Normalized defining polynomial
\( x^{20} - 6 x^{19} + 17 x^{18} - 30 x^{17} - 56 x^{16} + 282 x^{15} - 1314 x^{14} + 6402 x^{13} + \cdots + 541725625 \)
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: |
\(35635746108202593567757236917916961401\)
\(\medspace = 3^{10}\cdot 7^{10}\cdot 271^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(75.44\) | 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: | $3^{1/2}7^{1/2}271^{1/2}\approx 75.43871685016919$ | ||
| Ramified primes: |
\(3\), \(7\), \(271\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7}a^{11}+\frac{1}{7}a^{10}+\frac{2}{7}a^{9}+\frac{2}{7}a^{8}+\frac{3}{7}a^{7}+\frac{3}{7}a^{6}+\frac{2}{7}a^{5}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{1}{7}a^{2}$, $\frac{1}{7}a^{12}+\frac{1}{7}a^{10}+\frac{1}{7}a^{8}-\frac{1}{7}a^{6}-\frac{1}{7}a^{4}-\frac{1}{7}a^{2}$, $\frac{1}{133}a^{13}+\frac{9}{133}a^{12}-\frac{8}{133}a^{11}-\frac{7}{19}a^{10}+\frac{18}{133}a^{9}+\frac{5}{133}a^{8}-\frac{9}{19}a^{7}+\frac{20}{133}a^{6}-\frac{12}{133}a^{5}-\frac{41}{133}a^{4}-\frac{17}{133}a^{3}+\frac{3}{133}a^{2}-\frac{8}{19}a$, $\frac{1}{133}a^{14}+\frac{6}{133}a^{12}+\frac{4}{133}a^{11}+\frac{3}{133}a^{10}-\frac{62}{133}a^{9}-\frac{51}{133}a^{8}-\frac{2}{133}a^{7}+\frac{55}{133}a^{6}+\frac{29}{133}a^{5}-\frac{47}{133}a^{4}+\frac{4}{133}a^{3}-\frac{64}{133}a^{2}-\frac{4}{19}a$, $\frac{1}{4655}a^{15}-\frac{16}{4655}a^{14}+\frac{9}{4655}a^{13}-\frac{274}{4655}a^{12}+\frac{276}{4655}a^{11}+\frac{327}{665}a^{10}+\frac{1584}{4655}a^{9}+\frac{295}{931}a^{8}-\frac{1413}{4655}a^{7}-\frac{2159}{4655}a^{6}+\frac{9}{95}a^{5}+\frac{1963}{4655}a^{4}-\frac{37}{95}a^{3}-\frac{326}{665}a^{2}-\frac{16}{95}a$, $\frac{1}{4655}a^{16}-\frac{2}{4655}a^{14}+\frac{2}{931}a^{13}-\frac{48}{4655}a^{12}-\frac{17}{931}a^{11}-\frac{1167}{4655}a^{10}+\frac{849}{4655}a^{9}-\frac{1578}{4655}a^{8}+\frac{1833}{4655}a^{7}-\frac{1868}{4655}a^{6}+\frac{1144}{4655}a^{5}-\frac{458}{931}a^{4}-\frac{60}{133}a^{3}-\frac{3}{665}a^{2}+\frac{14}{95}a$, $\frac{1}{4655}a^{17}+\frac{13}{4655}a^{14}+\frac{1}{931}a^{13}-\frac{108}{4655}a^{12}-\frac{18}{931}a^{11}-\frac{173}{4655}a^{10}+\frac{276}{931}a^{9}-\frac{8}{245}a^{8}-\frac{319}{4655}a^{7}+\frac{1446}{4655}a^{6}+\frac{517}{4655}a^{5}+\frac{4}{245}a^{4}+\frac{174}{665}a^{3}-\frac{99}{665}a^{2}+\frac{3}{95}a$, $\frac{1}{81\!\cdots\!05}a^{18}-\frac{72522783994511}{16\!\cdots\!21}a^{17}+\frac{550715051593017}{81\!\cdots\!05}a^{16}+\frac{151632301165322}{81\!\cdots\!05}a^{15}-\frac{37\!\cdots\!49}{11\!\cdots\!15}a^{14}+\frac{20\!\cdots\!83}{81\!\cdots\!05}a^{13}+\frac{24\!\cdots\!88}{81\!\cdots\!05}a^{12}-\frac{31\!\cdots\!24}{81\!\cdots\!05}a^{11}-\frac{31\!\cdots\!38}{81\!\cdots\!05}a^{10}+\frac{42\!\cdots\!91}{11\!\cdots\!15}a^{9}+\frac{64\!\cdots\!73}{16\!\cdots\!21}a^{8}+\frac{83\!\cdots\!02}{95\!\cdots\!13}a^{7}-\frac{85\!\cdots\!88}{16\!\cdots\!21}a^{6}+\frac{52\!\cdots\!84}{11\!\cdots\!15}a^{5}-\frac{80\!\cdots\!59}{23\!\cdots\!03}a^{4}-\frac{978919041920633}{33\!\cdots\!29}a^{3}-\frac{37\!\cdots\!07}{16\!\cdots\!45}a^{2}+\frac{211147569738278}{12\!\cdots\!65}a-\frac{947143819820}{13177257549527}$, $\frac{1}{33\!\cdots\!75}a^{19}+\frac{37\!\cdots\!49}{33\!\cdots\!75}a^{18}-\frac{32\!\cdots\!73}{33\!\cdots\!75}a^{17}+\frac{35\!\cdots\!58}{66\!\cdots\!95}a^{16}+\frac{69\!\cdots\!92}{47\!\cdots\!25}a^{15}+\frac{99\!\cdots\!53}{11\!\cdots\!75}a^{14}-\frac{33\!\cdots\!89}{33\!\cdots\!75}a^{13}+\frac{12\!\cdots\!38}{17\!\cdots\!25}a^{12}+\frac{33\!\cdots\!43}{13\!\cdots\!99}a^{11}-\frac{48\!\cdots\!13}{16\!\cdots\!25}a^{10}-\frac{37\!\cdots\!41}{33\!\cdots\!75}a^{9}-\frac{11\!\cdots\!37}{66\!\cdots\!95}a^{8}-\frac{15\!\cdots\!52}{33\!\cdots\!75}a^{7}-\frac{12\!\cdots\!66}{39\!\cdots\!75}a^{6}+\frac{10\!\cdots\!64}{67\!\cdots\!75}a^{5}+\frac{23\!\cdots\!83}{67\!\cdots\!75}a^{4}+\frac{47\!\cdots\!10}{38\!\cdots\!93}a^{3}-\frac{56\!\cdots\!94}{13\!\cdots\!75}a^{2}-\frac{18\!\cdots\!96}{51\!\cdots\!75}a-\frac{76\!\cdots\!80}{15\!\cdots\!59}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $7$, $19$ |
Class group and class number
$C_{58}\times C_{174}$, which has order $10092$ (assuming GRH)
Unit group
| Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{227002405619765552532824}{42960463921711350908262596858225} a^{19} + \frac{56329089473098914311297}{2527086113041844171074270403425} a^{18} - \frac{1775019398771613940912408}{42960463921711350908262596858225} a^{17} + \frac{17308074879947558492179}{8592092784342270181652519371645} a^{16} + \frac{2611523155953604170428187}{6137209131673050129751799551175} a^{15} - \frac{16942992925497691105757518}{42960463921711350908262596858225} a^{14} + \frac{155875202678428069089698596}{42960463921711350908262596858225} a^{13} - \frac{872091234553851828579270773}{42960463921711350908262596858225} a^{12} + \frac{415784073135469142977646758}{8592092784342270181652519371645} a^{11} - \frac{2083400244464798955298444}{19000647466480031361460679725} a^{10} - \frac{7177648829836324721495546461}{42960463921711350908262596858225} a^{9} + \frac{5299556183691243077951842884}{8592092784342270181652519371645} a^{8} + \frac{86564836802462463047405632883}{42960463921711350908262596858225} a^{7} - \frac{1363807187573929475487690634}{6137209131673050129751799551175} a^{6} - \frac{8434774917975810525612764951}{876744161667578589964542793025} a^{5} + \frac{59743420119315602622227463548}{876744161667578589964542793025} a^{4} - \frac{1205767378303689843078506639}{5009966638100449085511673103} a^{3} + \frac{15389541296002707544936801557}{125249165952511227137791827575} a^{2} - \frac{94562633017542943175553438}{387768315642449619621646525} a + \frac{1184294886764986908035361}{1982574846893727378516689} \)
(order $6$)
| 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: |
$\frac{61\!\cdots\!69}{39\!\cdots\!35}a^{19}-\frac{17\!\cdots\!06}{22\!\cdots\!55}a^{18}+\frac{11\!\cdots\!73}{66\!\cdots\!95}a^{17}-\frac{21\!\cdots\!72}{66\!\cdots\!95}a^{16}-\frac{80\!\cdots\!69}{95\!\cdots\!85}a^{15}+\frac{13\!\cdots\!53}{66\!\cdots\!95}a^{14}-\frac{85\!\cdots\!47}{66\!\cdots\!95}a^{13}+\frac{51\!\cdots\!49}{66\!\cdots\!95}a^{12}-\frac{10\!\cdots\!34}{66\!\cdots\!95}a^{11}+\frac{55\!\cdots\!39}{95\!\cdots\!85}a^{10}-\frac{12\!\cdots\!77}{66\!\cdots\!95}a^{9}-\frac{12\!\cdots\!99}{66\!\cdots\!95}a^{8}-\frac{57\!\cdots\!32}{78\!\cdots\!47}a^{7}-\frac{15\!\cdots\!05}{38\!\cdots\!93}a^{6}+\frac{27\!\cdots\!94}{13\!\cdots\!55}a^{5}-\frac{88\!\cdots\!46}{71\!\cdots\!45}a^{4}+\frac{41\!\cdots\!61}{38\!\cdots\!93}a^{3}-\frac{20\!\cdots\!43}{19\!\cdots\!65}a^{2}+\frac{25\!\cdots\!32}{10\!\cdots\!35}a-\frac{54\!\cdots\!12}{15\!\cdots\!59}$, $\frac{95\!\cdots\!63}{33\!\cdots\!75}a^{19}-\frac{34\!\cdots\!88}{92\!\cdots\!75}a^{18}+\frac{51\!\cdots\!11}{33\!\cdots\!75}a^{17}-\frac{25\!\cdots\!99}{66\!\cdots\!95}a^{16}+\frac{50\!\cdots\!22}{11\!\cdots\!75}a^{15}+\frac{44\!\cdots\!76}{33\!\cdots\!75}a^{14}-\frac{75\!\cdots\!33}{11\!\cdots\!75}a^{13}+\frac{11\!\cdots\!91}{33\!\cdots\!75}a^{12}-\frac{83\!\cdots\!14}{66\!\cdots\!95}a^{11}+\frac{16\!\cdots\!79}{47\!\cdots\!25}a^{10}-\frac{16\!\cdots\!92}{17\!\cdots\!25}a^{9}+\frac{33\!\cdots\!78}{66\!\cdots\!95}a^{8}+\frac{55\!\cdots\!14}{33\!\cdots\!75}a^{7}+\frac{27\!\cdots\!53}{47\!\cdots\!25}a^{6}-\frac{81\!\cdots\!84}{97\!\cdots\!25}a^{5}-\frac{16\!\cdots\!98}{97\!\cdots\!25}a^{4}+\frac{14\!\cdots\!28}{38\!\cdots\!93}a^{3}-\frac{16\!\cdots\!39}{97\!\cdots\!25}a^{2}+\frac{15\!\cdots\!87}{51\!\cdots\!75}a-\frac{33\!\cdots\!94}{15\!\cdots\!59}$, $\frac{33\!\cdots\!76}{21\!\cdots\!75}a^{19}+\frac{68\!\cdots\!23}{49\!\cdots\!25}a^{18}-\frac{22\!\cdots\!18}{21\!\cdots\!75}a^{17}+\frac{73\!\cdots\!51}{42\!\cdots\!95}a^{16}+\frac{38\!\cdots\!91}{43\!\cdots\!75}a^{15}-\frac{37\!\cdots\!23}{21\!\cdots\!75}a^{14}+\frac{29\!\cdots\!96}{21\!\cdots\!75}a^{13}-\frac{10\!\cdots\!43}{21\!\cdots\!75}a^{12}+\frac{74\!\cdots\!72}{85\!\cdots\!39}a^{11}-\frac{10\!\cdots\!96}{43\!\cdots\!75}a^{10}+\frac{78\!\cdots\!69}{21\!\cdots\!75}a^{9}+\frac{23\!\cdots\!30}{85\!\cdots\!39}a^{8}-\frac{82\!\cdots\!47}{21\!\cdots\!75}a^{7}-\frac{21\!\cdots\!74}{30\!\cdots\!25}a^{6}+\frac{20\!\cdots\!91}{88\!\cdots\!75}a^{5}+\frac{32\!\cdots\!73}{43\!\cdots\!75}a^{4}-\frac{37\!\cdots\!11}{12\!\cdots\!65}a^{3}+\frac{22\!\cdots\!48}{32\!\cdots\!75}a^{2}-\frac{78\!\cdots\!02}{75\!\cdots\!25}a+\frac{19\!\cdots\!55}{98\!\cdots\!99}$, $\frac{13\!\cdots\!61}{47\!\cdots\!25}a^{19}+\frac{92\!\cdots\!24}{47\!\cdots\!25}a^{18}-\frac{26\!\cdots\!58}{47\!\cdots\!25}a^{17}+\frac{48\!\cdots\!23}{95\!\cdots\!85}a^{16}+\frac{16\!\cdots\!74}{47\!\cdots\!25}a^{15}-\frac{67\!\cdots\!68}{47\!\cdots\!25}a^{14}-\frac{13\!\cdots\!69}{47\!\cdots\!25}a^{13}-\frac{56\!\cdots\!58}{47\!\cdots\!25}a^{12}+\frac{56\!\cdots\!31}{95\!\cdots\!85}a^{11}-\frac{36\!\cdots\!43}{11\!\cdots\!75}a^{10}+\frac{93\!\cdots\!94}{47\!\cdots\!25}a^{9}+\frac{89\!\cdots\!52}{95\!\cdots\!85}a^{8}-\frac{14\!\cdots\!22}{47\!\cdots\!25}a^{7}-\frac{69\!\cdots\!83}{47\!\cdots\!25}a^{6}-\frac{80\!\cdots\!92}{67\!\cdots\!75}a^{5}+\frac{10\!\cdots\!49}{57\!\cdots\!25}a^{4}-\frac{12\!\cdots\!73}{20\!\cdots\!47}a^{3}+\frac{14\!\cdots\!94}{97\!\cdots\!25}a^{2}-\frac{33\!\cdots\!06}{72\!\cdots\!25}a-\frac{41\!\cdots\!82}{52\!\cdots\!71}$, $\frac{77\!\cdots\!59}{33\!\cdots\!75}a^{19}-\frac{20\!\cdots\!81}{17\!\cdots\!25}a^{18}+\frac{91\!\cdots\!78}{33\!\cdots\!75}a^{17}-\frac{33\!\cdots\!52}{66\!\cdots\!95}a^{16}-\frac{61\!\cdots\!37}{47\!\cdots\!25}a^{15}+\frac{10\!\cdots\!43}{33\!\cdots\!75}a^{14}-\frac{63\!\cdots\!21}{33\!\cdots\!75}a^{13}+\frac{38\!\cdots\!88}{33\!\cdots\!75}a^{12}-\frac{12\!\cdots\!40}{45\!\cdots\!31}a^{11}+\frac{43\!\cdots\!07}{47\!\cdots\!25}a^{10}-\frac{74\!\cdots\!46}{17\!\cdots\!25}a^{9}-\frac{93\!\cdots\!88}{39\!\cdots\!35}a^{8}-\frac{32\!\cdots\!03}{33\!\cdots\!75}a^{7}-\frac{69\!\cdots\!26}{47\!\cdots\!25}a^{6}+\frac{23\!\cdots\!06}{67\!\cdots\!75}a^{5}-\frac{14\!\cdots\!28}{67\!\cdots\!75}a^{4}+\frac{32\!\cdots\!91}{19\!\cdots\!65}a^{3}-\frac{19\!\cdots\!27}{97\!\cdots\!25}a^{2}+\frac{22\!\cdots\!16}{51\!\cdots\!75}a-\frac{51\!\cdots\!63}{90\!\cdots\!27}$, $\frac{77\!\cdots\!41}{33\!\cdots\!75}a^{19}+\frac{42\!\cdots\!14}{33\!\cdots\!75}a^{18}+\frac{14\!\cdots\!76}{19\!\cdots\!75}a^{17}-\frac{18\!\cdots\!54}{66\!\cdots\!95}a^{16}+\frac{10\!\cdots\!97}{47\!\cdots\!25}a^{15}-\frac{24\!\cdots\!68}{33\!\cdots\!75}a^{14}-\frac{19\!\cdots\!49}{33\!\cdots\!75}a^{13}-\frac{19\!\cdots\!88}{33\!\cdots\!75}a^{12}-\frac{71\!\cdots\!41}{66\!\cdots\!95}a^{11}+\frac{16\!\cdots\!83}{47\!\cdots\!25}a^{10}-\frac{27\!\cdots\!41}{33\!\cdots\!75}a^{9}+\frac{72\!\cdots\!76}{66\!\cdots\!95}a^{8}+\frac{32\!\cdots\!11}{77\!\cdots\!25}a^{7}-\frac{50\!\cdots\!59}{47\!\cdots\!25}a^{6}-\frac{81\!\cdots\!71}{67\!\cdots\!75}a^{5}-\frac{25\!\cdots\!36}{97\!\cdots\!25}a^{4}-\frac{16\!\cdots\!76}{19\!\cdots\!65}a^{3}-\frac{32\!\cdots\!38}{97\!\cdots\!25}a^{2}+\frac{65\!\cdots\!24}{51\!\cdots\!75}a-\frac{35\!\cdots\!75}{15\!\cdots\!59}$, $\frac{40\!\cdots\!31}{33\!\cdots\!75}a^{19}-\frac{29\!\cdots\!66}{33\!\cdots\!75}a^{18}+\frac{63\!\cdots\!81}{19\!\cdots\!75}a^{17}-\frac{34\!\cdots\!81}{66\!\cdots\!95}a^{16}-\frac{71\!\cdots\!76}{11\!\cdots\!75}a^{15}+\frac{17\!\cdots\!17}{33\!\cdots\!75}a^{14}-\frac{39\!\cdots\!51}{17\!\cdots\!25}a^{13}+\frac{29\!\cdots\!37}{33\!\cdots\!75}a^{12}-\frac{21\!\cdots\!06}{66\!\cdots\!95}a^{11}+\frac{38\!\cdots\!53}{47\!\cdots\!25}a^{10}-\frac{31\!\cdots\!71}{33\!\cdots\!75}a^{9}-\frac{94\!\cdots\!73}{66\!\cdots\!95}a^{8}+\frac{15\!\cdots\!88}{33\!\cdots\!75}a^{7}-\frac{77\!\cdots\!64}{47\!\cdots\!25}a^{6}+\frac{95\!\cdots\!79}{67\!\cdots\!75}a^{5}-\frac{21\!\cdots\!32}{67\!\cdots\!75}a^{4}+\frac{24\!\cdots\!02}{19\!\cdots\!65}a^{3}-\frac{36\!\cdots\!23}{97\!\cdots\!25}a^{2}+\frac{32\!\cdots\!74}{51\!\cdots\!75}a-\frac{91\!\cdots\!24}{15\!\cdots\!59}$, $\frac{20\!\cdots\!63}{33\!\cdots\!75}a^{19}-\frac{17\!\cdots\!08}{33\!\cdots\!75}a^{18}+\frac{44\!\cdots\!61}{33\!\cdots\!75}a^{17}+\frac{81\!\cdots\!96}{66\!\cdots\!95}a^{16}-\frac{52\!\cdots\!89}{47\!\cdots\!25}a^{15}+\frac{11\!\cdots\!76}{33\!\cdots\!75}a^{14}-\frac{30\!\cdots\!37}{33\!\cdots\!75}a^{13}+\frac{16\!\cdots\!66}{33\!\cdots\!75}a^{12}-\frac{22\!\cdots\!11}{13\!\cdots\!99}a^{11}+\frac{13\!\cdots\!37}{58\!\cdots\!25}a^{10}+\frac{92\!\cdots\!57}{33\!\cdots\!75}a^{9}-\frac{33\!\cdots\!67}{13\!\cdots\!99}a^{8}+\frac{97\!\cdots\!74}{33\!\cdots\!75}a^{7}+\frac{24\!\cdots\!13}{47\!\cdots\!25}a^{6}+\frac{23\!\cdots\!77}{67\!\cdots\!75}a^{5}-\frac{16\!\cdots\!41}{67\!\cdots\!75}a^{4}+\frac{11\!\cdots\!58}{19\!\cdots\!65}a^{3}-\frac{96\!\cdots\!99}{97\!\cdots\!25}a^{2}+\frac{81\!\cdots\!87}{51\!\cdots\!75}a-\frac{90\!\cdots\!98}{52\!\cdots\!71}$, $\frac{24\!\cdots\!77}{47\!\cdots\!25}a^{19}-\frac{21\!\cdots\!81}{67\!\cdots\!75}a^{18}+\frac{17\!\cdots\!29}{47\!\cdots\!25}a^{17}+\frac{34\!\cdots\!62}{95\!\cdots\!85}a^{16}-\frac{20\!\cdots\!67}{47\!\cdots\!25}a^{15}+\frac{33\!\cdots\!49}{47\!\cdots\!25}a^{14}-\frac{23\!\cdots\!39}{67\!\cdots\!75}a^{13}+\frac{15\!\cdots\!44}{47\!\cdots\!25}a^{12}-\frac{51\!\cdots\!73}{95\!\cdots\!85}a^{11}+\frac{32\!\cdots\!72}{47\!\cdots\!25}a^{10}+\frac{15\!\cdots\!03}{47\!\cdots\!25}a^{9}-\frac{14\!\cdots\!44}{19\!\cdots\!57}a^{8}-\frac{75\!\cdots\!99}{47\!\cdots\!25}a^{7}+\frac{74\!\cdots\!14}{47\!\cdots\!25}a^{6}+\frac{20\!\cdots\!06}{67\!\cdots\!75}a^{5}-\frac{93\!\cdots\!62}{13\!\cdots\!75}a^{4}+\frac{11\!\cdots\!50}{38\!\cdots\!93}a^{3}-\frac{47\!\cdots\!21}{13\!\cdots\!75}a^{2}+\frac{49\!\cdots\!93}{72\!\cdots\!25}a-\frac{83\!\cdots\!66}{15\!\cdots\!59}$
| 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: | \( 80801816.0664 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 80801816.0664 \cdot 10092}{6\cdot\sqrt{35635746108202593567757236917916961401}}\cr\approx \mathstrut & 2.18324698408 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 20 |
| The 8 conjugacy class representatives for $D_{10}$ |
| Character table for $D_{10}$ |
Intermediate fields
| \(\Q(\sqrt{1897}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-5691}) \), \(\Q(\sqrt{-3}, \sqrt{1897})\), 5.5.3598609.1 x5, 10.10.24566124836069257.1, 10.0.3146846776576083.1 x5, 10.0.5969568335164829451.1 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | 10.0.3146846776576083.1, 10.0.5969568335164829451.1 |
| Minimal sibling: | 10.0.3146846776576083.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{10}$ | ${\href{/padicField/19.1.0.1}{1} }^{20}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.2.0.1}{2} }^{10}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(271\)
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |