Normalized defining polynomial
\( x^{47} - 1081 x^{45} - 705 x^{44} + 534061 x^{43} + 683944 x^{42} - 160057842 x^{41} - 299647654 x^{40} + 32561335719 x^{39} + 78763416061 x^{38} + \cdots + 10\!\cdots\!49 \)
Invariants
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{3763}a^{33}+\frac{1769}{3763}a^{32}+\frac{1597}{3763}a^{31}+\frac{1239}{3763}a^{30}+\frac{323}{3763}a^{29}+\frac{905}{3763}a^{28}-\frac{1681}{3763}a^{27}-\frac{1291}{3763}a^{26}+\frac{1080}{3763}a^{25}+\frac{948}{3763}a^{24}-\frac{291}{3763}a^{23}+\frac{580}{3763}a^{22}+\frac{1261}{3763}a^{21}-\frac{836}{3763}a^{20}-\frac{1312}{3763}a^{19}-\frac{57}{3763}a^{18}-\frac{757}{3763}a^{17}+\frac{1572}{3763}a^{16}-\frac{87}{3763}a^{15}-\frac{336}{3763}a^{14}+\frac{1602}{3763}a^{13}+\frac{1831}{3763}a^{12}+\frac{1548}{3763}a^{11}-\frac{1678}{3763}a^{10}+\frac{1443}{3763}a^{9}-\frac{735}{3763}a^{8}+\frac{1321}{3763}a^{7}-\frac{1773}{3763}a^{6}-\frac{698}{3763}a^{5}+\frac{535}{3763}a^{4}+\frac{124}{3763}a^{3}+\frac{666}{3763}a^{2}-\frac{1128}{3763}a+\frac{19}{71}$, $\frac{1}{3763}a^{34}-\frac{711}{3763}a^{32}-\frac{1604}{3763}a^{31}-\frac{1402}{3763}a^{30}+\frac{1494}{3763}a^{29}+\frac{412}{3763}a^{28}-\frac{372}{3763}a^{27}+\frac{718}{3763}a^{26}-\frac{1731}{3763}a^{25}+\frac{995}{3763}a^{24}-\frac{172}{3763}a^{23}-\frac{1223}{3763}a^{22}-\frac{86}{3763}a^{21}-\frac{1287}{3763}a^{20}-\frac{900}{3763}a^{19}-\frac{1525}{3763}a^{18}+\frac{1077}{3763}a^{17}-\frac{98}{3763}a^{16}-\frac{716}{3763}a^{15}+\frac{1432}{3763}a^{14}+\frac{1432}{3763}a^{13}-\frac{1311}{3763}a^{12}-\frac{626}{3763}a^{11}+\frac{818}{3763}a^{10}+\frac{1675}{3763}a^{9}-\frac{462}{3763}a^{8}-\frac{1799}{3763}a^{7}+\frac{1160}{3763}a^{6}+\frac{1033}{3763}a^{5}-\frac{1778}{3763}a^{4}-\frac{436}{3763}a^{3}-\frac{1463}{3763}a^{2}-\frac{1714}{3763}a-\frac{28}{71}$, $\frac{1}{3763}a^{35}-\frac{687}{3763}a^{32}+\frac{1402}{3763}a^{31}+\frac{1881}{3763}a^{30}+\frac{522}{3763}a^{29}-\frac{390}{3763}a^{28}-\frac{1602}{3763}a^{27}-\frac{1460}{3763}a^{26}+\frac{1223}{3763}a^{25}+\frac{279}{3763}a^{24}-\frac{1159}{3763}a^{23}-\frac{1636}{3763}a^{22}-\frac{310}{3763}a^{21}-\frac{14}{71}a^{20}-\frac{1133}{3763}a^{19}-\frac{1820}{3763}a^{18}-\frac{216}{3763}a^{17}-\frac{635}{3763}a^{16}-\frac{217}{3763}a^{15}-\frac{395}{3763}a^{14}+\frac{1285}{3763}a^{13}-\frac{783}{3763}a^{12}-\frac{21}{71}a^{11}+\frac{1488}{3763}a^{10}-\frac{1788}{3763}a^{9}-\frac{1327}{3763}a^{8}-\frac{359}{3763}a^{7}+\frac{1035}{3763}a^{6}-\frac{1340}{3763}a^{5}-\frac{114}{3763}a^{4}+\frac{152}{3763}a^{3}+\frac{1437}{3763}a^{2}+\frac{1790}{3763}a+\frac{19}{71}$, $\frac{1}{3763}a^{36}+\frac{1256}{3763}a^{32}+\frac{224}{3763}a^{31}+\frac{1277}{3763}a^{30}-\frac{506}{3763}a^{29}-\frac{762}{3763}a^{28}-\frac{1066}{3763}a^{27}-\frac{1389}{3763}a^{26}+\frac{928}{3763}a^{25}-\frac{882}{3763}a^{24}+\frac{1649}{3763}a^{23}-\frac{728}{3763}a^{22}+\frac{75}{3763}a^{21}+\frac{274}{3763}a^{20}-\frac{44}{3763}a^{19}-\frac{1745}{3763}a^{18}-\frac{1400}{3763}a^{17}-\frac{234}{3763}a^{16}+\frac{44}{3763}a^{15}-\frac{4}{3763}a^{14}+\frac{995}{3763}a^{13}-\frac{58}{3763}a^{12}+\frac{35}{3763}a^{11}+\frac{667}{3763}a^{10}+\frac{345}{3763}a^{9}-\frac{1062}{3763}a^{8}+\frac{1679}{3763}a^{7}-\frac{179}{3763}a^{6}-\frac{1739}{3763}a^{5}-\frac{1077}{3763}a^{4}+\frac{76}{3763}a^{3}+\frac{246}{3763}a^{2}+\frac{1249}{3763}a-\frac{11}{71}$, $\frac{1}{3763}a^{37}-\frac{1470}{3763}a^{32}+\frac{1124}{3763}a^{31}+\frac{1192}{3763}a^{30}-\frac{46}{3763}a^{29}-\frac{1320}{3763}a^{28}-\frac{1096}{3763}a^{27}+\frac{571}{3763}a^{26}+\frac{1081}{3763}a^{25}+\frac{69}{3763}a^{24}-\frac{243}{3763}a^{23}+\frac{1617}{3763}a^{22}+\frac{681}{3763}a^{21}+\frac{95}{3763}a^{20}+\frac{32}{71}a^{19}-\frac{1305}{3763}a^{18}-\frac{1481}{3763}a^{17}+\frac{1187}{3763}a^{16}+\frac{141}{3763}a^{15}+\frac{1555}{3763}a^{14}+\frac{1035}{3763}a^{13}-\frac{508}{3763}a^{12}+\frac{1850}{3763}a^{11}+\frac{633}{3763}a^{10}+\frac{296}{3763}a^{9}-\frac{859}{3763}a^{8}+\frac{128}{3763}a^{7}+\frac{1216}{3763}a^{6}-\frac{1168}{3763}a^{5}+\frac{1693}{3763}a^{4}-\frac{1215}{3763}a^{3}+\frac{139}{3763}a^{2}+\frac{1297}{3763}a-\frac{8}{71}$, $\frac{1}{3763}a^{38}+\frac{1321}{3763}a^{32}+\frac{670}{3763}a^{31}-\frac{8}{3763}a^{30}-\frac{648}{3763}a^{29}+\frac{915}{3763}a^{28}+\frac{1792}{3763}a^{27}-\frac{137}{3763}a^{26}-\frac{317}{3763}a^{25}+\frac{19}{71}a^{24}-\frac{934}{3763}a^{23}-\frac{920}{3763}a^{22}-\frac{1394}{3763}a^{21}-\frac{486}{3763}a^{20}+\frac{474}{3763}a^{19}+\frac{18}{53}a^{18}-\frac{1518}{3763}a^{17}+\frac{499}{3763}a^{16}+\frac{1607}{3763}a^{15}+\frac{68}{3763}a^{14}-\frac{1206}{3763}a^{13}-\frac{888}{3763}a^{12}-\frac{422}{3763}a^{11}-\frac{1599}{3763}a^{10}+\frac{1782}{3763}a^{9}-\frac{341}{3763}a^{8}+\frac{26}{71}a^{7}+\frac{281}{3763}a^{6}-\frac{831}{3763}a^{5}-\frac{1232}{3763}a^{4}+\frac{1795}{3763}a^{3}-\frac{1826}{3763}a^{2}+\frac{899}{3763}a+\frac{27}{71}$, $\frac{1}{252121}a^{39}-\frac{19}{252121}a^{38}+\frac{30}{252121}a^{37}-\frac{20}{252121}a^{36}+\frac{20}{252121}a^{35}-\frac{13}{252121}a^{34}+\frac{29}{252121}a^{33}+\frac{84834}{252121}a^{32}+\frac{30327}{252121}a^{31}-\frac{7440}{252121}a^{30}-\frac{99528}{252121}a^{29}-\frac{40787}{252121}a^{28}+\frac{48068}{252121}a^{27}-\frac{81}{3763}a^{26}+\frac{4605}{252121}a^{25}+\frac{1736}{252121}a^{24}+\frac{79463}{252121}a^{23}+\frac{39234}{252121}a^{22}+\frac{1891}{4757}a^{21}+\frac{80597}{252121}a^{20}-\frac{435}{3763}a^{19}+\frac{4441}{252121}a^{18}+\frac{88329}{252121}a^{17}-\frac{45748}{252121}a^{16}-\frac{37685}{252121}a^{15}+\frac{86817}{252121}a^{14}-\frac{87816}{252121}a^{13}-\frac{36365}{252121}a^{12}-\frac{86471}{252121}a^{11}+\frac{98820}{252121}a^{10}-\frac{1370}{4757}a^{9}-\frac{79834}{252121}a^{8}-\frac{120550}{252121}a^{7}-\frac{108198}{252121}a^{6}-\frac{42288}{252121}a^{5}+\frac{6649}{252121}a^{4}+\frac{56824}{252121}a^{3}-\frac{74189}{252121}a^{2}-\frac{17803}{252121}a+\frac{1152}{4757}$, $\frac{1}{252121}a^{40}+\frac{4}{252121}a^{38}+\frac{14}{252121}a^{37}-\frac{25}{252121}a^{36}+\frac{32}{252121}a^{35}-\frac{17}{252121}a^{34}+\frac{27}{252121}a^{33}+\frac{125226}{252121}a^{32}+\frac{58702}{252121}a^{31}+\frac{26241}{252121}a^{30}+\frac{11315}{252121}a^{29}-\frac{106733}{252121}a^{28}-\frac{38711}{252121}a^{27}-\frac{9934}{252121}a^{26}+\frac{54994}{252121}a^{25}-\frac{16796}{252121}a^{24}-\frac{117393}{252121}a^{23}-\frac{1559}{3551}a^{22}+\frac{23141}{252121}a^{21}+\frac{118581}{252121}a^{20}-\frac{59209}{252121}a^{19}+\frac{85541}{252121}a^{18}+\frac{45809}{252121}a^{17}-\frac{55126}{252121}a^{16}-\frac{108273}{252121}a^{15}+\frac{99834}{252121}a^{14}-\frac{45681}{252121}a^{13}-\frac{42349}{252121}a^{12}+\frac{80085}{252121}a^{11}+\frac{89033}{252121}a^{10}-\frac{106426}{252121}a^{9}-\frac{74688}{252121}a^{8}+\frac{20521}{252121}a^{7}+\frac{11244}{252121}a^{6}-\frac{43743}{252121}a^{5}-\frac{84778}{252121}a^{4}-\frac{119463}{252121}a^{3}+\frac{97593}{252121}a^{2}+\frac{37230}{252121}a+\frac{850}{4757}$, $\frac{1}{252121}a^{41}+\frac{23}{252121}a^{38}-\frac{11}{252121}a^{37}-\frac{22}{252121}a^{36}-\frac{30}{252121}a^{35}+\frac{12}{252121}a^{34}+\frac{21}{252121}a^{33}+\frac{103343}{252121}a^{32}+\frac{94744}{252121}a^{31}+\frac{67205}{252121}a^{30}+\frac{14803}{252121}a^{29}-\frac{68590}{252121}a^{28}+\frac{1511}{3763}a^{27}+\frac{82933}{252121}a^{26}-\frac{6942}{252121}a^{25}+\frac{52275}{252121}a^{24}-\frac{85903}{252121}a^{23}+\frac{21511}{252121}a^{22}-\frac{32200}{252121}a^{21}+\frac{111925}{252121}a^{20}+\frac{121788}{252121}a^{19}+\frac{52299}{252121}a^{18}-\frac{5102}{252121}a^{17}-\frac{41928}{252121}a^{16}-\frac{21245}{252121}a^{15}+\frac{118931}{252121}a^{14}-\frac{75397}{252121}a^{13}-\frac{104095}{252121}a^{12}-\frac{91435}{252121}a^{11}+\frac{32552}{252121}a^{10}+\frac{125101}{252121}a^{9}-\frac{3920}{252121}a^{8}-\frac{65671}{252121}a^{7}-\frac{39148}{252121}a^{6}-\frac{116291}{252121}a^{5}+\frac{56482}{252121}a^{4}-\frac{10979}{252121}a^{3}+\frac{22704}{252121}a^{2}-\frac{42260}{252121}a-\frac{1057}{4757}$, $\frac{1}{17900591}a^{42}-\frac{32}{17900591}a^{41}+\frac{5}{17900591}a^{40}-\frac{13}{17900591}a^{39}+\frac{2101}{17900591}a^{38}-\frac{2221}{17900591}a^{37}-\frac{808}{17900591}a^{36}+\frac{1283}{17900591}a^{35}-\frac{2124}{17900591}a^{34}+\frac{793}{17900591}a^{33}-\frac{153521}{337747}a^{32}+\frac{4784526}{17900591}a^{31}+\frac{2890777}{17900591}a^{30}+\frac{4278038}{17900591}a^{29}-\frac{8910688}{17900591}a^{28}+\frac{4891425}{17900591}a^{27}+\frac{2692613}{17900591}a^{26}+\frac{1261175}{17900591}a^{25}-\frac{1840457}{17900591}a^{24}-\frac{3209558}{17900591}a^{23}+\frac{4817043}{17900591}a^{22}+\frac{7488416}{17900591}a^{21}-\frac{1933587}{17900591}a^{20}+\frac{6033524}{17900591}a^{19}+\frac{5832060}{17900591}a^{18}-\frac{6123719}{17900591}a^{17}-\frac{5123064}{17900591}a^{16}-\frac{8818169}{17900591}a^{15}+\frac{3104791}{17900591}a^{14}+\frac{6481214}{17900591}a^{13}+\frac{6152030}{17900591}a^{12}-\frac{1759030}{17900591}a^{11}-\frac{2963551}{17900591}a^{10}+\frac{392007}{17900591}a^{9}-\frac{2533657}{17900591}a^{8}-\frac{2790851}{17900591}a^{7}+\frac{4379603}{17900591}a^{6}-\frac{7646007}{17900591}a^{5}+\frac{1178620}{17900591}a^{4}+\frac{4447066}{17900591}a^{3}+\frac{1200664}{17900591}a^{2}+\frac{844394}{17900591}a-\frac{153813}{337747}$, $\frac{1}{17900591}a^{43}-\frac{25}{17900591}a^{41}+\frac{5}{17900591}a^{40}-\frac{19}{17900591}a^{39}+\frac{756}{17900591}a^{38}+\frac{2031}{17900591}a^{37}+\frac{703}{17900591}a^{36}-\frac{970}{17900591}a^{35}-\frac{2139}{17900591}a^{34}+\frac{7}{337747}a^{33}-\frac{122259}{267173}a^{32}+\frac{3507844}{17900591}a^{31}+\frac{199614}{17900591}a^{30}+\frac{3639613}{17900591}a^{29}+\frac{6772944}{17900591}a^{28}+\frac{7166246}{17900591}a^{27}+\frac{2245097}{17900591}a^{26}+\frac{6522839}{17900591}a^{25}+\frac{1861564}{17900591}a^{24}-\frac{3172044}{17900591}a^{23}-\frac{2325656}{17900591}a^{22}+\frac{6876997}{17900591}a^{21}+\frac{191514}{17900591}a^{20}-\frac{2284535}{17900591}a^{19}+\frac{2599347}{17900591}a^{18}-\frac{104124}{267173}a^{17}-\frac{8169626}{17900591}a^{16}+\frac{734312}{17900591}a^{15}-\frac{3968891}{17900591}a^{14}+\frac{2761037}{17900591}a^{13}-\frac{7940793}{17900591}a^{12}-\frac{3803641}{17900591}a^{11}-\frac{8325654}{17900591}a^{10}+\frac{3745385}{17900591}a^{9}+\frac{2236878}{17900591}a^{8}+\frac{3610720}{17900591}a^{7}-\frac{5182124}{17900591}a^{6}-\frac{2527343}{17900591}a^{5}-\frac{7589776}{17900591}a^{4}-\frac{6105276}{17900591}a^{3}+\frac{624815}{17900591}a^{2}-\frac{4938349}{17900591}a+\frac{149443}{337747}$, $\frac{1}{58194821341}a^{44}+\frac{306}{58194821341}a^{43}+\frac{979}{58194821341}a^{42}+\frac{65804}{58194821341}a^{41}+\frac{37203}{58194821341}a^{40}-\frac{66816}{58194821341}a^{39}+\frac{4406315}{58194821341}a^{38}-\frac{3734216}{58194821341}a^{37}+\frac{6126048}{58194821341}a^{36}-\frac{6590006}{58194821341}a^{35}-\frac{5713066}{58194821341}a^{34}+\frac{6191305}{58194821341}a^{33}-\frac{4650975295}{58194821341}a^{32}-\frac{16950920777}{58194821341}a^{31}+\frac{26049853216}{58194821341}a^{30}-\frac{7238201177}{58194821341}a^{29}+\frac{16905049964}{58194821341}a^{28}+\frac{28375571180}{58194821341}a^{27}+\frac{16890999478}{58194821341}a^{26}-\frac{27834053897}{58194821341}a^{25}-\frac{28634017869}{58194821341}a^{24}-\frac{13055807950}{58194821341}a^{23}-\frac{20959904954}{58194821341}a^{22}+\frac{11053509283}{58194821341}a^{21}+\frac{3122837419}{58194821341}a^{20}+\frac{28769349778}{58194821341}a^{19}-\frac{16644918760}{58194821341}a^{18}+\frac{25929521013}{58194821341}a^{17}-\frac{26682799204}{58194821341}a^{16}+\frac{7985367926}{58194821341}a^{15}-\frac{16646483413}{58194821341}a^{14}-\frac{26035244004}{58194821341}a^{13}+\frac{6852114667}{58194821341}a^{12}+\frac{16948016843}{58194821341}a^{11}-\frac{12955610389}{58194821341}a^{10}+\frac{21910767660}{58194821341}a^{9}+\frac{24029190317}{58194821341}a^{8}+\frac{2408090626}{58194821341}a^{7}-\frac{4843592981}{58194821341}a^{6}+\frac{10275904904}{58194821341}a^{5}+\frac{24652340149}{58194821341}a^{4}-\frac{11962704383}{58194821341}a^{3}-\frac{16585859502}{58194821341}a^{2}-\frac{6441696227}{58194821341}a-\frac{75721253}{1098015497}$, $\frac{1}{35\!\cdots\!47}a^{45}-\frac{2657193}{35\!\cdots\!47}a^{44}-\frac{6930008733}{35\!\cdots\!47}a^{43}-\frac{204546381}{35\!\cdots\!47}a^{42}+\frac{2673870763}{49\!\cdots\!57}a^{41}+\frac{263571318067}{35\!\cdots\!47}a^{40}-\frac{677130996371}{35\!\cdots\!47}a^{39}-\frac{38489790373294}{35\!\cdots\!47}a^{38}-\frac{5660630542343}{35\!\cdots\!47}a^{37}-\frac{91509342495}{66\!\cdots\!99}a^{36}+\frac{1597176005624}{35\!\cdots\!47}a^{35}+\frac{44445377245561}{35\!\cdots\!47}a^{34}-\frac{35051836607279}{35\!\cdots\!47}a^{33}+\frac{73\!\cdots\!43}{35\!\cdots\!47}a^{32}-\frac{16\!\cdots\!63}{35\!\cdots\!47}a^{31}-\frac{18\!\cdots\!46}{35\!\cdots\!47}a^{30}-\frac{17\!\cdots\!17}{35\!\cdots\!47}a^{29}+\frac{81\!\cdots\!47}{35\!\cdots\!47}a^{28}-\frac{80\!\cdots\!81}{35\!\cdots\!47}a^{27}-\frac{56\!\cdots\!87}{35\!\cdots\!47}a^{26}-\frac{95\!\cdots\!20}{35\!\cdots\!47}a^{25}+\frac{96\!\cdots\!73}{35\!\cdots\!47}a^{24}+\frac{54\!\cdots\!48}{35\!\cdots\!47}a^{23}-\frac{43\!\cdots\!74}{35\!\cdots\!47}a^{22}+\frac{25\!\cdots\!52}{66\!\cdots\!99}a^{21}+\frac{68\!\cdots\!51}{35\!\cdots\!47}a^{20}-\frac{12\!\cdots\!98}{35\!\cdots\!47}a^{19}+\frac{33\!\cdots\!73}{35\!\cdots\!47}a^{18}+\frac{52\!\cdots\!66}{35\!\cdots\!47}a^{17}-\frac{10\!\cdots\!09}{35\!\cdots\!47}a^{16}-\frac{13\!\cdots\!94}{35\!\cdots\!47}a^{15}+\frac{17\!\cdots\!66}{35\!\cdots\!47}a^{14}-\frac{16\!\cdots\!80}{35\!\cdots\!47}a^{13}+\frac{99\!\cdots\!30}{35\!\cdots\!47}a^{12}+\frac{10\!\cdots\!19}{35\!\cdots\!47}a^{11}-\frac{10\!\cdots\!14}{35\!\cdots\!47}a^{10}-\frac{40\!\cdots\!04}{35\!\cdots\!47}a^{9}+\frac{38\!\cdots\!62}{35\!\cdots\!47}a^{8}+\frac{14\!\cdots\!19}{35\!\cdots\!47}a^{7}+\frac{10\!\cdots\!99}{35\!\cdots\!47}a^{6}-\frac{14\!\cdots\!75}{35\!\cdots\!47}a^{5}+\frac{14\!\cdots\!37}{35\!\cdots\!47}a^{4}-\frac{73\!\cdots\!63}{35\!\cdots\!47}a^{3}+\frac{17\!\cdots\!14}{35\!\cdots\!47}a^{2}-\frac{14\!\cdots\!00}{66\!\cdots\!99}a-\frac{52818476998726}{125436192361783}$, $\frac{1}{23\!\cdots\!23}a^{46}+\frac{17\!\cdots\!56}{23\!\cdots\!23}a^{45}-\frac{92\!\cdots\!33}{23\!\cdots\!23}a^{44}-\frac{34\!\cdots\!69}{23\!\cdots\!23}a^{43}+\frac{33\!\cdots\!62}{23\!\cdots\!23}a^{42}+\frac{30\!\cdots\!92}{23\!\cdots\!23}a^{41}-\frac{26\!\cdots\!84}{23\!\cdots\!23}a^{40}-\frac{12\!\cdots\!80}{23\!\cdots\!23}a^{39}+\frac{12\!\cdots\!26}{23\!\cdots\!23}a^{38}-\frac{51\!\cdots\!79}{23\!\cdots\!23}a^{37}+\frac{85\!\cdots\!02}{23\!\cdots\!23}a^{36}+\frac{28\!\cdots\!22}{23\!\cdots\!23}a^{35}+\frac{31\!\cdots\!92}{23\!\cdots\!23}a^{34}-\frac{17\!\cdots\!58}{23\!\cdots\!23}a^{33}-\frac{30\!\cdots\!99}{23\!\cdots\!23}a^{32}+\frac{76\!\cdots\!27}{23\!\cdots\!23}a^{31}-\frac{15\!\cdots\!60}{23\!\cdots\!23}a^{30}+\frac{71\!\cdots\!47}{23\!\cdots\!23}a^{29}-\frac{27\!\cdots\!54}{23\!\cdots\!23}a^{28}-\frac{11\!\cdots\!16}{23\!\cdots\!23}a^{27}-\frac{71\!\cdots\!98}{23\!\cdots\!23}a^{26}-\frac{50\!\cdots\!65}{23\!\cdots\!23}a^{25}+\frac{65\!\cdots\!88}{23\!\cdots\!23}a^{24}+\frac{10\!\cdots\!99}{23\!\cdots\!23}a^{23}-\frac{86\!\cdots\!15}{23\!\cdots\!23}a^{22}+\frac{10\!\cdots\!05}{23\!\cdots\!23}a^{21}-\frac{45\!\cdots\!90}{23\!\cdots\!23}a^{20}+\frac{72\!\cdots\!02}{23\!\cdots\!23}a^{19}-\frac{15\!\cdots\!85}{35\!\cdots\!69}a^{18}-\frac{10\!\cdots\!39}{23\!\cdots\!23}a^{17}-\frac{11\!\cdots\!15}{23\!\cdots\!23}a^{16}-\frac{46\!\cdots\!67}{23\!\cdots\!23}a^{15}+\frac{81\!\cdots\!89}{23\!\cdots\!23}a^{14}-\frac{93\!\cdots\!39}{23\!\cdots\!23}a^{13}-\frac{78\!\cdots\!09}{23\!\cdots\!23}a^{12}-\frac{11\!\cdots\!45}{23\!\cdots\!23}a^{11}-\frac{22\!\cdots\!98}{23\!\cdots\!23}a^{10}-\frac{45\!\cdots\!65}{23\!\cdots\!23}a^{9}-\frac{99\!\cdots\!17}{23\!\cdots\!23}a^{8}+\frac{11\!\cdots\!05}{23\!\cdots\!23}a^{7}-\frac{68\!\cdots\!27}{23\!\cdots\!23}a^{6}-\frac{83\!\cdots\!00}{23\!\cdots\!23}a^{5}-\frac{25\!\cdots\!08}{23\!\cdots\!23}a^{4}+\frac{91\!\cdots\!48}{23\!\cdots\!23}a^{3}+\frac{65\!\cdots\!68}{23\!\cdots\!23}a^{2}-\frac{11\!\cdots\!51}{45\!\cdots\!91}a-\frac{28\!\cdots\!44}{84\!\cdots\!47}$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $46$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
Galois group
A cyclic group of order 47 |
The 47 conjugacy class representatives for $C_{47}$ |
Character table for $C_{47}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | R | ${\href{/padicField/53.1.0.1}{1} }^{47}$ | $47$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(47\) | Deg $47$ | $47$ | $1$ | $92$ |