Normalized defining polynomial
\( x^{20} - 5 x^{19} + 16 x^{18} - 34 x^{17} + 49 x^{16} - 74 x^{15} + 105 x^{14} - 152 x^{13} + 165 x^{12} + \cdots + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-97907636384234445551929843\) \(\medspace = -\,163\cdot 521^{2}\cdot 38569^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.93\) | 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: | $163^{1/2}521^{1/2}38569^{1/2}\approx 57231.068371995294$ | ||
Ramified primes: | \(163\), \(521\), \(38569\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-163}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{49}a^{18}+\frac{20}{49}a^{17}-\frac{5}{49}a^{16}+\frac{5}{49}a^{15}-\frac{2}{7}a^{14}+\frac{9}{49}a^{13}-\frac{20}{49}a^{12}+\frac{1}{49}a^{10}-\frac{8}{49}a^{9}+\frac{19}{49}a^{8}-\frac{3}{7}a^{7}-\frac{18}{49}a^{6}-\frac{16}{49}a^{5}+\frac{12}{49}a^{4}-\frac{8}{49}a^{3}-\frac{18}{49}a+\frac{19}{49}$, $\frac{1}{11\!\cdots\!51}a^{19}+\frac{84\!\cdots\!16}{11\!\cdots\!51}a^{18}+\frac{66\!\cdots\!45}{11\!\cdots\!51}a^{17}-\frac{34\!\cdots\!08}{11\!\cdots\!51}a^{16}-\frac{46\!\cdots\!35}{11\!\cdots\!51}a^{15}+\frac{23\!\cdots\!85}{11\!\cdots\!51}a^{14}+\frac{51\!\cdots\!02}{11\!\cdots\!51}a^{13}+\frac{65\!\cdots\!08}{11\!\cdots\!51}a^{12}+\frac{55\!\cdots\!77}{11\!\cdots\!51}a^{11}-\frac{72\!\cdots\!43}{11\!\cdots\!51}a^{10}-\frac{49\!\cdots\!34}{11\!\cdots\!51}a^{9}-\frac{13\!\cdots\!87}{11\!\cdots\!51}a^{8}+\frac{54\!\cdots\!99}{11\!\cdots\!51}a^{7}-\frac{25\!\cdots\!43}{11\!\cdots\!51}a^{6}-\frac{51\!\cdots\!11}{11\!\cdots\!51}a^{5}+\frac{32\!\cdots\!38}{11\!\cdots\!51}a^{4}+\frac{56\!\cdots\!02}{11\!\cdots\!51}a^{3}+\frac{32\!\cdots\!18}{11\!\cdots\!51}a^{2}+\frac{39\!\cdots\!03}{11\!\cdots\!51}a+\frac{24\!\cdots\!65}{11\!\cdots\!51}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | 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: | $\frac{21\!\cdots\!34}{11\!\cdots\!51}a^{19}-\frac{12\!\cdots\!59}{11\!\cdots\!51}a^{18}+\frac{41\!\cdots\!26}{11\!\cdots\!51}a^{17}-\frac{93\!\cdots\!64}{11\!\cdots\!51}a^{16}+\frac{14\!\cdots\!49}{11\!\cdots\!51}a^{15}-\frac{21\!\cdots\!89}{11\!\cdots\!51}a^{14}+\frac{42\!\cdots\!30}{16\!\cdots\!93}a^{13}-\frac{42\!\cdots\!62}{11\!\cdots\!51}a^{12}+\frac{50\!\cdots\!59}{11\!\cdots\!51}a^{11}-\frac{20\!\cdots\!51}{11\!\cdots\!51}a^{10}+\frac{46\!\cdots\!19}{11\!\cdots\!51}a^{9}-\frac{31\!\cdots\!85}{11\!\cdots\!51}a^{8}-\frac{92\!\cdots\!83}{11\!\cdots\!51}a^{7}+\frac{50\!\cdots\!41}{16\!\cdots\!93}a^{6}+\frac{82\!\cdots\!89}{11\!\cdots\!51}a^{5}+\frac{15\!\cdots\!03}{16\!\cdots\!93}a^{4}-\frac{14\!\cdots\!22}{11\!\cdots\!51}a^{3}-\frac{33\!\cdots\!42}{11\!\cdots\!51}a^{2}-\frac{24\!\cdots\!92}{16\!\cdots\!93}a-\frac{72\!\cdots\!18}{11\!\cdots\!51}$, $\frac{15\!\cdots\!65}{16\!\cdots\!93}a^{19}-\frac{37\!\cdots\!84}{11\!\cdots\!51}a^{18}+\frac{85\!\cdots\!37}{11\!\cdots\!51}a^{17}-\frac{73\!\cdots\!25}{11\!\cdots\!51}a^{16}-\frac{12\!\cdots\!55}{11\!\cdots\!51}a^{15}+\frac{29\!\cdots\!98}{16\!\cdots\!93}a^{14}-\frac{38\!\cdots\!51}{11\!\cdots\!51}a^{13}+\frac{51\!\cdots\!85}{11\!\cdots\!51}a^{12}-\frac{18\!\cdots\!73}{16\!\cdots\!93}a^{11}+\frac{31\!\cdots\!68}{11\!\cdots\!51}a^{10}-\frac{74\!\cdots\!48}{11\!\cdots\!51}a^{9}-\frac{63\!\cdots\!96}{11\!\cdots\!51}a^{8}-\frac{57\!\cdots\!16}{16\!\cdots\!93}a^{7}-\frac{10\!\cdots\!11}{11\!\cdots\!51}a^{6}+\frac{28\!\cdots\!80}{11\!\cdots\!51}a^{5}+\frac{26\!\cdots\!85}{11\!\cdots\!51}a^{4}+\frac{18\!\cdots\!53}{11\!\cdots\!51}a^{3}-\frac{10\!\cdots\!76}{16\!\cdots\!93}a^{2}-\frac{46\!\cdots\!77}{11\!\cdots\!51}a-\frac{32\!\cdots\!25}{11\!\cdots\!51}$, $\frac{30\!\cdots\!12}{11\!\cdots\!51}a^{19}-\frac{13\!\cdots\!66}{11\!\cdots\!51}a^{18}+\frac{10\!\cdots\!01}{16\!\cdots\!93}a^{17}-\frac{20\!\cdots\!60}{11\!\cdots\!51}a^{16}+\frac{40\!\cdots\!93}{11\!\cdots\!51}a^{15}-\frac{54\!\cdots\!38}{11\!\cdots\!51}a^{14}+\frac{21\!\cdots\!63}{11\!\cdots\!51}a^{13}-\frac{46\!\cdots\!33}{11\!\cdots\!51}a^{12}+\frac{77\!\cdots\!06}{11\!\cdots\!51}a^{11}-\frac{97\!\cdots\!59}{11\!\cdots\!51}a^{10}-\frac{14\!\cdots\!16}{11\!\cdots\!51}a^{9}+\frac{52\!\cdots\!63}{11\!\cdots\!51}a^{8}+\frac{12\!\cdots\!26}{11\!\cdots\!51}a^{7}-\frac{89\!\cdots\!80}{11\!\cdots\!51}a^{6}-\frac{72\!\cdots\!98}{11\!\cdots\!51}a^{5}+\frac{15\!\cdots\!24}{11\!\cdots\!51}a^{4}+\frac{94\!\cdots\!18}{11\!\cdots\!51}a^{3}+\frac{45\!\cdots\!46}{11\!\cdots\!51}a^{2}+\frac{93\!\cdots\!80}{11\!\cdots\!51}a-\frac{14\!\cdots\!34}{11\!\cdots\!51}$, $\frac{88\!\cdots\!46}{11\!\cdots\!51}a^{19}-\frac{12\!\cdots\!91}{11\!\cdots\!51}a^{18}+\frac{53\!\cdots\!55}{11\!\cdots\!51}a^{17}+\frac{10\!\cdots\!62}{11\!\cdots\!51}a^{16}-\frac{37\!\cdots\!93}{11\!\cdots\!51}a^{15}+\frac{47\!\cdots\!60}{11\!\cdots\!51}a^{14}-\frac{12\!\cdots\!49}{11\!\cdots\!51}a^{13}+\frac{17\!\cdots\!70}{11\!\cdots\!51}a^{12}-\frac{28\!\cdots\!04}{11\!\cdots\!51}a^{11}+\frac{40\!\cdots\!50}{11\!\cdots\!51}a^{10}-\frac{82\!\cdots\!24}{16\!\cdots\!93}a^{9}+\frac{50\!\cdots\!62}{11\!\cdots\!51}a^{8}-\frac{78\!\cdots\!83}{11\!\cdots\!51}a^{7}-\frac{94\!\cdots\!79}{11\!\cdots\!51}a^{6}-\frac{12\!\cdots\!43}{11\!\cdots\!51}a^{5}+\frac{10\!\cdots\!60}{11\!\cdots\!51}a^{4}+\frac{10\!\cdots\!71}{11\!\cdots\!51}a^{3}-\frac{30\!\cdots\!54}{11\!\cdots\!51}a^{2}-\frac{28\!\cdots\!26}{11\!\cdots\!51}a-\frac{10\!\cdots\!86}{11\!\cdots\!51}$, $\frac{19\!\cdots\!86}{11\!\cdots\!51}a^{19}-\frac{10\!\cdots\!94}{11\!\cdots\!51}a^{18}+\frac{33\!\cdots\!86}{11\!\cdots\!51}a^{17}-\frac{74\!\cdots\!63}{11\!\cdots\!51}a^{16}+\frac{16\!\cdots\!11}{16\!\cdots\!93}a^{15}-\frac{17\!\cdots\!48}{11\!\cdots\!51}a^{14}+\frac{23\!\cdots\!46}{11\!\cdots\!51}a^{13}-\frac{69\!\cdots\!66}{23\!\cdots\!99}a^{12}+\frac{38\!\cdots\!64}{11\!\cdots\!51}a^{11}-\frac{13\!\cdots\!22}{11\!\cdots\!51}a^{10}+\frac{12\!\cdots\!18}{11\!\cdots\!51}a^{9}-\frac{35\!\cdots\!99}{16\!\cdots\!93}a^{8}-\frac{23\!\cdots\!43}{11\!\cdots\!51}a^{7}+\frac{12\!\cdots\!55}{11\!\cdots\!51}a^{6}+\frac{11\!\cdots\!74}{11\!\cdots\!51}a^{5}+\frac{22\!\cdots\!64}{11\!\cdots\!51}a^{4}-\frac{71\!\cdots\!54}{23\!\cdots\!99}a^{3}-\frac{57\!\cdots\!69}{11\!\cdots\!51}a^{2}-\frac{12\!\cdots\!41}{11\!\cdots\!51}a+\frac{11\!\cdots\!29}{23\!\cdots\!99}$, $\frac{13\!\cdots\!12}{11\!\cdots\!51}a^{19}-\frac{69\!\cdots\!79}{11\!\cdots\!51}a^{18}+\frac{22\!\cdots\!15}{11\!\cdots\!51}a^{17}-\frac{46\!\cdots\!55}{11\!\cdots\!51}a^{16}+\frac{64\!\cdots\!00}{11\!\cdots\!51}a^{15}-\frac{88\!\cdots\!82}{11\!\cdots\!51}a^{14}+\frac{12\!\cdots\!52}{11\!\cdots\!51}a^{13}-\frac{17\!\cdots\!17}{11\!\cdots\!51}a^{12}+\frac{17\!\cdots\!71}{11\!\cdots\!51}a^{11}+\frac{24\!\cdots\!71}{11\!\cdots\!51}a^{10}-\frac{46\!\cdots\!13}{11\!\cdots\!51}a^{9}-\frac{14\!\cdots\!34}{11\!\cdots\!51}a^{8}-\frac{16\!\cdots\!15}{11\!\cdots\!51}a^{7}+\frac{18\!\cdots\!19}{11\!\cdots\!51}a^{6}+\frac{23\!\cdots\!06}{16\!\cdots\!93}a^{5}+\frac{52\!\cdots\!63}{11\!\cdots\!51}a^{4}-\frac{70\!\cdots\!27}{11\!\cdots\!51}a^{3}-\frac{65\!\cdots\!60}{11\!\cdots\!51}a^{2}+\frac{22\!\cdots\!43}{11\!\cdots\!51}a+\frac{22\!\cdots\!61}{11\!\cdots\!51}$, $\frac{33\!\cdots\!24}{11\!\cdots\!51}a^{19}-\frac{17\!\cdots\!12}{11\!\cdots\!51}a^{18}+\frac{56\!\cdots\!37}{11\!\cdots\!51}a^{17}-\frac{12\!\cdots\!66}{11\!\cdots\!51}a^{16}+\frac{18\!\cdots\!40}{11\!\cdots\!51}a^{15}-\frac{26\!\cdots\!58}{11\!\cdots\!51}a^{14}+\frac{38\!\cdots\!05}{11\!\cdots\!51}a^{13}-\frac{55\!\cdots\!08}{11\!\cdots\!51}a^{12}+\frac{61\!\cdots\!25}{11\!\cdots\!51}a^{11}-\frac{16\!\cdots\!04}{11\!\cdots\!51}a^{10}+\frac{13\!\cdots\!56}{11\!\cdots\!51}a^{9}-\frac{50\!\cdots\!59}{11\!\cdots\!51}a^{8}-\frac{31\!\cdots\!23}{11\!\cdots\!51}a^{7}+\frac{34\!\cdots\!49}{11\!\cdots\!51}a^{6}+\frac{25\!\cdots\!89}{11\!\cdots\!51}a^{5}+\frac{29\!\cdots\!82}{11\!\cdots\!51}a^{4}-\frac{10\!\cdots\!48}{11\!\cdots\!51}a^{3}-\frac{36\!\cdots\!64}{11\!\cdots\!51}a^{2}-\frac{32\!\cdots\!61}{11\!\cdots\!51}a-\frac{59\!\cdots\!80}{11\!\cdots\!51}$, $\frac{23\!\cdots\!43}{23\!\cdots\!99}a^{19}-\frac{13\!\cdots\!54}{23\!\cdots\!99}a^{18}+\frac{44\!\cdots\!60}{23\!\cdots\!99}a^{17}-\frac{98\!\cdots\!11}{23\!\cdots\!99}a^{16}+\frac{13\!\cdots\!87}{23\!\cdots\!99}a^{15}-\frac{17\!\cdots\!22}{23\!\cdots\!99}a^{14}+\frac{24\!\cdots\!78}{23\!\cdots\!99}a^{13}-\frac{36\!\cdots\!28}{23\!\cdots\!99}a^{12}+\frac{42\!\cdots\!88}{23\!\cdots\!99}a^{11}-\frac{33\!\cdots\!78}{23\!\cdots\!99}a^{10}-\frac{25\!\cdots\!53}{23\!\cdots\!99}a^{9}-\frac{35\!\cdots\!12}{23\!\cdots\!99}a^{8}+\frac{18\!\cdots\!03}{23\!\cdots\!99}a^{7}+\frac{57\!\cdots\!70}{23\!\cdots\!99}a^{6}+\frac{73\!\cdots\!06}{23\!\cdots\!99}a^{5}-\frac{22\!\cdots\!67}{23\!\cdots\!99}a^{4}-\frac{27\!\cdots\!88}{23\!\cdots\!99}a^{3}+\frac{36\!\cdots\!49}{23\!\cdots\!99}a^{2}+\frac{19\!\cdots\!59}{23\!\cdots\!99}a+\frac{31\!\cdots\!56}{23\!\cdots\!99}$, $\frac{61\!\cdots\!02}{11\!\cdots\!51}a^{19}-\frac{32\!\cdots\!18}{11\!\cdots\!51}a^{18}+\frac{10\!\cdots\!23}{11\!\cdots\!51}a^{17}-\frac{22\!\cdots\!75}{11\!\cdots\!51}a^{16}+\frac{33\!\cdots\!70}{11\!\cdots\!51}a^{15}-\frac{50\!\cdots\!15}{11\!\cdots\!51}a^{14}+\frac{73\!\cdots\!25}{11\!\cdots\!51}a^{13}-\frac{10\!\cdots\!90}{11\!\cdots\!51}a^{12}+\frac{12\!\cdots\!78}{11\!\cdots\!51}a^{11}-\frac{40\!\cdots\!92}{11\!\cdots\!51}a^{10}+\frac{41\!\cdots\!73}{11\!\cdots\!51}a^{9}-\frac{11\!\cdots\!37}{11\!\cdots\!51}a^{8}-\frac{53\!\cdots\!51}{11\!\cdots\!51}a^{7}+\frac{66\!\cdots\!80}{11\!\cdots\!51}a^{6}+\frac{83\!\cdots\!54}{16\!\cdots\!93}a^{5}+\frac{61\!\cdots\!75}{11\!\cdots\!51}a^{4}-\frac{31\!\cdots\!12}{11\!\cdots\!51}a^{3}-\frac{89\!\cdots\!89}{11\!\cdots\!51}a^{2}-\frac{98\!\cdots\!81}{11\!\cdots\!51}a+\frac{10\!\cdots\!13}{11\!\cdots\!51}$, $\frac{24\!\cdots\!99}{11\!\cdots\!51}a^{19}-\frac{13\!\cdots\!32}{11\!\cdots\!51}a^{18}+\frac{66\!\cdots\!55}{16\!\cdots\!93}a^{17}-\frac{10\!\cdots\!54}{11\!\cdots\!51}a^{16}+\frac{16\!\cdots\!71}{11\!\cdots\!51}a^{15}-\frac{24\!\cdots\!20}{11\!\cdots\!51}a^{14}+\frac{34\!\cdots\!01}{11\!\cdots\!51}a^{13}-\frac{50\!\cdots\!98}{11\!\cdots\!51}a^{12}+\frac{59\!\cdots\!37}{11\!\cdots\!51}a^{11}-\frac{27\!\cdots\!99}{11\!\cdots\!51}a^{10}+\frac{11\!\cdots\!24}{11\!\cdots\!51}a^{9}-\frac{45\!\cdots\!20}{11\!\cdots\!51}a^{8}-\frac{34\!\cdots\!91}{11\!\cdots\!51}a^{7}+\frac{38\!\cdots\!25}{11\!\cdots\!51}a^{6}+\frac{13\!\cdots\!01}{11\!\cdots\!51}a^{5}+\frac{10\!\cdots\!66}{11\!\cdots\!51}a^{4}-\frac{18\!\cdots\!13}{11\!\cdots\!51}a^{3}-\frac{23\!\cdots\!18}{11\!\cdots\!51}a^{2}-\frac{20\!\cdots\!65}{11\!\cdots\!51}a-\frac{65\!\cdots\!39}{11\!\cdots\!51}$, $\frac{26\!\cdots\!93}{11\!\cdots\!51}a^{19}-\frac{12\!\cdots\!92}{11\!\cdots\!51}a^{18}+\frac{38\!\cdots\!75}{11\!\cdots\!51}a^{17}-\frac{77\!\cdots\!93}{11\!\cdots\!51}a^{16}+\frac{10\!\cdots\!91}{11\!\cdots\!51}a^{15}-\frac{15\!\cdots\!42}{11\!\cdots\!51}a^{14}+\frac{22\!\cdots\!04}{11\!\cdots\!51}a^{13}-\frac{32\!\cdots\!79}{11\!\cdots\!51}a^{12}+\frac{32\!\cdots\!26}{11\!\cdots\!51}a^{11}+\frac{40\!\cdots\!17}{11\!\cdots\!51}a^{10}+\frac{19\!\cdots\!09}{16\!\cdots\!93}a^{9}-\frac{42\!\cdots\!53}{11\!\cdots\!51}a^{8}-\frac{43\!\cdots\!80}{11\!\cdots\!51}a^{7}+\frac{11\!\cdots\!89}{11\!\cdots\!51}a^{6}+\frac{35\!\cdots\!25}{11\!\cdots\!51}a^{5}+\frac{40\!\cdots\!51}{11\!\cdots\!51}a^{4}+\frac{28\!\cdots\!83}{11\!\cdots\!51}a^{3}-\frac{83\!\cdots\!41}{11\!\cdots\!51}a^{2}-\frac{74\!\cdots\!49}{11\!\cdots\!51}a-\frac{26\!\cdots\!62}{11\!\cdots\!51}$, $\frac{38\!\cdots\!97}{11\!\cdots\!51}a^{19}-\frac{11\!\cdots\!55}{11\!\cdots\!51}a^{18}+\frac{23\!\cdots\!60}{11\!\cdots\!51}a^{17}-\frac{13\!\cdots\!20}{11\!\cdots\!51}a^{16}-\frac{58\!\cdots\!14}{16\!\cdots\!93}a^{15}-\frac{15\!\cdots\!78}{11\!\cdots\!51}a^{14}+\frac{91\!\cdots\!91}{11\!\cdots\!51}a^{13}-\frac{29\!\cdots\!49}{16\!\cdots\!93}a^{12}+\frac{82\!\cdots\!85}{11\!\cdots\!51}a^{11}+\frac{30\!\cdots\!16}{11\!\cdots\!51}a^{10}+\frac{10\!\cdots\!42}{11\!\cdots\!51}a^{9}-\frac{31\!\cdots\!81}{23\!\cdots\!99}a^{8}-\frac{86\!\cdots\!07}{11\!\cdots\!51}a^{7}-\frac{72\!\cdots\!88}{11\!\cdots\!51}a^{6}+\frac{11\!\cdots\!90}{11\!\cdots\!51}a^{5}+\frac{13\!\cdots\!41}{11\!\cdots\!51}a^{4}+\frac{25\!\cdots\!67}{16\!\cdots\!93}a^{3}+\frac{56\!\cdots\!07}{11\!\cdots\!51}a^{2}-\frac{33\!\cdots\!40}{11\!\cdots\!51}a+\frac{12\!\cdots\!30}{16\!\cdots\!93}$ (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: | \( 173153.406765 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{7}\cdot 173153.406765 \cdot 1}{2\cdot\sqrt{97907636384234445551929843}}\cr\approx \mathstrut & 0.216486996090 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.C_2\wr S_5$ (as 20T1015):
A non-solvable group of order 3932160 |
The 506 conjugacy class representatives for $C_2^{10}.C_2\wr S_5$ |
Character table for $C_2^{10}.C_2\wr S_5$ |
Intermediate fields
5.5.38569.1, 10.6.775022803481.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 20.8.30631371843819989683233329.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }^{2}$ | $20$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(163\) | $\Q_{163}$ | $x + 161$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{163}$ | $x + 161$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
163.2.1.2 | $x^{2} + 163$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
\(521\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(38569\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |