Normalized defining polynomial
\( x^{47} - x^{46} - 460 x^{45} + 327 x^{44} + 94524 x^{43} - 38110 x^{42} - 11539413 x^{41} + 809932 x^{40} + 937957195 x^{39} + 272017844 x^{38} + \cdots - 19\!\cdots\!69 \)
Invariants
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23}a^{20}+\frac{6}{23}a^{19}-\frac{8}{23}a^{18}+\frac{4}{23}a^{17}-\frac{5}{23}a^{16}+\frac{3}{23}a^{15}+\frac{7}{23}a^{14}+\frac{9}{23}a^{13}+\frac{4}{23}a^{12}-\frac{1}{23}a^{9}-\frac{6}{23}a^{8}+\frac{8}{23}a^{7}-\frac{4}{23}a^{6}+\frac{5}{23}a^{5}-\frac{3}{23}a^{4}-\frac{7}{23}a^{3}-\frac{9}{23}a^{2}-\frac{4}{23}a$, $\frac{1}{23}a^{21}+\frac{2}{23}a^{19}+\frac{6}{23}a^{18}-\frac{6}{23}a^{17}+\frac{10}{23}a^{16}-\frac{11}{23}a^{15}-\frac{10}{23}a^{14}-\frac{4}{23}a^{13}-\frac{1}{23}a^{12}-\frac{1}{23}a^{10}-\frac{2}{23}a^{8}-\frac{6}{23}a^{7}+\frac{6}{23}a^{6}-\frac{10}{23}a^{5}+\frac{11}{23}a^{4}+\frac{10}{23}a^{3}+\frac{4}{23}a^{2}+\frac{1}{23}a$, $\frac{1}{23}a^{22}-\frac{6}{23}a^{19}+\frac{10}{23}a^{18}+\frac{2}{23}a^{17}-\frac{1}{23}a^{16}+\frac{7}{23}a^{15}+\frac{5}{23}a^{14}+\frac{4}{23}a^{13}-\frac{8}{23}a^{12}-\frac{1}{23}a^{11}+\frac{6}{23}a^{8}-\frac{10}{23}a^{7}-\frac{2}{23}a^{6}+\frac{1}{23}a^{5}-\frac{7}{23}a^{4}-\frac{5}{23}a^{3}-\frac{4}{23}a^{2}+\frac{8}{23}a$, $\frac{1}{23}a^{23}-\frac{1}{23}a$, $\frac{1}{23}a^{24}-\frac{1}{23}a^{2}$, $\frac{1}{23}a^{25}-\frac{1}{23}a^{3}$, $\frac{1}{23}a^{26}-\frac{1}{23}a^{4}$, $\frac{1}{23}a^{27}-\frac{1}{23}a^{5}$, $\frac{1}{23}a^{28}-\frac{1}{23}a^{6}$, $\frac{1}{23}a^{29}-\frac{1}{23}a^{7}$, $\frac{1}{23}a^{30}-\frac{1}{23}a^{8}$, $\frac{1}{23}a^{31}-\frac{1}{23}a^{9}$, $\frac{1}{23}a^{32}-\frac{1}{23}a^{10}$, $\frac{1}{23}a^{33}-\frac{1}{23}a^{11}$, $\frac{1}{23}a^{34}-\frac{1}{23}a^{12}$, $\frac{1}{529}a^{35}-\frac{8}{529}a^{34}-\frac{4}{529}a^{33}-\frac{9}{529}a^{32}-\frac{10}{529}a^{31}+\frac{2}{529}a^{30}+\frac{4}{529}a^{29}-\frac{1}{529}a^{28}-\frac{11}{529}a^{27}+\frac{6}{529}a^{26}+\frac{8}{529}a^{25}-\frac{5}{529}a^{24}+\frac{6}{529}a^{23}-\frac{1}{529}a^{22}+\frac{8}{529}a^{21}+\frac{6}{529}a^{20}+\frac{104}{529}a^{19}+\frac{36}{529}a^{18}-\frac{95}{529}a^{17}-\frac{225}{529}a^{16}+\frac{38}{529}a^{15}+\frac{256}{529}a^{14}+\frac{224}{529}a^{13}+\frac{55}{529}a^{12}+\frac{5}{529}a^{11}+\frac{231}{529}a^{10}-\frac{226}{529}a^{9}-\frac{60}{529}a^{8}+\frac{52}{529}a^{7}-\frac{157}{529}a^{6}+\frac{190}{529}a^{5}+\frac{209}{529}a^{4}+\frac{127}{529}a^{3}+\frac{194}{529}a^{2}-\frac{76}{529}a-\frac{8}{23}$, $\frac{1}{529}a^{36}+\frac{1}{529}a^{34}+\frac{5}{529}a^{33}+\frac{10}{529}a^{32}-\frac{9}{529}a^{31}-\frac{3}{529}a^{30}+\frac{8}{529}a^{29}+\frac{4}{529}a^{28}+\frac{10}{529}a^{27}+\frac{10}{529}a^{26}-\frac{10}{529}a^{25}-\frac{11}{529}a^{24}+\frac{1}{529}a^{23}+\frac{1}{529}a^{21}-\frac{9}{529}a^{20}-\frac{236}{529}a^{19}+\frac{9}{529}a^{18}-\frac{157}{529}a^{17}-\frac{60}{529}a^{16}-\frac{222}{529}a^{15}+\frac{248}{529}a^{14}+\frac{145}{529}a^{13}-\frac{199}{529}a^{12}+\frac{225}{529}a^{11}+\frac{12}{529}a^{10}-\frac{189}{529}a^{9}+\frac{170}{529}a^{8}-\frac{63}{529}a^{7}+\frac{199}{529}a^{6}-\frac{65}{529}a^{5}-\frac{18}{529}a^{4}+\frac{129}{529}a^{3}-\frac{19}{529}a^{2}-\frac{171}{529}a+\frac{5}{23}$, $\frac{1}{529}a^{37}-\frac{10}{529}a^{34}-\frac{9}{529}a^{33}+\frac{7}{529}a^{31}+\frac{6}{529}a^{30}+\frac{11}{529}a^{28}-\frac{2}{529}a^{27}+\frac{7}{529}a^{26}+\frac{4}{529}a^{25}+\frac{6}{529}a^{24}-\frac{6}{529}a^{23}+\frac{2}{529}a^{22}+\frac{6}{529}a^{21}+\frac{11}{529}a^{20}-\frac{118}{529}a^{19}+\frac{37}{529}a^{18}-\frac{149}{529}a^{17}+\frac{26}{529}a^{16}+\frac{187}{529}a^{15}-\frac{157}{529}a^{14}+\frac{175}{529}a^{13}+\frac{124}{529}a^{12}+\frac{30}{529}a^{11}+\frac{86}{529}a^{10}+\frac{143}{529}a^{9}+\frac{20}{529}a^{8}-\frac{83}{529}a^{7}-\frac{11}{23}a^{6}-\frac{208}{529}a^{5}-\frac{80}{529}a^{4}-\frac{123}{529}a^{3}+\frac{95}{529}a^{2}+\frac{260}{529}a+\frac{8}{23}$, $\frac{1}{51313}a^{38}+\frac{11}{51313}a^{37}+\frac{11}{51313}a^{36}+\frac{17}{51313}a^{35}-\frac{439}{51313}a^{34}+\frac{308}{51313}a^{33}-\frac{149}{51313}a^{32}-\frac{700}{51313}a^{31}+\frac{248}{51313}a^{30}+\frac{34}{2231}a^{29}-\frac{301}{51313}a^{28}+\frac{557}{51313}a^{27}-\frac{452}{51313}a^{26}-\frac{1017}{51313}a^{25}+\frac{379}{51313}a^{24}-\frac{236}{51313}a^{23}+\frac{990}{51313}a^{22}-\frac{1053}{51313}a^{21}-\frac{992}{51313}a^{20}+\frac{14108}{51313}a^{19}-\frac{9090}{51313}a^{18}+\frac{10563}{51313}a^{17}+\frac{20441}{51313}a^{16}+\frac{17044}{51313}a^{15}+\frac{16023}{51313}a^{14}-\frac{196}{529}a^{13}+\frac{394}{2231}a^{12}+\frac{20621}{51313}a^{11}-\frac{20257}{51313}a^{10}-\frac{21515}{51313}a^{9}+\frac{3584}{51313}a^{8}-\frac{662}{51313}a^{7}-\frac{17898}{51313}a^{6}+\frac{160}{2231}a^{5}-\frac{73}{529}a^{4}-\frac{5288}{51313}a^{3}-\frac{22140}{51313}a^{2}-\frac{19381}{51313}a+\frac{778}{2231}$, $\frac{1}{51313}a^{39}-\frac{13}{51313}a^{37}-\frac{7}{51313}a^{36}-\frac{44}{51313}a^{35}-\frac{392}{51313}a^{34}+\frac{440}{51313}a^{33}-\frac{1098}{51313}a^{32}-\frac{297}{51313}a^{31}-\frac{491}{51313}a^{30}+\frac{894}{51313}a^{29}+\frac{279}{51313}a^{28}-\frac{1050}{51313}a^{27}+\frac{172}{51313}a^{26}+\frac{1}{2231}a^{25}-\frac{1107}{51313}a^{24}-\frac{100}{51313}a^{23}+\frac{1055}{51313}a^{22}+\frac{309}{51313}a^{21}-\frac{297}{51313}a^{20}-\frac{13152}{51313}a^{19}-\frac{2355}{51313}a^{18}-\frac{22323}{51313}a^{17}-\frac{7405}{51313}a^{16}-\frac{25573}{51313}a^{15}-\frac{17367}{51313}a^{14}-\frac{21978}{51313}a^{13}-\frac{228}{2231}a^{12}+\frac{741}{2231}a^{11}-\frac{545}{51313}a^{10}-\frac{18450}{51313}a^{9}+\frac{23740}{51313}a^{8}-\frac{16824}{51313}a^{7}+\frac{1320}{51313}a^{6}-\frac{1389}{51313}a^{5}+\frac{15179}{51313}a^{4}+\frac{3436}{51313}a^{3}+\frac{20944}{51313}a^{2}-\frac{842}{51313}a-\frac{798}{2231}$, $\frac{1}{51313}a^{40}+\frac{39}{51313}a^{37}+\frac{2}{51313}a^{36}+\frac{1}{2231}a^{35}+\frac{747}{51313}a^{34}+\frac{287}{51313}a^{33}-\frac{488}{51313}a^{32}-\frac{182}{51313}a^{31}-\frac{247}{51313}a^{30}-\frac{710}{51313}a^{29}+\frac{81}{51313}a^{28}+\frac{41}{51313}a^{27}+\frac{355}{51313}a^{26}-\frac{1039}{51313}a^{25}-\frac{120}{51313}a^{24}-\frac{364}{51313}a^{23}-\frac{595}{51313}a^{22}+\frac{273}{51313}a^{21}-\frac{537}{51313}a^{20}-\frac{23233}{51313}a^{19}+\frac{22661}{51313}a^{18}-\frac{21697}{51313}a^{17}+\frac{19097}{51313}a^{16}-\frac{3666}{51313}a^{15}-\frac{7097}{51313}a^{14}-\frac{23577}{51313}a^{13}-\frac{21224}{51313}a^{12}+\frac{2815}{51313}a^{11}-\frac{7766}{51313}a^{10}-\frac{23155}{51313}a^{9}+\frac{6391}{51313}a^{8}+\frac{21426}{51313}a^{7}-\frac{487}{51313}a^{6}+\frac{3655}{51313}a^{5}+\frac{19441}{51313}a^{4}-\frac{5896}{51313}a^{3}-\frac{8526}{51313}a^{2}-\frac{23733}{51313}a+\frac{608}{2231}$, $\frac{1}{51313}a^{41}-\frac{39}{51313}a^{37}-\frac{18}{51313}a^{36}-\frac{13}{51313}a^{35}-\frac{925}{51313}a^{34}-\frac{278}{51313}a^{33}-\frac{773}{51313}a^{32}+\frac{475}{51313}a^{31}-\frac{488}{51313}a^{30}-\frac{929}{51313}a^{29}-\frac{151}{51313}a^{28}+\frac{651}{51313}a^{27}+\frac{293}{51313}a^{26}+\frac{743}{51313}a^{25}-\frac{983}{51313}a^{24}-\frac{606}{51313}a^{23}+\frac{463}{51313}a^{22}+\frac{81}{51313}a^{21}+\frac{32}{51313}a^{20}-\frac{3460}{51313}a^{19}-\frac{18715}{51313}a^{18}+\frac{21912}{51313}a^{17}+\frac{4235}{51313}a^{16}-\frac{4162}{51313}a^{15}+\frac{20147}{51313}a^{14}-\frac{22873}{51313}a^{13}+\frac{25466}{51313}a^{12}+\frac{4852}{51313}a^{11}+\frac{15021}{51313}a^{10}+\frac{19618}{51313}a^{9}-\frac{883}{51313}a^{8}-\frac{5127}{51313}a^{7}-\frac{15735}{51313}a^{6}+\frac{14825}{51313}a^{5}+\frac{11176}{51313}a^{4}-\frac{24230}{51313}a^{3}-\frac{13000}{51313}a^{2}+\frac{13049}{51313}a+\frac{19}{2231}$, $\frac{1}{51313}a^{42}+\frac{1}{2231}a^{37}+\frac{28}{51313}a^{36}+\frac{29}{51313}a^{35}-\frac{618}{51313}a^{34}+\frac{472}{51313}a^{33}-\frac{680}{51313}a^{32}-\frac{919}{51313}a^{31}-\frac{763}{51313}a^{30}-\frac{596}{51313}a^{29}+\frac{649}{51313}a^{28}+\frac{94}{51313}a^{27}+\frac{25}{2231}a^{26}-\frac{294}{51313}a^{25}-\frac{957}{51313}a^{24}-\frac{593}{51313}a^{23}-\frac{303}{51313}a^{22}+\frac{42}{2231}a^{21}-\frac{1020}{51313}a^{20}+\frac{20889}{51313}a^{19}-\frac{16475}{51313}a^{18}-\frac{21472}{51313}a^{17}-\frac{24479}{51313}a^{16}-\frac{4419}{51313}a^{15}+\frac{915}{51313}a^{14}-\frac{16438}{51313}a^{13}-\frac{11591}{51313}a^{12}+\frac{1142}{51313}a^{11}-\frac{25057}{51313}a^{10}+\frac{17900}{51313}a^{9}+\frac{12235}{51313}a^{8}-\frac{7700}{51313}a^{7}+\frac{8219}{51313}a^{6}+\frac{19187}{51313}a^{5}-\frac{23066}{51313}a^{4}+\frac{9494}{51313}a^{3}+\frac{24335}{51313}a^{2}-\frac{6679}{51313}a+\frac{660}{2231}$, $\frac{1}{29607601}a^{43}+\frac{202}{29607601}a^{42}-\frac{275}{29607601}a^{41}+\frac{221}{29607601}a^{40}-\frac{169}{29607601}a^{39}-\frac{90}{29607601}a^{38}+\frac{10325}{29607601}a^{37}+\frac{24209}{29607601}a^{36}+\frac{1856}{29607601}a^{35}-\frac{152326}{29607601}a^{34}-\frac{369614}{29607601}a^{33}+\frac{128592}{29607601}a^{32}+\frac{598679}{29607601}a^{31}+\frac{439723}{29607601}a^{30}+\frac{638075}{29607601}a^{29}-\frac{450983}{29607601}a^{28}-\frac{9508}{29607601}a^{27}-\frac{455568}{29607601}a^{26}+\frac{20648}{1287287}a^{25}-\frac{424027}{29607601}a^{24}+\frac{429307}{29607601}a^{23}+\frac{307134}{29607601}a^{22}+\frac{818}{51313}a^{21}-\frac{490906}{29607601}a^{20}+\frac{7383498}{29607601}a^{19}-\frac{5740073}{29607601}a^{18}-\frac{7475787}{29607601}a^{17}+\frac{6456762}{29607601}a^{16}-\frac{12090597}{29607601}a^{15}+\frac{693042}{29607601}a^{14}+\frac{8207541}{29607601}a^{13}-\frac{6235124}{29607601}a^{12}-\frac{3851591}{29607601}a^{11}-\frac{5771025}{29607601}a^{10}+\frac{1735}{29607601}a^{9}-\frac{14072046}{29607601}a^{8}+\frac{12522704}{29607601}a^{7}-\frac{12092399}{29607601}a^{6}-\frac{684008}{29607601}a^{5}+\frac{2543099}{29607601}a^{4}-\frac{1979882}{29607601}a^{3}+\frac{8329051}{29607601}a^{2}-\frac{2634634}{29607601}a-\frac{348734}{1287287}$, $\frac{1}{10333052749}a^{44}+\frac{8}{10333052749}a^{43}-\frac{39463}{10333052749}a^{42}-\frac{62406}{10333052749}a^{41}-\frac{94396}{10333052749}a^{40}+\frac{57507}{10333052749}a^{39}+\frac{73}{29607601}a^{38}-\frac{869847}{10333052749}a^{37}-\frac{6658798}{10333052749}a^{36}-\frac{152153}{449263163}a^{35}+\frac{205130856}{10333052749}a^{34}-\frac{173809617}{10333052749}a^{33}+\frac{111261256}{10333052749}a^{32}-\frac{90746445}{10333052749}a^{31}+\frac{47024562}{10333052749}a^{30}-\frac{82062295}{10333052749}a^{29}-\frac{6704844}{449263163}a^{28}-\frac{98181598}{10333052749}a^{27}+\frac{867757}{449263163}a^{26}-\frac{8107878}{449263163}a^{25}+\frac{68400563}{10333052749}a^{24}+\frac{176388269}{10333052749}a^{23}-\frac{114948300}{10333052749}a^{22}+\frac{8636080}{10333052749}a^{21}-\frac{177301325}{10333052749}a^{20}-\frac{4747014}{449263163}a^{19}-\frac{3977152186}{10333052749}a^{18}-\frac{4037721329}{10333052749}a^{17}+\frac{3603919715}{10333052749}a^{16}-\frac{3535923597}{10333052749}a^{15}+\frac{4516624244}{10333052749}a^{14}-\frac{3194772617}{10333052749}a^{13}+\frac{3293767944}{10333052749}a^{12}+\frac{144195868}{449263163}a^{11}-\frac{216357072}{449263163}a^{10}-\frac{4265202256}{10333052749}a^{9}-\frac{3832014357}{10333052749}a^{8}+\frac{3276649766}{10333052749}a^{7}-\frac{1368437924}{10333052749}a^{6}+\frac{22324968}{106526317}a^{5}+\frac{48589427}{449263163}a^{4}-\frac{4819321725}{10333052749}a^{3}+\frac{4306430892}{10333052749}a^{2}+\frac{2802020173}{10333052749}a-\frac{60238231}{449263163}$, $\frac{1}{14\!\cdots\!33}a^{45}-\frac{50843566}{14\!\cdots\!33}a^{44}-\frac{8219797658}{14\!\cdots\!33}a^{43}+\frac{8853321985640}{14\!\cdots\!33}a^{42}+\frac{12461523134527}{14\!\cdots\!33}a^{41}+\frac{8270961724154}{14\!\cdots\!33}a^{40}-\frac{6660767441529}{14\!\cdots\!33}a^{39}-\frac{602732747360}{14\!\cdots\!33}a^{38}-\frac{346618820570517}{14\!\cdots\!33}a^{37}+\frac{11\!\cdots\!50}{14\!\cdots\!33}a^{36}+\frac{646982985906245}{14\!\cdots\!33}a^{35}-\frac{24\!\cdots\!11}{14\!\cdots\!33}a^{34}+\frac{24\!\cdots\!35}{14\!\cdots\!33}a^{33}+\frac{20\!\cdots\!07}{14\!\cdots\!33}a^{32}+\frac{22\!\cdots\!40}{14\!\cdots\!33}a^{31}+\frac{82\!\cdots\!45}{14\!\cdots\!33}a^{30}-\frac{17\!\cdots\!55}{14\!\cdots\!33}a^{29}-\frac{19\!\cdots\!32}{14\!\cdots\!33}a^{28}-\frac{26\!\cdots\!04}{14\!\cdots\!33}a^{27}-\frac{28\!\cdots\!24}{14\!\cdots\!33}a^{26}+\frac{19\!\cdots\!50}{14\!\cdots\!33}a^{25}+\frac{18\!\cdots\!25}{14\!\cdots\!33}a^{24}+\frac{14\!\cdots\!88}{14\!\cdots\!33}a^{23}+\frac{28\!\cdots\!88}{14\!\cdots\!33}a^{22}-\frac{91\!\cdots\!60}{14\!\cdots\!33}a^{21}-\frac{29\!\cdots\!06}{14\!\cdots\!33}a^{20}-\frac{55\!\cdots\!52}{14\!\cdots\!33}a^{19}-\frac{62\!\cdots\!46}{14\!\cdots\!33}a^{18}+\frac{50\!\cdots\!92}{14\!\cdots\!33}a^{17}-\frac{87\!\cdots\!33}{14\!\cdots\!33}a^{16}+\frac{51\!\cdots\!59}{14\!\cdots\!33}a^{15}-\frac{67\!\cdots\!23}{14\!\cdots\!33}a^{14}-\frac{42\!\cdots\!12}{14\!\cdots\!33}a^{13}+\frac{30\!\cdots\!84}{14\!\cdots\!33}a^{12}-\frac{49\!\cdots\!28}{14\!\cdots\!33}a^{11}+\frac{19\!\cdots\!71}{14\!\cdots\!33}a^{10}+\frac{44\!\cdots\!58}{14\!\cdots\!33}a^{9}-\frac{23\!\cdots\!69}{14\!\cdots\!33}a^{8}+\frac{19\!\cdots\!02}{14\!\cdots\!33}a^{7}-\frac{15\!\cdots\!71}{14\!\cdots\!33}a^{6}-\frac{33\!\cdots\!96}{14\!\cdots\!33}a^{5}-\frac{62\!\cdots\!32}{14\!\cdots\!33}a^{4}+\frac{11\!\cdots\!90}{14\!\cdots\!33}a^{3}-\frac{52\!\cdots\!38}{14\!\cdots\!33}a^{2}+\frac{32\!\cdots\!05}{14\!\cdots\!33}a+\frac{28\!\cdots\!75}{60\!\cdots\!71}$, $\frac{1}{24\!\cdots\!99}a^{46}+\frac{77\!\cdots\!85}{24\!\cdots\!99}a^{45}+\frac{93\!\cdots\!02}{24\!\cdots\!99}a^{44}-\frac{40\!\cdots\!95}{24\!\cdots\!99}a^{43}+\frac{10\!\cdots\!45}{24\!\cdots\!99}a^{42}-\frac{10\!\cdots\!72}{24\!\cdots\!99}a^{41}+\frac{22\!\cdots\!35}{24\!\cdots\!99}a^{40}+\frac{12\!\cdots\!00}{24\!\cdots\!99}a^{39}-\frac{16\!\cdots\!83}{24\!\cdots\!99}a^{38}-\frac{89\!\cdots\!28}{10\!\cdots\!13}a^{37}+\frac{79\!\cdots\!98}{24\!\cdots\!99}a^{36}-\frac{66\!\cdots\!80}{24\!\cdots\!99}a^{35}-\frac{34\!\cdots\!92}{24\!\cdots\!99}a^{34}+\frac{16\!\cdots\!50}{24\!\cdots\!99}a^{33}-\frac{11\!\cdots\!39}{24\!\cdots\!99}a^{32}-\frac{33\!\cdots\!24}{24\!\cdots\!99}a^{31}+\frac{49\!\cdots\!51}{24\!\cdots\!99}a^{30}+\frac{27\!\cdots\!19}{24\!\cdots\!99}a^{29}-\frac{65\!\cdots\!83}{24\!\cdots\!99}a^{28}+\frac{40\!\cdots\!79}{24\!\cdots\!99}a^{27}-\frac{38\!\cdots\!22}{24\!\cdots\!99}a^{26}-\frac{72\!\cdots\!23}{24\!\cdots\!99}a^{25}-\frac{45\!\cdots\!63}{24\!\cdots\!99}a^{24}+\frac{19\!\cdots\!95}{24\!\cdots\!99}a^{23}-\frac{21\!\cdots\!73}{24\!\cdots\!99}a^{22}-\frac{37\!\cdots\!23}{24\!\cdots\!99}a^{21}-\frac{15\!\cdots\!27}{24\!\cdots\!99}a^{20}+\frac{43\!\cdots\!18}{10\!\cdots\!13}a^{19}-\frac{13\!\cdots\!63}{24\!\cdots\!99}a^{18}+\frac{95\!\cdots\!59}{24\!\cdots\!99}a^{17}-\frac{10\!\cdots\!38}{24\!\cdots\!99}a^{16}+\frac{59\!\cdots\!82}{24\!\cdots\!99}a^{15}-\frac{11\!\cdots\!76}{24\!\cdots\!99}a^{14}-\frac{76\!\cdots\!13}{24\!\cdots\!99}a^{13}-\frac{75\!\cdots\!01}{24\!\cdots\!99}a^{12}+\frac{53\!\cdots\!70}{24\!\cdots\!99}a^{11}-\frac{49\!\cdots\!08}{24\!\cdots\!99}a^{10}-\frac{89\!\cdots\!28}{24\!\cdots\!99}a^{9}-\frac{52\!\cdots\!67}{24\!\cdots\!99}a^{8}-\frac{79\!\cdots\!84}{24\!\cdots\!99}a^{7}-\frac{11\!\cdots\!36}{24\!\cdots\!99}a^{6}-\frac{38\!\cdots\!82}{24\!\cdots\!99}a^{5}+\frac{12\!\cdots\!88}{10\!\cdots\!13}a^{4}+\frac{22\!\cdots\!52}{24\!\cdots\!99}a^{3}-\frac{16\!\cdots\!05}{24\!\cdots\!99}a^{2}-\frac{77\!\cdots\!18}{24\!\cdots\!99}a-\frac{19\!\cdots\!93}{10\!\cdots\!13}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $23$ |
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 | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | ${\href{/padicField/23.1.0.1}{1} }^{47}$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ | $47$ |
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 |
---|---|---|---|---|---|---|---|
\(941\) | Deg $47$ | $47$ | $1$ | $46$ |