Normalized defining polynomial

\( x^{20} - 4 x^{19} - 16 x^{18} + 12 x^{17} + 222 x^{16} + 378 x^{15} + 1750 x^{14} + 4112 x^{13} + \cdots + 1497569 \) x^20 - 4*x^19 - 16*x^18 + 12*x^17 + 222*x^16 + 378*x^15 + 1750*x^14 + 4112*x^13 + 13433*x^12 + 23190*x^11 + 70881*x^10 + 89262*x^9 + 287947*x^8 + 230252*x^7 + 856214*x^6 + 418974*x^5 + 1783158*x^4 + 462756*x^3 + 2333308*x^2 + 204212*x + 1497569

Invariants

Degree:		$20$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[0, 10]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(115436618418847759115953789493837824\) 115436618418847759115953789493837824 \(\medspace = 2^{30}\cdot 401^{10}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(56.64\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		$2^{3/2}401^{1/2}\approx 56.639209034025185$
Ramified primes:		\(2\), \(401\) 2, 401	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$20$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is Galois over $\Q$.
This is a CM field.
Reflex fields:		unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{10}+\frac{1}{9}a^{6}+\frac{2}{9}a^{2}+\frac{1}{9}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{11}+\frac{1}{9}a^{7}+\frac{2}{9}a^{3}+\frac{1}{9}a$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{10}+\frac{1}{9}a^{8}+\frac{1}{9}a^{6}+\frac{2}{9}a^{4}+\frac{1}{3}a^{2}+\frac{1}{9}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{11}+\frac{1}{9}a^{9}+\frac{1}{9}a^{7}+\frac{2}{9}a^{5}+\frac{1}{3}a^{3}+\frac{1}{9}a$, $\frac{1}{81}a^{16}-\frac{1}{27}a^{15}-\frac{1}{81}a^{14}+\frac{1}{27}a^{13}+\frac{1}{81}a^{12}-\frac{4}{27}a^{11}+\frac{1}{27}a^{9}+\frac{1}{9}a^{8}+\frac{1}{27}a^{7}+\frac{4}{27}a^{6}-\frac{1}{27}a^{5}-\frac{17}{81}a^{4}+\frac{4}{27}a^{3}+\frac{11}{81}a^{2}-\frac{1}{27}a-\frac{26}{81}$, $\frac{1}{81}a^{17}-\frac{1}{81}a^{15}+\frac{1}{81}a^{13}-\frac{1}{9}a^{11}-\frac{2}{27}a^{10}+\frac{1}{27}a^{8}-\frac{2}{27}a^{7}-\frac{4}{27}a^{6}-\frac{35}{81}a^{5}-\frac{4}{27}a^{4}+\frac{2}{81}a^{3}-\frac{2}{27}a^{2}-\frac{35}{81}a-\frac{5}{27}$, $\frac{1}{3159}a^{18}-\frac{5}{1053}a^{17}+\frac{4}{1053}a^{16}-\frac{44}{1053}a^{15}-\frac{55}{1053}a^{14}+\frac{38}{1053}a^{13}+\frac{10}{243}a^{12}-\frac{55}{351}a^{11}-\frac{20}{351}a^{10}-\frac{166}{1053}a^{9}-\frac{101}{1053}a^{8}-\frac{58}{351}a^{7}-\frac{239}{3159}a^{6}+\frac{26}{81}a^{5}+\frac{335}{1053}a^{4}+\frac{487}{1053}a^{3}+\frac{11}{351}a^{2}-\frac{47}{1053}a+\frac{211}{3159}$, $\frac{1}{46\!\cdots\!87}a^{19}-\frac{20\!\cdots\!61}{46\!\cdots\!87}a^{18}-\frac{21\!\cdots\!15}{17\!\cdots\!81}a^{17}-\frac{88\!\cdots\!17}{15\!\cdots\!29}a^{16}-\frac{11\!\cdots\!60}{15\!\cdots\!29}a^{15}+\frac{10\!\cdots\!78}{15\!\cdots\!29}a^{14}-\frac{10\!\cdots\!81}{46\!\cdots\!87}a^{13}+\frac{12\!\cdots\!86}{46\!\cdots\!87}a^{12}+\frac{25\!\cdots\!43}{51\!\cdots\!43}a^{11}+\frac{15\!\cdots\!24}{15\!\cdots\!29}a^{10}+\frac{72\!\cdots\!28}{15\!\cdots\!29}a^{9}+\frac{13\!\cdots\!02}{15\!\cdots\!29}a^{8}+\frac{58\!\cdots\!06}{46\!\cdots\!87}a^{7}+\frac{24\!\cdots\!15}{46\!\cdots\!87}a^{6}-\frac{16\!\cdots\!56}{51\!\cdots\!43}a^{5}-\frac{13\!\cdots\!81}{51\!\cdots\!43}a^{4}-\frac{65\!\cdots\!11}{18\!\cdots\!63}a^{3}-\frac{24\!\cdots\!98}{51\!\cdots\!43}a^{2}+\frac{45\!\cdots\!80}{12\!\cdots\!51}a-\frac{87\!\cdots\!52}{55\!\cdots\!89}$ 1, a, a^2, a^3, a^4, a^5, 1/3*a^6 + 1/3*a^4 + 1/3*a^2 + 1/3, 1/3*a^7 + 1/3*a^5 + 1/3*a^3 + 1/3*a, 1/3*a^8 - 1/3, 1/3*a^9 - 1/3*a, 1/3*a^10 - 1/3*a^2, 1/3*a^11 - 1/3*a^3, 1/9*a^12 - 1/9*a^10 + 1/9*a^6 + 2/9*a^2 + 1/9, 1/9*a^13 - 1/9*a^11 + 1/9*a^7 + 2/9*a^3 + 1/9*a, 1/9*a^14 - 1/9*a^10 + 1/9*a^8 + 1/9*a^6 + 2/9*a^4 + 1/3*a^2 + 1/9, 1/9*a^15 - 1/9*a^11 + 1/9*a^9 + 1/9*a^7 + 2/9*a^5 + 1/3*a^3 + 1/9*a, 1/81*a^16 - 1/27*a^15 - 1/81*a^14 + 1/27*a^13 + 1/81*a^12 - 4/27*a^11 + 1/27*a^9 + 1/9*a^8 + 1/27*a^7 + 4/27*a^6 - 1/27*a^5 - 17/81*a^4 + 4/27*a^3 + 11/81*a^2 - 1/27*a - 26/81, 1/81*a^17 - 1/81*a^15 + 1/81*a^13 - 1/9*a^11 - 2/27*a^10 + 1/27*a^8 - 2/27*a^7 - 4/27*a^6 - 35/81*a^5 - 4/27*a^4 + 2/81*a^3 - 2/27*a^2 - 35/81*a - 5/27, 1/3159*a^18 - 5/1053*a^17 + 4/1053*a^16 - 44/1053*a^15 - 55/1053*a^14 + 38/1053*a^13 + 10/243*a^12 - 55/351*a^11 - 20/351*a^10 - 166/1053*a^9 - 101/1053*a^8 - 58/351*a^7 - 239/3159*a^6 + 26/81*a^5 + 335/1053*a^4 + 487/1053*a^3 + 11/351*a^2 - 47/1053*a + 211/3159, 1/4622622013433723557839120206012538748450020087*a^19 - 204012882564854622185756306162134692266161/4622622013433723557839120206012538748450020087*a^18 - 213660789585975930567132040843113748911115/171208222719767539179226674296760694387037781*a^17 - 8825070444788564411125233311798407047170117/1540874004477907852613040068670846249483340029*a^16 - 11165235893898027442828027168165657922410460/1540874004477907852613040068670846249483340029*a^15 + 10958875094033743214871708736153850740750778/1540874004477907852613040068670846249483340029*a^14 - 108056567873376776225293150666085140780238681/4622622013433723557839120206012538748450020087*a^13 + 121846648292001061997346325430297829234277886/4622622013433723557839120206012538748450020087*a^12 + 25957647404461607123186498463622512885258943/513624668159302617537680022890282083161113343*a^11 + 152693641800869951136831769417235478491821124/1540874004477907852613040068670846249483340029*a^10 + 72705056866566787514547367456752942312323828/1540874004477907852613040068670846249483340029*a^9 + 133240573384771003685244442809786370099495502/1540874004477907852613040068670846249483340029*a^8 + 583522128941149922296343944703454316240262806/4622622013433723557839120206012538748450020087*a^7 + 244781639913691986647919445878257032834949015/4622622013433723557839120206012538748450020087*a^6 - 162819263179169069002296902109074060445881956/513624668159302617537680022890282083161113343*a^5 - 13695838506354039230042085448029149149210981/513624668159302617537680022890282083161113343*a^4 - 6549584563642203869442262857888744697246811/18564747041902504248349880345431882523895663*a^3 - 244811961930464369474322497590411064864794298/513624668159302617537680022890282083161113343*a^2 + 45254341861693549212233647805462089610787280/124935730092803339401057302865203749958108651*a - 8704400197926551918725854258820379805327452/55694241125707512745049641036295647571686989

Class group and class number

$C_{29}\times C_{174}$, which has order $5046$ (assuming GRH)

Unit group

Rank:		$9$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -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:		$\frac{41\!\cdots\!84}{48\!\cdots\!87}a^{19}-\frac{17\!\cdots\!82}{48\!\cdots\!87}a^{18}-\frac{86\!\cdots\!68}{53\!\cdots\!43}a^{17}+\frac{38\!\cdots\!78}{16\!\cdots\!29}a^{16}+\frac{39\!\cdots\!36}{16\!\cdots\!29}a^{15}+\frac{38\!\cdots\!04}{16\!\cdots\!29}a^{14}+\frac{25\!\cdots\!28}{37\!\cdots\!99}a^{13}+\frac{10\!\cdots\!79}{48\!\cdots\!87}a^{12}+\frac{41\!\cdots\!08}{53\!\cdots\!43}a^{11}+\frac{14\!\cdots\!58}{16\!\cdots\!29}a^{10}+\frac{15\!\cdots\!44}{53\!\cdots\!43}a^{9}+\frac{36\!\cdots\!20}{16\!\cdots\!29}a^{8}+\frac{51\!\cdots\!08}{48\!\cdots\!87}a^{7}+\frac{62\!\cdots\!20}{37\!\cdots\!99}a^{6}+\frac{36\!\cdots\!68}{16\!\cdots\!29}a^{5}-\frac{59\!\cdots\!63}{16\!\cdots\!29}a^{4}+\frac{16\!\cdots\!96}{53\!\cdots\!43}a^{3}-\frac{35\!\cdots\!66}{16\!\cdots\!29}a^{2}+\frac{65\!\cdots\!48}{48\!\cdots\!87}a-\frac{72\!\cdots\!08}{37\!\cdots\!99}$, $\frac{23\!\cdots\!90}{32\!\cdots\!11}a^{19}-\frac{90\!\cdots\!77}{25\!\cdots\!47}a^{18}-\frac{11\!\cdots\!22}{10\!\cdots\!37}a^{17}+\frac{30\!\cdots\!14}{10\!\cdots\!37}a^{16}+\frac{15\!\cdots\!56}{83\!\cdots\!49}a^{15}+\frac{80\!\cdots\!40}{10\!\cdots\!37}a^{14}+\frac{17\!\cdots\!16}{32\!\cdots\!11}a^{13}+\frac{42\!\cdots\!03}{32\!\cdots\!11}a^{12}+\frac{14\!\cdots\!86}{27\!\cdots\!83}a^{11}+\frac{31\!\cdots\!73}{83\!\cdots\!49}a^{10}+\frac{22\!\cdots\!30}{10\!\cdots\!37}a^{9}-\frac{13\!\cdots\!26}{10\!\cdots\!37}a^{8}+\frac{27\!\cdots\!94}{32\!\cdots\!11}a^{7}-\frac{18\!\cdots\!79}{32\!\cdots\!11}a^{6}+\frac{23\!\cdots\!12}{10\!\cdots\!37}a^{5}-\frac{58\!\cdots\!07}{27\!\cdots\!83}a^{4}+\frac{50\!\cdots\!10}{14\!\cdots\!71}a^{3}-\frac{15\!\cdots\!28}{36\!\cdots\!79}a^{2}+\frac{19\!\cdots\!06}{88\!\cdots\!03}a-\frac{12\!\cdots\!20}{39\!\cdots\!17}$, $\frac{11\!\cdots\!88}{26\!\cdots\!29}a^{19}-\frac{26\!\cdots\!93}{80\!\cdots\!87}a^{18}-\frac{65\!\cdots\!12}{26\!\cdots\!29}a^{17}+\frac{11\!\cdots\!74}{29\!\cdots\!81}a^{16}+\frac{96\!\cdots\!02}{89\!\cdots\!43}a^{15}-\frac{17\!\cdots\!89}{68\!\cdots\!11}a^{14}-\frac{11\!\cdots\!90}{26\!\cdots\!29}a^{13}-\frac{12\!\cdots\!95}{80\!\cdots\!87}a^{12}-\frac{74\!\cdots\!94}{29\!\cdots\!81}a^{11}-\frac{12\!\cdots\!89}{89\!\cdots\!43}a^{10}-\frac{53\!\cdots\!02}{26\!\cdots\!29}a^{9}-\frac{22\!\cdots\!46}{26\!\cdots\!29}a^{8}-\frac{13\!\cdots\!26}{20\!\cdots\!33}a^{7}-\frac{26\!\cdots\!01}{80\!\cdots\!87}a^{6}-\frac{14\!\cdots\!06}{89\!\cdots\!43}a^{5}-\frac{23\!\cdots\!79}{26\!\cdots\!29}a^{4}-\frac{31\!\cdots\!52}{10\!\cdots\!21}a^{3}-\frac{39\!\cdots\!79}{26\!\cdots\!29}a^{2}-\frac{74\!\cdots\!28}{24\!\cdots\!39}a-\frac{12\!\cdots\!11}{97\!\cdots\!89}$, $\frac{28\!\cdots\!26}{21\!\cdots\!53}a^{19}+\frac{20\!\cdots\!95}{65\!\cdots\!59}a^{18}-\frac{11\!\cdots\!10}{21\!\cdots\!53}a^{17}-\frac{30\!\cdots\!40}{72\!\cdots\!51}a^{16}+\frac{48\!\cdots\!60}{80\!\cdots\!39}a^{15}+\frac{39\!\cdots\!52}{24\!\cdots\!17}a^{14}+\frac{27\!\cdots\!48}{21\!\cdots\!53}a^{13}+\frac{61\!\cdots\!96}{65\!\cdots\!59}a^{12}+\frac{20\!\cdots\!46}{80\!\cdots\!39}a^{11}+\frac{12\!\cdots\!25}{24\!\cdots\!17}a^{10}+\frac{16\!\cdots\!54}{21\!\cdots\!53}a^{9}+\frac{51\!\cdots\!34}{21\!\cdots\!53}a^{8}+\frac{47\!\cdots\!90}{21\!\cdots\!53}a^{7}+\frac{47\!\cdots\!71}{65\!\cdots\!59}a^{6}+\frac{10\!\cdots\!24}{80\!\cdots\!39}a^{5}+\frac{37\!\cdots\!46}{21\!\cdots\!53}a^{4}-\frac{41\!\cdots\!86}{87\!\cdots\!97}a^{3}+\frac{45\!\cdots\!34}{21\!\cdots\!53}a^{2}-\frac{40\!\cdots\!74}{45\!\cdots\!13}a+\frac{25\!\cdots\!91}{78\!\cdots\!73}$, $\frac{92\!\cdots\!76}{46\!\cdots\!87}a^{19}-\frac{31\!\cdots\!72}{46\!\cdots\!87}a^{18}-\frac{23\!\cdots\!04}{51\!\cdots\!43}a^{17}+\frac{53\!\cdots\!10}{15\!\cdots\!29}a^{16}+\frac{10\!\cdots\!80}{15\!\cdots\!29}a^{15}+\frac{14\!\cdots\!02}{15\!\cdots\!29}a^{14}+\frac{46\!\cdots\!28}{46\!\cdots\!87}a^{13}+\frac{23\!\cdots\!77}{46\!\cdots\!87}a^{12}+\frac{10\!\cdots\!62}{51\!\cdots\!43}a^{11}+\frac{24\!\cdots\!88}{15\!\cdots\!29}a^{10}+\frac{22\!\cdots\!12}{51\!\cdots\!43}a^{9}+\frac{49\!\cdots\!91}{15\!\cdots\!29}a^{8}+\frac{66\!\cdots\!70}{46\!\cdots\!87}a^{7}-\frac{48\!\cdots\!83}{46\!\cdots\!87}a^{6}+\frac{40\!\cdots\!62}{15\!\cdots\!29}a^{5}-\frac{13\!\cdots\!15}{15\!\cdots\!29}a^{4}+\frac{39\!\cdots\!72}{61\!\cdots\!21}a^{3}-\frac{40\!\cdots\!16}{15\!\cdots\!29}a^{2}+\frac{67\!\cdots\!78}{12\!\cdots\!51}a-\frac{15\!\cdots\!47}{55\!\cdots\!89}$, $\frac{27\!\cdots\!00}{44\!\cdots\!89}a^{19}-\frac{21\!\cdots\!32}{13\!\cdots\!67}a^{18}-\frac{66\!\cdots\!54}{44\!\cdots\!89}a^{17}-\frac{21\!\cdots\!91}{14\!\cdots\!63}a^{16}+\frac{10\!\cdots\!32}{55\!\cdots\!69}a^{15}+\frac{19\!\cdots\!66}{49\!\cdots\!21}a^{14}+\frac{39\!\cdots\!96}{44\!\cdots\!89}a^{13}+\frac{41\!\cdots\!57}{13\!\cdots\!67}a^{12}+\frac{50\!\cdots\!56}{55\!\cdots\!69}a^{11}+\frac{84\!\cdots\!80}{49\!\cdots\!21}a^{10}+\frac{17\!\cdots\!50}{44\!\cdots\!89}a^{9}+\frac{31\!\cdots\!64}{44\!\cdots\!89}a^{8}+\frac{63\!\cdots\!16}{44\!\cdots\!89}a^{7}+\frac{28\!\cdots\!38}{13\!\cdots\!67}a^{6}+\frac{15\!\cdots\!46}{49\!\cdots\!21}a^{5}+\frac{22\!\cdots\!51}{44\!\cdots\!89}a^{4}+\frac{85\!\cdots\!98}{18\!\cdots\!61}a^{3}+\frac{23\!\cdots\!99}{34\!\cdots\!53}a^{2}+\frac{38\!\cdots\!30}{12\!\cdots\!97}a+\frac{10\!\cdots\!43}{16\!\cdots\!49}$, $\frac{11\!\cdots\!84}{65\!\cdots\!59}a^{19}-\frac{24\!\cdots\!63}{65\!\cdots\!59}a^{18}-\frac{20\!\cdots\!08}{72\!\cdots\!51}a^{17}-\frac{17\!\cdots\!76}{21\!\cdots\!53}a^{16}+\frac{31\!\cdots\!70}{21\!\cdots\!53}a^{15}+\frac{36\!\cdots\!39}{21\!\cdots\!53}a^{14}+\frac{54\!\cdots\!50}{65\!\cdots\!59}a^{13}+\frac{12\!\cdots\!26}{65\!\cdots\!59}a^{12}+\frac{35\!\cdots\!02}{72\!\cdots\!51}a^{11}+\frac{28\!\cdots\!75}{21\!\cdots\!53}a^{10}+\frac{23\!\cdots\!10}{72\!\cdots\!51}a^{9}+\frac{12\!\cdots\!94}{21\!\cdots\!53}a^{8}+\frac{84\!\cdots\!14}{65\!\cdots\!59}a^{7}+\frac{12\!\cdots\!35}{65\!\cdots\!59}a^{6}+\frac{79\!\cdots\!06}{21\!\cdots\!53}a^{5}+\frac{10\!\cdots\!20}{21\!\cdots\!53}a^{4}+\frac{60\!\cdots\!20}{87\!\cdots\!97}a^{3}+\frac{14\!\cdots\!21}{16\!\cdots\!81}a^{2}+\frac{10\!\cdots\!64}{17\!\cdots\!07}a+\frac{70\!\cdots\!56}{78\!\cdots\!73}$, $\frac{22\!\cdots\!64}{10\!\cdots\!37}a^{19}-\frac{11\!\cdots\!89}{10\!\cdots\!37}a^{18}-\frac{11\!\cdots\!46}{36\!\cdots\!79}a^{17}+\frac{32\!\cdots\!74}{36\!\cdots\!79}a^{16}+\frac{20\!\cdots\!74}{36\!\cdots\!79}a^{15}+\frac{50\!\cdots\!49}{36\!\cdots\!79}a^{14}+\frac{10\!\cdots\!42}{10\!\cdots\!37}a^{13}+\frac{23\!\cdots\!82}{10\!\cdots\!37}a^{12}+\frac{44\!\cdots\!46}{40\!\cdots\!31}a^{11}-\frac{68\!\cdots\!93}{36\!\cdots\!79}a^{10}+\frac{10\!\cdots\!28}{36\!\cdots\!79}a^{9}-\frac{25\!\cdots\!53}{36\!\cdots\!79}a^{8}+\frac{12\!\cdots\!14}{10\!\cdots\!37}a^{7}-\frac{45\!\cdots\!85}{10\!\cdots\!37}a^{6}+\frac{82\!\cdots\!48}{36\!\cdots\!79}a^{5}-\frac{16\!\cdots\!21}{12\!\cdots\!93}a^{4}+\frac{38\!\cdots\!38}{14\!\cdots\!71}a^{3}-\frac{33\!\cdots\!20}{12\!\cdots\!93}a^{2}-\frac{93\!\cdots\!58}{29\!\cdots\!01}a-\frac{35\!\cdots\!16}{13\!\cdots\!39}$, $\frac{53\!\cdots\!88}{80\!\cdots\!87}a^{19}-\frac{26\!\cdots\!62}{80\!\cdots\!87}a^{18}-\frac{24\!\cdots\!20}{26\!\cdots\!29}a^{17}+\frac{62\!\cdots\!69}{26\!\cdots\!29}a^{16}+\frac{41\!\cdots\!36}{26\!\cdots\!29}a^{15}+\frac{22\!\cdots\!78}{26\!\cdots\!29}a^{14}+\frac{52\!\cdots\!08}{80\!\cdots\!87}a^{13}+\frac{96\!\cdots\!65}{80\!\cdots\!87}a^{12}+\frac{40\!\cdots\!50}{89\!\cdots\!43}a^{11}+\frac{78\!\cdots\!96}{26\!\cdots\!29}a^{10}+\frac{50\!\cdots\!64}{26\!\cdots\!29}a^{9}-\frac{19\!\cdots\!95}{26\!\cdots\!29}a^{8}+\frac{59\!\cdots\!34}{80\!\cdots\!87}a^{7}-\frac{72\!\cdots\!43}{80\!\cdots\!87}a^{6}+\frac{49\!\cdots\!46}{26\!\cdots\!29}a^{5}-\frac{32\!\cdots\!11}{89\!\cdots\!43}a^{4}+\frac{10\!\cdots\!04}{35\!\cdots\!07}a^{3}-\frac{22\!\cdots\!80}{29\!\cdots\!81}a^{2}+\frac{30\!\cdots\!62}{16\!\cdots\!27}a-\frac{64\!\cdots\!68}{97\!\cdots\!89}$ 417540645965264959316587761643530784/48556443875943777458630898898252526217687*a^19 - 1740216665048574380425275285304008182/48556443875943777458630898898252526217687*a^18 - 865675948318399171314864749851460668/5395160430660419717625655433139169579743*a^17 + 3856727584121698524437794490559206878/16185481291981259152876966299417508739229*a^16 + 39311251817934166090239688437279056336/16185481291981259152876966299417508739229*a^15 + 38485654917433659629202212169721601504/16185481291981259152876966299417508739229*a^14 + 25365938400507194671643640419320218028/3735111067380290573740838376788655862899*a^13 + 1081015157969710057621178301027657575779/48556443875943777458630898898252526217687*a^12 + 412874433401426246731090788389334174208/5395160430660419717625655433139169579743*a^11 + 1440043154552139231488479543057409641358/16185481291981259152876966299417508739229*a^10 + 1507522602497249103150611190356674469744/5395160430660419717625655433139169579743*a^9 + 3656953075576062296980417106249240940220/16185481291981259152876966299417508739229*a^8 + 51681647035783682757103862441227123907008/48556443875943777458630898898252526217687*a^7 + 620773525477041476433421058852364345820/3735111067380290573740838376788655862899*a^6 + 36564109832611840731564108669939710542268/16185481291981259152876966299417508739229*a^5 - 5905269449960671100784028807896253343863/16185481291981259152876966299417508739229*a^4 + 16355496618164817837867817233822573739796/5395160430660419717625655433139169579743*a^3 - 35625214543806048086435305193105319432766/16185481291981259152876966299417508739229*a^2 + 65105057637753153096515732038448176914148/48556443875943777458630898898252526217687*a - 7264627610580061226350905994257806638808/3735111067380290573740838376788655862899, 23784943011949103603349031878256490/3259920024536887468795592863846158930711*a^19 - 9066882687637032566451234192893177/250763078810529805291968681834319917747*a^18 - 116181119778652075144696556020093922/1086640008178962489598530954615386310237*a^17 + 304485002539824206889235339294241114/1086640008178962489598530954615386310237*a^16 + 152469822063194755703607595018422856/83587692936843268430656227278106639249*a^15 + 808305282951409938970856700682076440/1086640008178962489598530954615386310237*a^14 + 17979793744635270670589195863333924916/3259920024536887468795592863846158930711*a^13 + 42130084031633202243658517293168279603/3259920024536887468795592863846158930711*a^12 + 1408401531560266017280775418087191186/27862564312281089476885409092702213083*a^11 + 3169956607412725084070754131791144873/83587692936843268430656227278106639249*a^10 + 221126937848006375865628750490680272430/1086640008178962489598530954615386310237*a^9 - 13789847627404114788891675597801262826/1086640008178962489598530954615386310237*a^8 + 2757198169453092541497798891083918596394/3259920024536887468795592863846158930711*a^7 - 1846237354191566073079284767191416301679/3259920024536887468795592863846158930711*a^6 + 2306704187468722452254534850873047806912/1086640008178962489598530954615386310237*a^5 - 58194627177226024824264680106020557807/27862564312281089476885409092702213083*a^4 + 5095955060053032771638322561105031810/1454672032368089008833374771908147671*a^3 - 1553960581651456218364745975650534393028/362213336059654163199510318205128770079*a^2 + 197570021292506762783775398204313822806/88105946609105066724205212536382673803*a - 128295002971437463529173778045436825820/39276144873938403238501118841519987117, 113155652892252755774989277037379388/26844027185552653309402973269992617715429*a^19 - 2636069388594745951601724291290815093/80532081556657959928208919809977853146287*a^18 - 658730815782369256933403835911897312/26844027185552653309402973269992617715429*a^17 + 1193910615737020244617635354304165774/2982669687283628145489219252221401968381*a^16 + 9615091684431788091074914259211175802/8948009061850884436467657756664205905143*a^15 - 1761502551342859061623518964127302189/688308389373144956651358288974169685011*a^14 - 111410852647715661779423441557104221890/26844027185552653309402973269992617715429*a^13 - 1232415443158721711054789885361116604295/80532081556657959928208919809977853146287*a^12 - 74521891640616385607378065786454700494/2982669687283628145489219252221401968381*a^11 - 1273083415756336223689793980756817501389/8948009061850884436467657756664205905143*a^10 - 5360197286487846907759771279482341805602/26844027185552653309402973269992617715429*a^9 - 22421117256286893459116497870281854558146/26844027185552653309402973269992617715429*a^8 - 1377728731962244032899103618719028535426/2064925168119434869954074866922509055033*a^7 - 260604790928457344820133911635787700601401/80532081556657959928208919809977853146287*a^6 - 14774630704954688588750919340614365609206/8948009061850884436467657756664205905143*a^5 - 234225479879999783095645228981445120348479/26844027185552653309402973269992617715429*a^4 - 319601506376473400760447543815035702352/107807338094588969114068165742942239821*a^3 - 396228704542067464402451266165629601235479/26844027185552653309402973269992617715429*a^2 - 741892307300999545177434446497459958128/241838082752726606391017777207140700139*a - 12161968246193525665504576457278236308711/970266042851300722026613491686480158389, 2837240005466493378532709886105026/2177627540800866956870743589423363783253*a^19 + 2078246273897136522397203474769495/6532882622402600870612230768270091349759*a^18 - 116582336092096556755906489852049410/2177627540800866956870743589423363783253*a^17 - 30432083184610400402232335454738740/725875846933622318956914529807787927751*a^16 + 48772967917974109163066991956680760/80652871881513590995212725534198658639*a^15 + 395343745373351373039781420901338952/241958615644540772985638176602595975917*a^14 + 2778227001152940606940251905515821848/2177627540800866956870743589423363783253*a^13 + 61013250110713289489364473248926182896/6532882622402600870612230768270091349759*a^12 + 2095252472463181759021694523557142646/80652871881513590995212725534198658639*a^11 + 12351398441235411154423969660604858825/241958615644540772985638176602595975917*a^10 + 165336882663629202834500516419600435154/2177627540800866956870743589423363783253*a^9 + 519648764347563369585951947077205879734/2177627540800866956870743589423363783253*a^8 + 475975409027636157952363631884957608290/2177627540800866956870743589423363783253*a^7 + 4785623005689541418616158843611965868771/6532882622402600870612230768270091349759*a^6 + 10991570136894684012849611227990591124/80652871881513590995212725534198658639*a^5 + 3753754345064276040351167267292819410146/2177627540800866956870743589423363783253*a^4 - 4124826133297839372614470977238228186/8745492131730389385023066624190215997*a^3 + 4549326401688653496168049278007713431134/2177627540800866956870743589423363783253*a^2 - 4081816828180189898568949921869808574/4527292184617187020521296443707617013*a + 25307474496613211309439766795293889391/78709429185573504465207599617711943973, 9261204270078693771584496240389636586476/4622622013433723557839120206012538748450020087*a^19 - 31920343176701029425669863769500769974572/4622622013433723557839120206012538748450020087*a^18 - 23696059811872854380838803331860176985504/513624668159302617537680022890282083161113343*a^17 + 53359947248052373083793975452799113368810/1540874004477907852613040068670846249483340029*a^16 + 1061538175825808325624180946831280247999280/1540874004477907852613040068670846249483340029*a^15 + 1405563919542879917910563445560250344347702/1540874004477907852613040068670846249483340029*a^14 + 4678442276363125201124121098157264271847828/4622622013433723557839120206012538748450020087*a^13 + 23072487755376965640990948419761935458979977/4622622013433723557839120206012538748450020087*a^12 + 10031883622050776534118257125065609977031062/513624668159302617537680022890282083161113343*a^11 + 24145098046017349376191772775185424179285788/1540874004477907852613040068670846249483340029*a^10 + 22581440469340759302165841910681863228305412/513624668159302617537680022890282083161113343*a^9 + 49985069358366512970552077193672646093670291/1540874004477907852613040068670846249483340029*a^8 + 661387678953050749571898919373316261634670270/4622622013433723557839120206012538748450020087*a^7 - 480520346937414559744164728575209777943409383/4622622013433723557839120206012538748450020087*a^6 + 403391098020013926870494182734908021135689462/1540874004477907852613040068670846249483340029*a^5 - 1337435815773423628177716640239956853569286515/1540874004477907852613040068670846249483340029*a^4 + 3900644158750655905839758243054444235569072/6188249013967501416116626781810627507965221*a^3 - 4000988389082105049622424982993986376734611316/1540874004477907852613040068670846249483340029*a^2 + 67139812164561891183643540100060684786869778/124935730092803339401057302865203749958108651*a - 154957917219241832052957541873839850227834947/55694241125707512745049641036295647571686989, 270813565082529546046474322997945700/44843689196411857996363320877472898037989*a^19 - 2117267785734457986002387541089859832/134531067589235573989089962632418694113967*a^18 - 6659219736148720284683217470535986654/44843689196411857996363320877472898037989*a^17 - 21897173795845089736462187499376591/14947896398803952665454440292490966012663*a^16 + 1010553897582546267639113857796773232/553625792548294543164979270092258000469*a^15 + 19659314199699391403911306709193842866/4982632132934650888484813430830322004221*a^14 + 393981343166623438458219410089585088296/44843689196411857996363320877472898037989*a^13 + 4168900840666076570057558915826751067657/134531067589235573989089962632418694113967*a^12 + 50076361610910452462098373430786147856/553625792548294543164979270092258000469*a^11 + 849219306860987923287284984257425399580/4982632132934650888484813430830322004221*a^10 + 17239157149765576111988145921533124442450/44843689196411857996363320877472898037989*a^9 + 31991023797616684696395131303061595231664/44843689196411857996363320877472898037989*a^8 + 63161637561200010383435128299513457512316/44843689196411857996363320877472898037989*a^7 + 280154250199024268178543438899102923488938/134531067589235573989089962632418694113967*a^6 + 15540899997084786472756747602714405990746/4982632132934650888484813430830322004221*a^5 + 220337119523391541640062320860733848211551/44843689196411857996363320877472898037989*a^4 + 850119004458842860402132603428993971998/180095137334987381511499280632421277261*a^3 + 23434286073944777454779024588840813997899/3449514553570142922797178529036376772153*a^2 + 3871579036234342510459127775460700576630/1211991599903023189090900564256024271297*a + 10175351704695295927604170842018188828443/1620856236014886433603493525691791495349, 11615260663130179911335214316599584/6532882622402600870612230768270091349759*a^19 - 24446896430481520124649714155578063/6532882622402600870612230768270091349759*a^18 - 20935834783567306365213110653009808/725875846933622318956914529807787927751*a^17 - 170181830979430701004346848834632376/2177627540800866956870743589423363783253*a^16 + 315303675748429442855347309258099570/2177627540800866956870743589423363783253*a^15 + 3634683696801471709518618304346574139/2177627540800866956870743589423363783253*a^14 + 54810285446012912235195816885209760050/6532882622402600870612230768270091349759*a^13 + 123390855386427303312149835013196334626/6532882622402600870612230768270091349759*a^12 + 35970401842557403052867986446933781502/725875846933622318956914529807787927751*a^11 + 281244439119498051245739534582535397875/2177627540800866956870743589423363783253*a^10 + 237542966391461371587384834305980292410/725875846933622318956914529807787927751*a^9 + 1271550856430278229760206490182463704594/2177627540800866956870743589423363783253*a^8 + 8456274362019290030060133717991237746914/6532882622402600870612230768270091349759*a^7 + 12655165201381074177879055939585973105135/6532882622402600870612230768270091349759*a^6 + 7921485111800856142896096124446932744806/2177627540800866956870743589423363783253*a^5 + 10698704788080194614088868123189947058220/2177627540800866956870743589423363783253*a^4 + 60688272615512352224312024785597392320/8745492131730389385023066624190215997*a^3 + 1403917091684732733002022420095891856721/167509810830835919759287968417181829481*a^2 + 1096772053209678919381559641074250424264/176564395200070293800330561304597063507*a + 704186237384659962682360293411027895756/78709429185573504465207599617711943973, 22855558106207464798372947705172064/1086640008178962489598530954615386310237*a^19 - 116343897885817443553351443944142989/1086640008178962489598530954615386310237*a^18 - 113977947596902355371420216969877846/362213336059654163199510318205128770079*a^17 + 326852820418450151527406512018449374/362213336059654163199510318205128770079*a^16 + 2050641017191749394939232076960430174/362213336059654163199510318205128770079*a^15 + 506243693337608334258701093226810049/362213336059654163199510318205128770079*a^14 + 10249522605522867791359753052407387742/1086640008178962489598530954615386310237*a^13 + 23983671505737178141544826093115194382/1086640008178962489598530954615386310237*a^12 + 4452657302503764903454821005372128846/40245926228850462577723368689458752231*a^11 - 6808017192053525421356747675191019693/362213336059654163199510318205128770079*a^10 + 101262374784627465091516066342564433728/362213336059654163199510318205128770079*a^9 - 250586674229398913867184576178968374153/362213336059654163199510318205128770079*a^8 + 1244425569795967880387078150761257235814/1086640008178962489598530954615386310237*a^7 - 4547214821811167960738419430898401374085/1086640008178962489598530954615386310237*a^6 + 825497911256107003482523673204105067148/362213336059654163199510318205128770079*a^5 - 1671072241941056434613603863596191511121/120737778686551387733170106068376256693*a^4 + 3819242644506354845163117805039563938/1454672032368089008833374771908147671*a^3 - 3308530691104931125069540859965001166320/120737778686551387733170106068376256693*a^2 - 9369106295236857108373452380929836658/29368648869701688908068404178794224601*a - 354069414096658513468938382380859005916/13092048291312801079500372947173329039, 535975548164597753964060860354284888/80532081556657959928208919809977853146287*a^19 - 2688416054046911428366448442490800262/80532081556657959928208919809977853146287*a^18 - 2423777222148017739816243672278729620/26844027185552653309402973269992617715429*a^17 + 6289675844715587665237837463722518769/26844027185552653309402973269992617715429*a^16 + 41990907171519567150392819499918123836/26844027185552653309402973269992617715429*a^15 + 22216658704237664853981181607317128578/26844027185552653309402973269992617715429*a^14 + 529064523018751576498043930649473408308/80532081556657959928208919809977853146287*a^13 + 960306349315038943250362270243075767665/80532081556657959928208919809977853146287*a^12 + 408369255581841022341299157305411327650/8948009061850884436467657756664205905143*a^11 + 784666671406280563641177355620375468296/26844027185552653309402973269992617715429*a^10 + 5063310791613668331218185841554309386164/26844027185552653309402973269992617715429*a^9 - 1992737273759798169570886995940693784695/26844027185552653309402973269992617715429*a^8 + 59757581359792554734829534347091869436934/80532081556657959928208919809977853146287*a^7 - 72772426084021141691818137420221014622743/80532081556657959928208919809977853146287*a^6 + 49373472973938424775381058623802297217046/26844027185552653309402973269992617715429*a^5 - 32964944090000600478148976316931847045011/8948009061850884436467657756664205905143*a^4 + 105222792905470520768161469973538245004/35935779364862989704689388580980746607*a^3 - 22727203298347417271478916378484719429280/2982669687283628145489219252221401968381*a^2 + 309932842059773544370042115679838531762/167426364982656881347627691912635869327*a - 6477024702380510164522307002905971412068/970266042851300722026613491686480158389 (assuming GRH)	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:		\( 795087.603907 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 795087.603907 \cdot 5046}{2\cdot\sqrt{115436618418847759115953789493837824}}\cr\approx \mathstrut & 0.566185989705 \end{aligned}\] (assuming GRH)

Galois group

$D_{10}$ (as 20T4):

Intermediate fields

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

Sibling fields

Frobenius cycle types

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
\(2\) 2	2.10.15.9	$x^{10} + 20 x^{9} + 206 x^{8} + 1376 x^{7} + 6504 x^{6} + 22496 x^{5} + 57328 x^{4} + 105856 x^{3} + 135504 x^{2} + 108864 x + 9568$	$2$	$5$	$15$	$C_{10}$	$[3]^{5}$
	2.10.15.9	$x^{10} + 20 x^{9} + 206 x^{8} + 1376 x^{7} + 6504 x^{6} + 22496 x^{5} + 57328 x^{4} + 105856 x^{3} + 135504 x^{2} + 108864 x + 9568$	$2$	$5$	$15$	$C_{10}$	$[3]^{5}$
\(401\) 401		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$