Number field 32.0.24788700386255228556692600250244140625.1

• Label: 32.0.247...625.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $2.479\times 10^{37}$
• Root discriminant: \(14.74\)
• Ramified primes: $5,13$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_8.A_4$ (as 32T402)

Normalized defining polynomial

x^32 - 2*x^31 + x^29 + 14*x^28 - 23*x^27 - 25*x^26 + 77*x^25 + 32*x^24 - 233*x^23 + 130*x^22 + 275*x^21 - 204*x^20 - 500*x^19 + 369*x^18 + 976*x^17 - 775*x^16 - 1988*x^15 + 2706*x^14 + 1255*x^13 - 4074*x^12 + 870*x^11 + 3435*x^10 - 2784*x^9 - 528*x^8 + 1624*x^7 - 465*x^6 - 564*x^5 + 639*x^4 - 293*x^3 + 55*x^2 - x + 1

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(24788700386255228556692600250244140625\) \(\medspace = 5^{28}\cdot 13^{16}\)
Root discriminant:		\(14.74\)
Galois root discriminant:		$5^{7/8}13^{2/3}\approx 22.606204673819228$
Ramified primes:		\(5\), \(13\)
Discriminant root field:		\(\Q\)
$\card{ \Aut(K/\Q) }$:		$8$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{421}a^{30}+\frac{189}{421}a^{29}-\frac{50}{421}a^{28}-\frac{39}{421}a^{27}-\frac{181}{421}a^{26}-\frac{190}{421}a^{25}+\frac{99}{421}a^{24}+\frac{157}{421}a^{23}-\frac{123}{421}a^{22}-\frac{42}{421}a^{21}-\frac{168}{421}a^{20}-\frac{106}{421}a^{19}-\frac{135}{421}a^{18}+\frac{90}{421}a^{17}+\frac{181}{421}a^{16}-\frac{160}{421}a^{15}+\frac{32}{421}a^{14}+\frac{56}{421}a^{13}+\frac{70}{421}a^{12}+\frac{135}{421}a^{11}+\frac{20}{421}a^{10}+\frac{176}{421}a^{9}-\frac{120}{421}a^{8}-\frac{95}{421}a^{7}+\frac{168}{421}a^{6}+\frac{90}{421}a^{5}+\frac{199}{421}a^{4}+\frac{54}{421}a^{3}-\frac{17}{421}a^{2}-\frac{41}{421}a+\frac{96}{421}$, $\frac{1}{51\!\cdots\!29}a^{31}+\frac{43\!\cdots\!58}{51\!\cdots\!29}a^{30}+\frac{27\!\cdots\!97}{51\!\cdots\!29}a^{29}-\frac{46\!\cdots\!25}{12\!\cdots\!49}a^{28}+\frac{24\!\cdots\!76}{51\!\cdots\!29}a^{27}+\frac{21\!\cdots\!70}{51\!\cdots\!29}a^{26}+\frac{23\!\cdots\!61}{51\!\cdots\!29}a^{25}+\frac{88\!\cdots\!62}{51\!\cdots\!29}a^{24}+\frac{56\!\cdots\!12}{51\!\cdots\!29}a^{23}-\frac{17\!\cdots\!09}{51\!\cdots\!29}a^{22}+\frac{65\!\cdots\!62}{51\!\cdots\!29}a^{21}-\frac{91\!\cdots\!43}{51\!\cdots\!29}a^{20}+\frac{11\!\cdots\!78}{47\!\cdots\!81}a^{19}+\frac{21\!\cdots\!42}{51\!\cdots\!29}a^{18}+\frac{25\!\cdots\!02}{51\!\cdots\!29}a^{17}+\frac{15\!\cdots\!39}{51\!\cdots\!29}a^{16}+\frac{21\!\cdots\!56}{51\!\cdots\!29}a^{15}-\frac{42\!\cdots\!76}{51\!\cdots\!29}a^{14}-\frac{21\!\cdots\!32}{51\!\cdots\!29}a^{13}-\frac{16\!\cdots\!72}{51\!\cdots\!29}a^{12}-\frac{11\!\cdots\!79}{51\!\cdots\!29}a^{11}-\frac{14\!\cdots\!12}{51\!\cdots\!29}a^{10}-\frac{17\!\cdots\!99}{51\!\cdots\!29}a^{9}+\frac{12\!\cdots\!12}{51\!\cdots\!29}a^{8}+\frac{16\!\cdots\!15}{51\!\cdots\!29}a^{7}+\frac{16\!\cdots\!63}{51\!\cdots\!29}a^{6}-\frac{18\!\cdots\!27}{51\!\cdots\!29}a^{5}-\frac{21\!\cdots\!13}{51\!\cdots\!29}a^{4}+\frac{35\!\cdots\!47}{51\!\cdots\!29}a^{3}-\frac{60\!\cdots\!38}{51\!\cdots\!29}a^{2}+\frac{46\!\cdots\!13}{51\!\cdots\!29}a+\frac{14\!\cdots\!58}{51\!\cdots\!29}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( -\frac{6423504503897763274593831789201417}{13582138582017456975740811356110849} a^{31} + \frac{7178582452064768517573258172256749}{13582138582017456975740811356110849} a^{30} + \frac{7067206241421796320356437201706777}{13582138582017456975740811356110849} a^{29} - \frac{1387658770165721849722628031402285}{13582138582017456975740811356110849} a^{28} - \frac{91687916282481827262138240324224439}{13582138582017456975740811356110849} a^{27} + \frac{67505959129681174008770660844121277}{13582138582017456975740811356110849} a^{26} + \frac{230816759281668032679765042160439415}{13582138582017456975740811356110849} a^{25} - \frac{303917237527655503407448652478540044}{13582138582017456975740811356110849} a^{24} - \frac{498333409341672780997534407394437763}{13582138582017456975740811356110849} a^{23} + \frac{1105613185044314640357390839560857893}{13582138582017456975740811356110849} a^{22} + \frac{185939608096098732574143187822123175}{13582138582017456975740811356110849} a^{21} - \frac{1761136541240886568117589419166090260}{13582138582017456975740811356110849} a^{20} - \frac{1975658090272192291210630468746397}{124606775981811531887530379413861} a^{19} + \frac{3250542113034486814507257709811605278}{13582138582017456975740811356110849} a^{18} + \frac{438925690650856033356595708791793706}{13582138582017456975740811356110849} a^{17} - \frac{6297357585930183706348102420950697120}{13582138582017456975740811356110849} a^{16} - \frac{472959949491997546585316064864329249}{13582138582017456975740811356110849} a^{15} + \frac{13162977646406243048540825239190696596}{13582138582017456975740811356110849} a^{14} - \frac{6016079853381329578031803061953102364}{13582138582017456975740811356110849} a^{13} - \frac{15029858183900166867855796226263769568}{13582138582017456975740811356110849} a^{12} + \frac{14248454465299087378515964043834627856}{13582138582017456975740811356110849} a^{11} + \frac{8617334512911968181073810535918181761}{13582138582017456975740811356110849} a^{10} - \frac{16888081068548984322439802448482682320}{13582138582017456975740811356110849} a^{9} + \frac{2468421839294359809519159076309269927}{13582138582017456975740811356110849} a^{8} + \frac{8022833735917993911655954850849516188}{13582138582017456975740811356110849} a^{7} - \frac{4285955043396325216690318456045600163}{13582138582017456975740811356110849} a^{6} - \frac{1687070555771401240397734324093192233}{13582138582017456975740811356110849} a^{5} + \frac{2913089983939363212288096549397022023}{13582138582017456975740811356110849} a^{4} - \frac{1487993229528095286120908123732764408}{13582138582017456975740811356110849} a^{3} + \frac{195403714016035807910534405552108474}{13582138582017456975740811356110849} a^{2} + \frac{97567936121317121148810131759033208}{13582138582017456975740811356110849} a + \frac{3023404456381992601855054844715558}{13582138582017456975740811356110849} \)  (order $10$)
Fundamental units:		$\frac{51\!\cdots\!21}{51\!\cdots\!29}a^{31}-\frac{58\!\cdots\!99}{51\!\cdots\!29}a^{30}-\frac{53\!\cdots\!30}{51\!\cdots\!29}a^{29}+\frac{96\!\cdots\!05}{51\!\cdots\!29}a^{28}+\frac{72\!\cdots\!35}{51\!\cdots\!29}a^{27}-\frac{55\!\cdots\!23}{51\!\cdots\!29}a^{26}-\frac{18\!\cdots\!34}{51\!\cdots\!29}a^{25}+\frac{24\!\cdots\!68}{51\!\cdots\!29}a^{24}+\frac{38\!\cdots\!59}{51\!\cdots\!29}a^{23}-\frac{87\!\cdots\!96}{51\!\cdots\!29}a^{22}-\frac{11\!\cdots\!66}{51\!\cdots\!29}a^{21}+\frac{13\!\cdots\!84}{51\!\cdots\!29}a^{20}+\frac{12\!\cdots\!20}{47\!\cdots\!81}a^{19}-\frac{25\!\cdots\!23}{51\!\cdots\!29}a^{18}-\frac{29\!\cdots\!30}{51\!\cdots\!29}a^{17}+\frac{49\!\cdots\!90}{51\!\cdots\!29}a^{16}+\frac{27\!\cdots\!40}{51\!\cdots\!29}a^{15}-\frac{10\!\cdots\!57}{51\!\cdots\!29}a^{14}+\frac{49\!\cdots\!87}{51\!\cdots\!29}a^{13}+\frac{11\!\cdots\!69}{51\!\cdots\!29}a^{12}-\frac{11\!\cdots\!80}{51\!\cdots\!29}a^{11}-\frac{61\!\cdots\!62}{51\!\cdots\!29}a^{10}+\frac{13\!\cdots\!61}{51\!\cdots\!29}a^{9}-\frac{24\!\cdots\!99}{51\!\cdots\!29}a^{8}-\frac{58\!\cdots\!47}{51\!\cdots\!29}a^{7}+\frac{34\!\cdots\!58}{51\!\cdots\!29}a^{6}+\frac{10\!\cdots\!17}{51\!\cdots\!29}a^{5}-\frac{22\!\cdots\!24}{51\!\cdots\!29}a^{4}+\frac{12\!\cdots\!67}{51\!\cdots\!29}a^{3}-\frac{23\!\cdots\!23}{51\!\cdots\!29}a^{2}-\frac{22\!\cdots\!25}{51\!\cdots\!29}a-\frac{11\!\cdots\!30}{51\!\cdots\!29}$, $\frac{52\!\cdots\!20}{51\!\cdots\!29}a^{31}-\frac{63\!\cdots\!61}{51\!\cdots\!29}a^{30}-\frac{48\!\cdots\!16}{51\!\cdots\!29}a^{29}+\frac{12\!\cdots\!55}{51\!\cdots\!29}a^{28}+\frac{73\!\cdots\!56}{51\!\cdots\!29}a^{27}-\frac{61\!\cdots\!85}{51\!\cdots\!29}a^{26}-\frac{17\!\cdots\!96}{51\!\cdots\!29}a^{25}+\frac{26\!\cdots\!35}{51\!\cdots\!29}a^{24}+\frac{36\!\cdots\!04}{51\!\cdots\!29}a^{23}-\frac{91\!\cdots\!47}{51\!\cdots\!29}a^{22}-\frac{33\!\cdots\!19}{51\!\cdots\!29}a^{21}+\frac{13\!\cdots\!48}{51\!\cdots\!29}a^{20}+\frac{20\!\cdots\!90}{47\!\cdots\!81}a^{19}-\frac{25\!\cdots\!41}{51\!\cdots\!29}a^{18}-\frac{80\!\cdots\!52}{51\!\cdots\!29}a^{17}+\frac{49\!\cdots\!74}{51\!\cdots\!29}a^{16}-\frac{15\!\cdots\!51}{51\!\cdots\!29}a^{15}-\frac{10\!\cdots\!96}{51\!\cdots\!29}a^{14}+\frac{59\!\cdots\!24}{51\!\cdots\!29}a^{13}+\frac{10\!\cdots\!03}{51\!\cdots\!29}a^{12}-\frac{12\!\cdots\!02}{51\!\cdots\!29}a^{11}-\frac{49\!\cdots\!85}{51\!\cdots\!29}a^{10}+\frac{13\!\cdots\!66}{51\!\cdots\!29}a^{9}-\frac{38\!\cdots\!45}{51\!\cdots\!29}a^{8}-\frac{54\!\cdots\!83}{51\!\cdots\!29}a^{7}+\frac{40\!\cdots\!80}{51\!\cdots\!29}a^{6}+\frac{62\!\cdots\!74}{51\!\cdots\!29}a^{5}-\frac{22\!\cdots\!98}{51\!\cdots\!29}a^{4}+\frac{15\!\cdots\!40}{51\!\cdots\!29}a^{3}-\frac{41\!\cdots\!95}{51\!\cdots\!29}a^{2}+\frac{24\!\cdots\!07}{51\!\cdots\!29}a-\frac{58\!\cdots\!48}{51\!\cdots\!29}$, $\frac{64\!\cdots\!17}{13\!\cdots\!49}a^{31}-\frac{71\!\cdots\!49}{13\!\cdots\!49}a^{30}-\frac{70\!\cdots\!77}{13\!\cdots\!49}a^{29}+\frac{13\!\cdots\!85}{13\!\cdots\!49}a^{28}+\frac{91\!\cdots\!39}{13\!\cdots\!49}a^{27}-\frac{67\!\cdots\!77}{13\!\cdots\!49}a^{26}-\frac{23\!\cdots\!15}{13\!\cdots\!49}a^{25}+\frac{30\!\cdots\!44}{13\!\cdots\!49}a^{24}+\frac{49\!\cdots\!63}{13\!\cdots\!49}a^{23}-\frac{11\!\cdots\!93}{13\!\cdots\!49}a^{22}-\frac{18\!\cdots\!75}{13\!\cdots\!49}a^{21}+\frac{17\!\cdots\!60}{13\!\cdots\!49}a^{20}+\frac{19\!\cdots\!97}{12\!\cdots\!61}a^{19}-\frac{32\!\cdots\!78}{13\!\cdots\!49}a^{18}-\frac{43\!\cdots\!06}{13\!\cdots\!49}a^{17}+\frac{62\!\cdots\!20}{13\!\cdots\!49}a^{16}+\frac{47\!\cdots\!49}{13\!\cdots\!49}a^{15}-\frac{13\!\cdots\!96}{13\!\cdots\!49}a^{14}+\frac{60\!\cdots\!64}{13\!\cdots\!49}a^{13}+\frac{15\!\cdots\!68}{13\!\cdots\!49}a^{12}-\frac{14\!\cdots\!56}{13\!\cdots\!49}a^{11}-\frac{86\!\cdots\!61}{13\!\cdots\!49}a^{10}+\frac{16\!\cdots\!20}{13\!\cdots\!49}a^{9}-\frac{24\!\cdots\!27}{13\!\cdots\!49}a^{8}-\frac{80\!\cdots\!88}{13\!\cdots\!49}a^{7}+\frac{42\!\cdots\!63}{13\!\cdots\!49}a^{6}+\frac{16\!\cdots\!33}{13\!\cdots\!49}a^{5}-\frac{29\!\cdots\!23}{13\!\cdots\!49}a^{4}+\frac{14\!\cdots\!08}{13\!\cdots\!49}a^{3}-\frac{19\!\cdots\!74}{13\!\cdots\!49}a^{2}-\frac{97\!\cdots\!08}{13\!\cdots\!49}a+\frac{10\!\cdots\!91}{13\!\cdots\!49}$, $\frac{11\!\cdots\!05}{51\!\cdots\!29}a^{31}-\frac{12\!\cdots\!75}{51\!\cdots\!29}a^{30}-\frac{10\!\cdots\!74}{51\!\cdots\!29}a^{29}+\frac{79\!\cdots\!58}{51\!\cdots\!29}a^{28}+\frac{16\!\cdots\!41}{51\!\cdots\!29}a^{27}-\frac{11\!\cdots\!90}{51\!\cdots\!29}a^{26}-\frac{38\!\cdots\!00}{51\!\cdots\!29}a^{25}+\frac{52\!\cdots\!83}{51\!\cdots\!29}a^{24}+\frac{82\!\cdots\!65}{51\!\cdots\!29}a^{23}-\frac{19\!\cdots\!30}{51\!\cdots\!29}a^{22}-\frac{15\!\cdots\!08}{51\!\cdots\!29}a^{21}+\frac{28\!\cdots\!22}{51\!\cdots\!29}a^{20}+\frac{22\!\cdots\!26}{47\!\cdots\!81}a^{19}-\frac{52\!\cdots\!61}{51\!\cdots\!29}a^{18}-\frac{58\!\cdots\!34}{51\!\cdots\!29}a^{17}+\frac{10\!\cdots\!40}{51\!\cdots\!29}a^{16}+\frac{41\!\cdots\!49}{51\!\cdots\!29}a^{15}-\frac{21\!\cdots\!82}{51\!\cdots\!29}a^{14}+\frac{11\!\cdots\!10}{51\!\cdots\!29}a^{13}+\frac{22\!\cdots\!41}{51\!\cdots\!29}a^{12}-\frac{59\!\cdots\!57}{12\!\cdots\!49}a^{11}-\frac{10\!\cdots\!45}{51\!\cdots\!29}a^{10}+\frac{27\!\cdots\!47}{51\!\cdots\!29}a^{9}-\frac{83\!\cdots\!92}{51\!\cdots\!29}a^{8}-\frac{10\!\cdots\!57}{51\!\cdots\!29}a^{7}+\frac{84\!\cdots\!96}{51\!\cdots\!29}a^{6}+\frac{12\!\cdots\!21}{51\!\cdots\!29}a^{5}-\frac{49\!\cdots\!25}{51\!\cdots\!29}a^{4}+\frac{30\!\cdots\!48}{51\!\cdots\!29}a^{3}-\frac{89\!\cdots\!48}{51\!\cdots\!29}a^{2}+\frac{17\!\cdots\!22}{51\!\cdots\!29}a-\frac{65\!\cdots\!87}{51\!\cdots\!29}$, $\frac{21\!\cdots\!68}{51\!\cdots\!29}a^{31}-\frac{25\!\cdots\!17}{51\!\cdots\!29}a^{30}-\frac{20\!\cdots\!21}{51\!\cdots\!29}a^{29}+\frac{35\!\cdots\!41}{51\!\cdots\!29}a^{28}+\frac{30\!\cdots\!97}{51\!\cdots\!29}a^{27}-\frac{24\!\cdots\!79}{51\!\cdots\!29}a^{26}-\frac{73\!\cdots\!89}{51\!\cdots\!29}a^{25}+\frac{10\!\cdots\!94}{51\!\cdots\!29}a^{24}+\frac{15\!\cdots\!84}{51\!\cdots\!29}a^{23}-\frac{37\!\cdots\!98}{51\!\cdots\!29}a^{22}-\frac{22\!\cdots\!77}{51\!\cdots\!29}a^{21}+\frac{56\!\cdots\!98}{51\!\cdots\!29}a^{20}+\frac{22\!\cdots\!43}{47\!\cdots\!81}a^{19}-\frac{10\!\cdots\!41}{51\!\cdots\!29}a^{18}-\frac{65\!\cdots\!10}{51\!\cdots\!29}a^{17}+\frac{20\!\cdots\!99}{51\!\cdots\!29}a^{16}-\frac{23\!\cdots\!94}{51\!\cdots\!29}a^{15}-\frac{42\!\cdots\!54}{51\!\cdots\!29}a^{14}+\frac{23\!\cdots\!66}{51\!\cdots\!29}a^{13}+\frac{45\!\cdots\!51}{51\!\cdots\!29}a^{12}-\frac{50\!\cdots\!61}{51\!\cdots\!29}a^{11}-\frac{20\!\cdots\!89}{51\!\cdots\!29}a^{10}+\frac{55\!\cdots\!65}{51\!\cdots\!29}a^{9}-\frac{15\!\cdots\!25}{51\!\cdots\!29}a^{8}-\frac{22\!\cdots\!93}{51\!\cdots\!29}a^{7}+\frac{16\!\cdots\!61}{51\!\cdots\!29}a^{6}+\frac{28\!\cdots\!02}{51\!\cdots\!29}a^{5}-\frac{97\!\cdots\!09}{51\!\cdots\!29}a^{4}+\frac{60\!\cdots\!72}{51\!\cdots\!29}a^{3}-\frac{15\!\cdots\!32}{51\!\cdots\!29}a^{2}+\frac{83\!\cdots\!79}{51\!\cdots\!29}a-\frac{42\!\cdots\!62}{51\!\cdots\!29}$, $\frac{38\!\cdots\!57}{51\!\cdots\!29}a^{31}-\frac{36\!\cdots\!95}{51\!\cdots\!29}a^{30}-\frac{43\!\cdots\!06}{51\!\cdots\!29}a^{29}-\frac{16\!\cdots\!21}{51\!\cdots\!29}a^{28}+\frac{53\!\cdots\!48}{51\!\cdots\!29}a^{27}-\frac{31\!\cdots\!84}{51\!\cdots\!29}a^{26}-\frac{13\!\cdots\!06}{51\!\cdots\!29}a^{25}+\frac{15\!\cdots\!35}{51\!\cdots\!29}a^{24}+\frac{30\!\cdots\!31}{51\!\cdots\!29}a^{23}-\frac{59\!\cdots\!30}{51\!\cdots\!29}a^{22}-\frac{16\!\cdots\!84}{51\!\cdots\!29}a^{21}+\frac{96\!\cdots\!95}{51\!\cdots\!29}a^{20}+\frac{22\!\cdots\!34}{47\!\cdots\!81}a^{19}-\frac{17\!\cdots\!30}{51\!\cdots\!29}a^{18}-\frac{48\!\cdots\!46}{51\!\cdots\!29}a^{17}+\frac{34\!\cdots\!55}{51\!\cdots\!29}a^{16}+\frac{71\!\cdots\!71}{51\!\cdots\!29}a^{15}-\frac{72\!\cdots\!68}{51\!\cdots\!29}a^{14}+\frac{26\!\cdots\!16}{51\!\cdots\!29}a^{13}+\frac{85\!\cdots\!18}{51\!\cdots\!29}a^{12}-\frac{70\!\cdots\!95}{51\!\cdots\!29}a^{11}-\frac{51\!\cdots\!14}{51\!\cdots\!29}a^{10}+\frac{87\!\cdots\!48}{51\!\cdots\!29}a^{9}-\frac{89\!\cdots\!19}{51\!\cdots\!29}a^{8}-\frac{41\!\cdots\!27}{51\!\cdots\!29}a^{7}+\frac{20\!\cdots\!98}{51\!\cdots\!29}a^{6}+\frac{91\!\cdots\!59}{51\!\cdots\!29}a^{5}-\frac{15\!\cdots\!92}{51\!\cdots\!29}a^{4}+\frac{77\!\cdots\!59}{51\!\cdots\!29}a^{3}-\frac{96\!\cdots\!34}{51\!\cdots\!29}a^{2}-\frac{39\!\cdots\!01}{51\!\cdots\!29}a+\frac{25\!\cdots\!73}{51\!\cdots\!29}$, $\frac{24\!\cdots\!71}{51\!\cdots\!29}a^{31}-\frac{23\!\cdots\!89}{51\!\cdots\!29}a^{30}-\frac{29\!\cdots\!24}{51\!\cdots\!29}a^{29}-\frac{28\!\cdots\!69}{51\!\cdots\!29}a^{28}+\frac{35\!\cdots\!38}{51\!\cdots\!29}a^{27}-\frac{19\!\cdots\!42}{51\!\cdots\!29}a^{26}-\frac{90\!\cdots\!96}{51\!\cdots\!29}a^{25}+\frac{98\!\cdots\!42}{51\!\cdots\!29}a^{24}+\frac{20\!\cdots\!21}{51\!\cdots\!29}a^{23}-\frac{38\!\cdots\!34}{51\!\cdots\!29}a^{22}-\frac{12\!\cdots\!98}{51\!\cdots\!29}a^{21}+\frac{62\!\cdots\!24}{51\!\cdots\!29}a^{20}+\frac{17\!\cdots\!50}{47\!\cdots\!81}a^{19}-\frac{11\!\cdots\!33}{51\!\cdots\!29}a^{18}-\frac{37\!\cdots\!91}{51\!\cdots\!29}a^{17}+\frac{22\!\cdots\!36}{51\!\cdots\!29}a^{16}+\frac{57\!\cdots\!47}{51\!\cdots\!29}a^{15}-\frac{47\!\cdots\!53}{51\!\cdots\!29}a^{14}+\frac{14\!\cdots\!21}{51\!\cdots\!29}a^{13}+\frac{56\!\cdots\!43}{51\!\cdots\!29}a^{12}-\frac{43\!\cdots\!62}{51\!\cdots\!29}a^{11}-\frac{35\!\cdots\!65}{51\!\cdots\!29}a^{10}+\frac{54\!\cdots\!88}{51\!\cdots\!29}a^{9}-\frac{36\!\cdots\!14}{51\!\cdots\!29}a^{8}-\frac{26\!\cdots\!72}{51\!\cdots\!29}a^{7}+\frac{12\!\cdots\!27}{51\!\cdots\!29}a^{6}+\frac{60\!\cdots\!27}{51\!\cdots\!29}a^{5}-\frac{95\!\cdots\!11}{51\!\cdots\!29}a^{4}+\frac{48\!\cdots\!00}{51\!\cdots\!29}a^{3}-\frac{65\!\cdots\!40}{51\!\cdots\!29}a^{2}-\frac{20\!\cdots\!94}{51\!\cdots\!29}a-\frac{27\!\cdots\!08}{51\!\cdots\!29}$, $\frac{17\!\cdots\!98}{51\!\cdots\!29}a^{31}-\frac{28\!\cdots\!58}{51\!\cdots\!29}a^{30}-\frac{85\!\cdots\!21}{51\!\cdots\!29}a^{29}+\frac{10\!\cdots\!96}{51\!\cdots\!29}a^{28}+\frac{24\!\cdots\!20}{51\!\cdots\!29}a^{27}-\frac{30\!\cdots\!31}{51\!\cdots\!29}a^{26}-\frac{52\!\cdots\!80}{51\!\cdots\!29}a^{25}+\frac{11\!\cdots\!89}{51\!\cdots\!29}a^{24}+\frac{91\!\cdots\!91}{51\!\cdots\!29}a^{23}-\frac{35\!\cdots\!62}{51\!\cdots\!29}a^{22}+\frac{10\!\cdots\!32}{51\!\cdots\!29}a^{21}+\frac{47\!\cdots\!28}{51\!\cdots\!29}a^{20}-\frac{15\!\cdots\!22}{47\!\cdots\!81}a^{19}-\frac{87\!\cdots\!21}{51\!\cdots\!29}a^{18}+\frac{29\!\cdots\!29}{51\!\cdots\!29}a^{17}+\frac{17\!\cdots\!84}{51\!\cdots\!29}a^{16}-\frac{67\!\cdots\!40}{51\!\cdots\!29}a^{15}-\frac{35\!\cdots\!27}{51\!\cdots\!29}a^{14}+\frac{33\!\cdots\!83}{51\!\cdots\!29}a^{13}+\frac{29\!\cdots\!25}{51\!\cdots\!29}a^{12}-\frac{55\!\cdots\!64}{51\!\cdots\!29}a^{11}-\frac{16\!\cdots\!85}{51\!\cdots\!29}a^{10}+\frac{52\!\cdots\!70}{51\!\cdots\!29}a^{9}-\frac{29\!\cdots\!56}{51\!\cdots\!29}a^{8}-\frac{13\!\cdots\!25}{51\!\cdots\!29}a^{7}+\frac{19\!\cdots\!33}{51\!\cdots\!29}a^{6}-\frac{30\!\cdots\!57}{51\!\cdots\!29}a^{5}-\frac{82\!\cdots\!34}{51\!\cdots\!29}a^{4}+\frac{80\!\cdots\!35}{51\!\cdots\!29}a^{3}-\frac{33\!\cdots\!66}{51\!\cdots\!29}a^{2}+\frac{62\!\cdots\!59}{51\!\cdots\!29}a-\frac{99\!\cdots\!92}{51\!\cdots\!29}$, $\frac{49\!\cdots\!23}{51\!\cdots\!29}a^{31}-\frac{53\!\cdots\!20}{51\!\cdots\!29}a^{30}-\frac{48\!\cdots\!45}{51\!\cdots\!29}a^{29}+\frac{21\!\cdots\!85}{51\!\cdots\!29}a^{28}+\frac{69\!\cdots\!98}{51\!\cdots\!29}a^{27}-\frac{49\!\cdots\!22}{51\!\cdots\!29}a^{26}-\frac{16\!\cdots\!54}{51\!\cdots\!29}a^{25}+\frac{22\!\cdots\!09}{51\!\cdots\!29}a^{24}+\frac{36\!\cdots\!20}{51\!\cdots\!29}a^{23}-\frac{80\!\cdots\!08}{51\!\cdots\!29}a^{22}-\frac{10\!\cdots\!76}{51\!\cdots\!29}a^{21}+\frac{12\!\cdots\!76}{51\!\cdots\!29}a^{20}+\frac{13\!\cdots\!83}{47\!\cdots\!81}a^{19}-\frac{22\!\cdots\!95}{51\!\cdots\!29}a^{18}-\frac{32\!\cdots\!85}{51\!\cdots\!29}a^{17}+\frac{44\!\cdots\!01}{51\!\cdots\!29}a^{16}+\frac{33\!\cdots\!92}{51\!\cdots\!29}a^{15}-\frac{93\!\cdots\!72}{51\!\cdots\!29}a^{14}+\frac{46\!\cdots\!72}{51\!\cdots\!29}a^{13}+\frac{10\!\cdots\!50}{51\!\cdots\!29}a^{12}-\frac{10\!\cdots\!27}{51\!\cdots\!29}a^{11}-\frac{50\!\cdots\!51}{51\!\cdots\!29}a^{10}+\frac{11\!\cdots\!50}{51\!\cdots\!29}a^{9}-\frac{29\!\cdots\!40}{51\!\cdots\!29}a^{8}-\frac{48\!\cdots\!94}{51\!\cdots\!29}a^{7}+\frac{34\!\cdots\!42}{51\!\cdots\!29}a^{6}+\frac{64\!\cdots\!40}{51\!\cdots\!29}a^{5}-\frac{20\!\cdots\!99}{51\!\cdots\!29}a^{4}+\frac{12\!\cdots\!73}{51\!\cdots\!29}a^{3}-\frac{31\!\cdots\!08}{51\!\cdots\!29}a^{2}-\frac{13\!\cdots\!69}{51\!\cdots\!29}a-\frac{15\!\cdots\!30}{51\!\cdots\!29}$, $\frac{15\!\cdots\!92}{51\!\cdots\!29}a^{31}-\frac{96\!\cdots\!62}{51\!\cdots\!29}a^{30}-\frac{28\!\cdots\!62}{51\!\cdots\!29}a^{29}-\frac{44\!\cdots\!03}{51\!\cdots\!29}a^{28}+\frac{23\!\cdots\!26}{51\!\cdots\!29}a^{27}-\frac{50\!\cdots\!96}{51\!\cdots\!29}a^{26}-\frac{68\!\cdots\!14}{51\!\cdots\!29}a^{25}+\frac{46\!\cdots\!71}{51\!\cdots\!29}a^{24}+\frac{16\!\cdots\!61}{51\!\cdots\!29}a^{23}-\frac{21\!\cdots\!00}{51\!\cdots\!29}a^{22}-\frac{19\!\cdots\!09}{51\!\cdots\!29}a^{21}+\frac{43\!\cdots\!44}{51\!\cdots\!29}a^{20}+\frac{25\!\cdots\!94}{47\!\cdots\!81}a^{19}-\frac{81\!\cdots\!05}{51\!\cdots\!29}a^{18}-\frac{53\!\cdots\!62}{51\!\cdots\!29}a^{17}+\frac{15\!\cdots\!68}{51\!\cdots\!29}a^{16}+\frac{93\!\cdots\!29}{51\!\cdots\!29}a^{15}-\frac{33\!\cdots\!40}{51\!\cdots\!29}a^{14}-\frac{21\!\cdots\!59}{51\!\cdots\!29}a^{13}+\frac{47\!\cdots\!57}{51\!\cdots\!29}a^{12}-\frac{17\!\cdots\!79}{51\!\cdots\!29}a^{11}-\frac{42\!\cdots\!65}{51\!\cdots\!29}a^{10}+\frac{34\!\cdots\!41}{51\!\cdots\!29}a^{9}+\frac{16\!\cdots\!66}{51\!\cdots\!29}a^{8}-\frac{26\!\cdots\!67}{51\!\cdots\!29}a^{7}+\frac{15\!\cdots\!95}{51\!\cdots\!29}a^{6}+\frac{11\!\cdots\!58}{51\!\cdots\!29}a^{5}-\frac{63\!\cdots\!54}{51\!\cdots\!29}a^{4}-\frac{36\!\cdots\!06}{51\!\cdots\!29}a^{3}+\frac{21\!\cdots\!19}{51\!\cdots\!29}a^{2}-\frac{82\!\cdots\!45}{51\!\cdots\!29}a-\frac{35\!\cdots\!38}{51\!\cdots\!29}$, $\frac{12\!\cdots\!62}{51\!\cdots\!29}a^{31}-\frac{92\!\cdots\!20}{51\!\cdots\!29}a^{30}-\frac{16\!\cdots\!27}{51\!\cdots\!29}a^{29}-\frac{31\!\cdots\!41}{51\!\cdots\!29}a^{28}+\frac{17\!\cdots\!04}{51\!\cdots\!29}a^{27}-\frac{66\!\cdots\!35}{51\!\cdots\!29}a^{26}-\frac{46\!\cdots\!04}{51\!\cdots\!29}a^{25}+\frac{43\!\cdots\!62}{51\!\cdots\!29}a^{24}+\frac{11\!\cdots\!01}{51\!\cdots\!29}a^{23}-\frac{17\!\cdots\!35}{51\!\cdots\!29}a^{22}-\frac{90\!\cdots\!78}{51\!\cdots\!29}a^{21}+\frac{30\!\cdots\!73}{51\!\cdots\!29}a^{20}+\frac{12\!\cdots\!74}{47\!\cdots\!81}a^{19}-\frac{57\!\cdots\!34}{51\!\cdots\!29}a^{18}-\frac{26\!\cdots\!74}{51\!\cdots\!29}a^{17}+\frac{11\!\cdots\!77}{51\!\cdots\!29}a^{16}+\frac{43\!\cdots\!29}{51\!\cdots\!29}a^{15}-\frac{23\!\cdots\!32}{51\!\cdots\!29}a^{14}+\frac{44\!\cdots\!17}{51\!\cdots\!29}a^{13}+\frac{29\!\cdots\!29}{51\!\cdots\!29}a^{12}-\frac{18\!\cdots\!88}{51\!\cdots\!29}a^{11}-\frac{21\!\cdots\!34}{51\!\cdots\!29}a^{10}+\frac{26\!\cdots\!90}{51\!\cdots\!29}a^{9}+\frac{17\!\cdots\!43}{51\!\cdots\!29}a^{8}-\frac{15\!\cdots\!27}{51\!\cdots\!29}a^{7}+\frac{50\!\cdots\!64}{51\!\cdots\!29}a^{6}+\frac{45\!\cdots\!06}{51\!\cdots\!29}a^{5}-\frac{47\!\cdots\!23}{51\!\cdots\!29}a^{4}+\frac{16\!\cdots\!40}{51\!\cdots\!29}a^{3}+\frac{17\!\cdots\!57}{51\!\cdots\!29}a^{2}-\frac{23\!\cdots\!07}{51\!\cdots\!29}a-\frac{56\!\cdots\!58}{51\!\cdots\!29}$, $\frac{14\!\cdots\!17}{51\!\cdots\!29}a^{31}-\frac{85\!\cdots\!10}{51\!\cdots\!29}a^{30}-\frac{21\!\cdots\!03}{51\!\cdots\!29}a^{29}-\frac{73\!\cdots\!68}{51\!\cdots\!29}a^{28}+\frac{20\!\cdots\!74}{51\!\cdots\!29}a^{27}-\frac{43\!\cdots\!99}{51\!\cdots\!29}a^{26}-\frac{56\!\cdots\!80}{51\!\cdots\!29}a^{25}+\frac{39\!\cdots\!44}{51\!\cdots\!29}a^{24}+\frac{13\!\cdots\!67}{51\!\cdots\!29}a^{23}-\frac{18\!\cdots\!24}{51\!\cdots\!29}a^{22}-\frac{14\!\cdots\!33}{51\!\cdots\!29}a^{21}+\frac{33\!\cdots\!72}{51\!\cdots\!29}a^{20}+\frac{21\!\cdots\!65}{47\!\cdots\!81}a^{19}-\frac{64\!\cdots\!28}{51\!\cdots\!29}a^{18}-\frac{44\!\cdots\!56}{51\!\cdots\!29}a^{17}+\frac{12\!\cdots\!40}{51\!\cdots\!29}a^{16}+\frac{77\!\cdots\!68}{51\!\cdots\!29}a^{15}-\frac{26\!\cdots\!59}{51\!\cdots\!29}a^{14}-\frac{62\!\cdots\!70}{51\!\cdots\!29}a^{13}+\frac{35\!\cdots\!04}{51\!\cdots\!29}a^{12}-\frac{14\!\cdots\!88}{51\!\cdots\!29}a^{11}-\frac{29\!\cdots\!28}{51\!\cdots\!29}a^{10}+\frac{25\!\cdots\!16}{51\!\cdots\!29}a^{9}+\frac{85\!\cdots\!17}{51\!\cdots\!29}a^{8}-\frac{16\!\cdots\!72}{51\!\cdots\!29}a^{7}+\frac{21\!\cdots\!24}{51\!\cdots\!29}a^{6}+\frac{60\!\cdots\!85}{51\!\cdots\!29}a^{5}-\frac{45\!\cdots\!96}{51\!\cdots\!29}a^{4}+\frac{85\!\cdots\!01}{51\!\cdots\!29}a^{3}+\frac{63\!\cdots\!78}{51\!\cdots\!29}a^{2}-\frac{22\!\cdots\!15}{51\!\cdots\!29}a-\frac{55\!\cdots\!78}{51\!\cdots\!29}$, $\frac{11\!\cdots\!02}{51\!\cdots\!29}a^{31}-\frac{45\!\cdots\!80}{51\!\cdots\!29}a^{30}-\frac{28\!\cdots\!57}{51\!\cdots\!29}a^{29}-\frac{94\!\cdots\!31}{51\!\cdots\!29}a^{28}+\frac{16\!\cdots\!02}{51\!\cdots\!29}a^{27}+\frac{57\!\cdots\!35}{51\!\cdots\!29}a^{26}-\frac{57\!\cdots\!51}{51\!\cdots\!29}a^{25}+\frac{11\!\cdots\!26}{51\!\cdots\!29}a^{24}+\frac{15\!\cdots\!04}{51\!\cdots\!29}a^{23}-\frac{10\!\cdots\!77}{51\!\cdots\!29}a^{22}-\frac{26\!\cdots\!35}{51\!\cdots\!29}a^{21}+\frac{31\!\cdots\!95}{51\!\cdots\!29}a^{20}+\frac{35\!\cdots\!47}{47\!\cdots\!81}a^{19}-\frac{59\!\cdots\!96}{51\!\cdots\!29}a^{18}-\frac{72\!\cdots\!82}{51\!\cdots\!29}a^{17}+\frac{11\!\cdots\!62}{51\!\cdots\!29}a^{16}+\frac{13\!\cdots\!18}{51\!\cdots\!29}a^{15}-\frac{24\!\cdots\!69}{51\!\cdots\!29}a^{14}-\frac{15\!\cdots\!46}{51\!\cdots\!29}a^{13}+\frac{42\!\cdots\!06}{51\!\cdots\!29}a^{12}+\frac{22\!\cdots\!18}{51\!\cdots\!29}a^{11}-\frac{47\!\cdots\!44}{51\!\cdots\!29}a^{10}+\frac{17\!\cdots\!46}{51\!\cdots\!29}a^{9}+\frac{30\!\cdots\!45}{51\!\cdots\!29}a^{8}-\frac{23\!\cdots\!48}{51\!\cdots\!29}a^{7}-\frac{69\!\cdots\!14}{51\!\cdots\!29}a^{6}+\frac{13\!\cdots\!82}{51\!\cdots\!29}a^{5}-\frac{33\!\cdots\!50}{51\!\cdots\!29}a^{4}-\frac{35\!\cdots\!68}{51\!\cdots\!29}a^{3}+\frac{34\!\cdots\!88}{51\!\cdots\!29}a^{2}-\frac{97\!\cdots\!94}{51\!\cdots\!29}a-\frac{73\!\cdots\!85}{51\!\cdots\!29}$, $\frac{53\!\cdots\!35}{51\!\cdots\!29}a^{31}-\frac{57\!\cdots\!65}{51\!\cdots\!29}a^{30}-\frac{55\!\cdots\!34}{51\!\cdots\!29}a^{29}+\frac{37\!\cdots\!60}{51\!\cdots\!29}a^{28}+\frac{75\!\cdots\!76}{51\!\cdots\!29}a^{27}-\frac{52\!\cdots\!23}{51\!\cdots\!29}a^{26}-\frac{18\!\cdots\!26}{51\!\cdots\!29}a^{25}+\frac{23\!\cdots\!64}{51\!\cdots\!29}a^{24}+\frac{40\!\cdots\!81}{51\!\cdots\!29}a^{23}-\frac{87\!\cdots\!03}{51\!\cdots\!29}a^{22}-\frac{14\!\cdots\!36}{51\!\cdots\!29}a^{21}+\frac{13\!\cdots\!62}{51\!\cdots\!29}a^{20}+\frac{18\!\cdots\!80}{47\!\cdots\!81}a^{19}-\frac{25\!\cdots\!42}{51\!\cdots\!29}a^{18}-\frac{42\!\cdots\!82}{51\!\cdots\!29}a^{17}+\frac{49\!\cdots\!09}{51\!\cdots\!29}a^{16}+\frac{50\!\cdots\!06}{51\!\cdots\!29}a^{15}-\frac{10\!\cdots\!87}{51\!\cdots\!29}a^{14}+\frac{47\!\cdots\!93}{51\!\cdots\!29}a^{13}+\frac{11\!\cdots\!58}{51\!\cdots\!29}a^{12}-\frac{11\!\cdots\!71}{51\!\cdots\!29}a^{11}-\frac{62\!\cdots\!29}{51\!\cdots\!29}a^{10}+\frac{12\!\cdots\!08}{51\!\cdots\!29}a^{9}-\frac{24\!\cdots\!49}{51\!\cdots\!29}a^{8}-\frac{55\!\cdots\!32}{51\!\cdots\!29}a^{7}+\frac{33\!\cdots\!98}{51\!\cdots\!29}a^{6}+\frac{95\!\cdots\!99}{51\!\cdots\!29}a^{5}-\frac{21\!\cdots\!11}{51\!\cdots\!29}a^{4}+\frac{12\!\cdots\!82}{51\!\cdots\!29}a^{3}-\frac{28\!\cdots\!83}{51\!\cdots\!29}a^{2}+\frac{53\!\cdots\!44}{51\!\cdots\!29}a-\frac{78\!\cdots\!39}{51\!\cdots\!29}$, $\frac{93\!\cdots\!76}{51\!\cdots\!29}a^{31}-\frac{12\!\cdots\!66}{51\!\cdots\!29}a^{30}-\frac{45\!\cdots\!36}{51\!\cdots\!29}a^{29}+\frac{23\!\cdots\!58}{51\!\cdots\!29}a^{28}+\frac{12\!\cdots\!08}{51\!\cdots\!29}a^{27}-\frac{13\!\cdots\!33}{51\!\cdots\!29}a^{26}-\frac{26\!\cdots\!68}{51\!\cdots\!29}a^{25}+\frac{51\!\cdots\!97}{51\!\cdots\!29}a^{24}+\frac{49\!\cdots\!99}{51\!\cdots\!29}a^{23}-\frac{16\!\cdots\!91}{51\!\cdots\!29}a^{22}+\frac{41\!\cdots\!58}{51\!\cdots\!29}a^{21}+\frac{22\!\cdots\!10}{51\!\cdots\!29}a^{20}-\frac{54\!\cdots\!80}{47\!\cdots\!81}a^{19}-\frac{40\!\cdots\!96}{51\!\cdots\!29}a^{18}+\frac{95\!\cdots\!54}{51\!\cdots\!29}a^{17}+\frac{79\!\cdots\!95}{51\!\cdots\!29}a^{16}-\frac{24\!\cdots\!22}{51\!\cdots\!29}a^{15}-\frac{16\!\cdots\!28}{51\!\cdots\!29}a^{14}+\frac{14\!\cdots\!09}{51\!\cdots\!29}a^{13}+\frac{13\!\cdots\!33}{51\!\cdots\!29}a^{12}-\frac{25\!\cdots\!24}{51\!\cdots\!29}a^{11}+\frac{15\!\cdots\!92}{51\!\cdots\!29}a^{10}+\frac{24\!\cdots\!87}{51\!\cdots\!29}a^{9}-\frac{15\!\cdots\!37}{51\!\cdots\!29}a^{8}-\frac{50\!\cdots\!47}{51\!\cdots\!29}a^{7}+\frac{10\!\cdots\!80}{51\!\cdots\!29}a^{6}-\frac{23\!\cdots\!03}{51\!\cdots\!29}a^{5}-\frac{42\!\cdots\!73}{51\!\cdots\!29}a^{4}+\frac{43\!\cdots\!36}{51\!\cdots\!29}a^{3}-\frac{17\!\cdots\!91}{51\!\cdots\!29}a^{2}+\frac{25\!\cdots\!21}{51\!\cdots\!29}a+\frac{39\!\cdots\!46}{51\!\cdots\!29}$ (assuming GRH)
Regulator:		\( 1271231.1521675275 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 1271231.1521675275 \cdot 1}{10\cdot\sqrt{24788700386255228556692600250244140625}}\cr\approx \mathstrut & 0.150652221984326 \end{aligned}\] (assuming GRH)

Galois group

$C_8.A_4$ (as 32T402):

A solvable group of order 96

The 28 conjugacy class representatives for $C_8.A_4$

Character table for $C_8.A_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.4225.1, 8.0.17850625.1, 16.0.199153008056640625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$24{,}\,{\href{/padicField/2.8.0.1}{8} }$	$24{,}\,{\href{/padicField/3.8.0.1}{8} }$	R	$24{,}\,{\href{/padicField/7.8.0.1}{8} }$	${\href{/padicField/11.6.0.1}{6} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$	R	$24{,}\,{\href{/padicField/17.8.0.1}{8} }$	${\href{/padicField/19.12.0.1}{12} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$	$24{,}\,{\href{/padicField/23.8.0.1}{8} }$	${\href{/padicField/29.12.0.1}{12} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$	${\href{/padicField/31.4.0.1}{4} }^{8}$	$24{,}\,{\href{/padicField/37.8.0.1}{8} }$	${\href{/padicField/41.3.0.1}{3} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$	$24{,}\,{\href{/padicField/43.8.0.1}{8} }$	${\href{/padicField/47.8.0.1}{8} }^{4}$	${\href{/padicField/53.8.0.1}{8} }^{4}$	${\href{/padicField/59.12.0.1}{12} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

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
\(5\)	5.16.14.1	$x^{16} - 20 x^{8} - 100$	$8$	$2$	$14$	$C_8: C_2$	$[\ ]_{8}^{2}$
	5.16.14.1	$x^{16} - 20 x^{8} - 100$	$8$	$2$	$14$	$C_8: C_2$	$[\ ]_{8}^{2}$
\(13\)	13.8.0.1	$x^{8} + 8 x^{4} + 12 x^{3} + 2 x^{2} + 3 x + 2$	$1$	$8$	$0$	$C_8$	$[\ ]^{8}$
		Deg $24$	$3$	$8$	$16$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.5.2t1.a.a	$1$	$ 5 $	\(\Q(\sqrt{5}) \)	$C_2$ (as 2T1)	$1$	$1$
	1.65.6t1.b.a	$1$	$ 5 \cdot 13 $	6.6.3570125.1	$C_6$ (as 6T1)	$0$	$1$
	1.65.6t1.b.b	$1$	$ 5 \cdot 13 $	6.6.3570125.1	$C_6$ (as 6T1)	$0$	$1$
	1.13.3t1.a.a	$1$	$ 13 $	3.3.169.1	$C_3$ (as 3T1)	$0$	$1$
	1.13.3t1.a.b	$1$	$ 13 $	3.3.169.1	$C_3$ (as 3T1)	$0$	$1$
*	1.5.4t1.a.a	$1$	$ 5 $	\(\Q(\zeta_{5})\)	$C_4$ (as 4T1)	$0$	$-1$
*	1.5.4t1.a.b	$1$	$ 5 $	\(\Q(\zeta_{5})\)	$C_4$ (as 4T1)	$0$	$-1$
	1.65.12t1.a.a	$1$	$ 5 \cdot 13 $	12.0.1593224064453125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	1.65.12t1.a.b	$1$	$ 5 \cdot 13 $	12.0.1593224064453125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	1.65.12t1.a.c	$1$	$ 5 \cdot 13 $	12.0.1593224064453125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	1.65.12t1.a.d	$1$	$ 5 \cdot 13 $	12.0.1593224064453125.1	$C_{12}$ (as 12T1)	$0$	$-1$
	2.4225.48.a.a	$2$	$ 5^{2} \cdot 13^{2}$	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
	2.4225.48.a.b	$2$	$ 5^{2} \cdot 13^{2}$	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
	2.4225.48.a.c	$2$	$ 5^{2} \cdot 13^{2}$	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
	2.4225.48.a.d	$2$	$ 5^{2} \cdot 13^{2}$	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.325.32t402.a.a	$2$	$ 5^{2} \cdot 13 $	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.325.32t402.a.b	$2$	$ 5^{2} \cdot 13 $	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.325.32t402.a.c	$2$	$ 5^{2} \cdot 13 $	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.325.32t402.a.d	$2$	$ 5^{2} \cdot 13 $	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.325.32t402.a.e	$2$	$ 5^{2} \cdot 13 $	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.325.32t402.a.f	$2$	$ 5^{2} \cdot 13 $	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.325.32t402.a.g	$2$	$ 5^{2} \cdot 13 $	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	2.325.32t402.a.h	$2$	$ 5^{2} \cdot 13 $	32.0.24788700386255228556692600250244140625.1	$C_8.A_4$ (as 32T402)	$0$	$0$
*	3.4225.4t4.a.a	$3$	$ 5^{2} \cdot 13^{2}$	4.0.4225.1	$A_4$ (as 4T4)	$1$	$-1$
*	3.845.6t6.a.a	$3$	$ 5 \cdot 13^{2}$	6.2.142805.1	$A_4\times C_2$ (as 6T6)	$1$	$-1$
*	3.21125.12t29.a.a	$3$	$ 5^{3} \cdot 13^{2}$	12.8.1593224064453125.1	$C_4\times A_4$ (as 12T29)	$0$	$1$
*	3.21125.12t29.a.b	$3$	$ 5^{3} \cdot 13^{2}$	12.8.1593224064453125.1	$C_4\times A_4$ (as 12T29)	$0$	$1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.