 x^4+191 x^3+75 x^2+74 x+15$
• $y^2=147 x^6+152 x^5+63 x^4+144 x^3+131 x^2+60 x+165$
• $y^2=151 x^6+76 x^5+182 x^4+x^3+101 x^2+24 x+75$
• $y^2=74 x^6+181 x^5+57 x^4+40 x^3+72 x^2+136 x+149$
• $y^2=187 x^6+57 x^5+160 x^4+39 x^3+133 x^2+17 x+174$
• $y^2=43 x^6+190 x^5+10 x^4+46 x^3+109 x^2+175 x+4$
• $y^2=155 x^6+99 x^5+147 x^4+24 x^3+109 x^2+93 x+16$
• $y^2=120 x^6+161 x^5+68 x^4+55 x^3+65 x^2+105 x+44$
• $y^2=149 x^6+181 x^5+160 x^4+85 x^3+136 x^2+83 x+123$
• $y^2=31 x^6+189 x^5+120 x^4+39 x^3+66 x^2+167 x+27$
• $y^2=5 x^6+156 x^5+132 x^4+31 x^3+95 x^2+20 x+143$
• $y^2=128 x^6+66 x^5+185 x^4+142 x^3+76 x^2+190 x+193$
• $y^2=67 x^6+90 x^5+113 x^4+188 x^3+50 x^2+54 x+159$
• $y^2=77 x^6+56 x^5+28 x^4+75 x^3+25 x^2+115 x+179$
• $y^2=95 x^6+150 x^5+4 x^4+108 x^3+15 x^2+69 x+79$
• $y^2=154 x^6+57 x^5+73 x^4+170 x^3+52 x^2+191 x+94$
• $y^2=80 x^6+155 x^5+165 x^4+43 x^3+141 x^2+93 x+84$
• $y^2=189 x^6+53 x^5+171 x^4+95 x^3+187 x^2+16 x+98$
• $y^2=29 x^6+65 x^5+35 x^4+115 x^3+149 x^2+33 x+100$
• $y^2=144 x^6+90 x^5+155 x^4+88 x^3+73 x^2+48 x+68$
• $y^2=63 x^6+48 x^5+102 x^4+195 x^3+153 x^2+145 x+100$
• $y^2=184 x^6+53 x^5+37 x^4+185 x^3+104 x^2+64 x+180$
• $y^2=18 x^6+192 x^5+130 x^4+153 x^3+75 x^2+64 x+39$
• $y^2=56 x^6+99 x^5+134 x^4+9 x^3+25 x^2+184 x+140$
• $y^2=154 x^6+53 x^5+150 x^4+67 x^3+48 x^2+184 x+37$
• $y^2=188 x^6+169 x^5+142 x^4+35 x^3+147 x^2+69 x+162$
• $y^2=123 x^6+40 x^5+168 x^4+57 x^3+169 x^2+23 x+35$
• $y^2=120 x^6+35 x^5+132 x^4+85 x^3+169 x^2+79 x+50$
• $y^2=195 x^6+122 x^5+12 x^4+21 x^3+75 x^2+13 x+35$
• $y^2=56 x^6+39 x^5+133 x^4+189 x^3+102 x^2+157 x+111$
• $y^2=140 x^6+90 x^5+148 x^4+114 x^3+51 x^2+76 x+149$
• $y^2=16 x^6+142 x^5+151 x^4+138 x^3+7 x^2+6 x+81$
• $y^2=103 x^6+124 x^5+85 x^4+33 x^3+9 x^2+77 x+130$
• $y^2=158 x^6+14 x^5+82 x^4+125 x^3+119 x^2+124 x+181$
• $y^2=152 x^6+127 x^5+30 x^4+107 x^3+175 x^2+82 x+136$
• $y^2=82 x^6+158 x^5+53 x^4+170 x^3+107 x^2+40 x+89$

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.ax and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.197.abb : \(\Q(\sqrt{-59}) \).
• 1.197.ax : \(\Q(\sqrt{-259}) \).

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.ae_ait	$2$	(not in LMFDB)
2.197.e_ait	$2$	(not in LMFDB)
2.197.by_bnb	$2$	(not in LMFDB)