Normalized defining polynomial
\( x^{20} - x^{19} - 11 x^{18} + 12 x^{17} + 76 x^{16} - 88 x^{15} - 318 x^{14} + 406 x^{13} + 946 x^{12} - 1352 x^{11} - 1783 x^{10} + 3334 x^{9} + 1837 x^{8} - 9549 x^{7} + 4445 x^{6} + 13860 x^{5} - 16127 x^{4} + 17590 x^{3} - 2068 x^{2} - 37412 x + 39601 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3206128490667995866421572265625=3^{10}\cdot 5^{10}\cdot 11^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.52$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(165=3\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{165}(1,·)$, $\chi_{165}(134,·)$, $\chi_{165}(71,·)$, $\chi_{165}(136,·)$, $\chi_{165}(74,·)$, $\chi_{165}(139,·)$, $\chi_{165}(79,·)$, $\chi_{165}(16,·)$, $\chi_{165}(146,·)$, $\chi_{165}(19,·)$, $\chi_{165}(149,·)$, $\chi_{165}(86,·)$, $\chi_{165}(26,·)$, $\chi_{165}(91,·)$, $\chi_{165}(29,·)$, $\chi_{165}(94,·)$, $\chi_{165}(31,·)$, $\chi_{165}(164,·)$, $\chi_{165}(109,·)$, $\chi_{165}(56,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{89} a^{12} + \frac{34}{89} a^{11} - \frac{7}{89} a^{10} + \frac{42}{89} a^{9} + \frac{32}{89} a^{8} + \frac{44}{89} a^{7} + \frac{19}{89} a^{6} + \frac{3}{89} a^{5} + \frac{5}{89} a^{4} - \frac{40}{89} a^{3} - \frac{43}{89} a^{2} - \frac{16}{89} a + \frac{4}{89}$, $\frac{1}{89} a^{13} - \frac{6}{89} a^{11} + \frac{13}{89} a^{10} + \frac{28}{89} a^{9} + \frac{24}{89} a^{8} + \frac{36}{89} a^{7} - \frac{20}{89} a^{6} - \frac{8}{89} a^{5} - \frac{32}{89} a^{4} - \frac{18}{89} a^{3} + \frac{22}{89} a^{2} + \frac{14}{89} a + \frac{42}{89}$, $\frac{1}{89} a^{14} + \frac{39}{89} a^{11} - \frac{14}{89} a^{10} + \frac{9}{89} a^{9} - \frac{39}{89} a^{8} - \frac{23}{89} a^{7} + \frac{17}{89} a^{6} - \frac{14}{89} a^{5} + \frac{12}{89} a^{4} - \frac{40}{89} a^{3} + \frac{23}{89} a^{2} + \frac{35}{89} a + \frac{24}{89}$, $\frac{1}{89} a^{15} - \frac{5}{89} a^{11} + \frac{15}{89} a^{10} + \frac{14}{89} a^{9} - \frac{25}{89} a^{8} - \frac{8}{89} a^{7} - \frac{43}{89} a^{6} - \frac{16}{89} a^{5} + \frac{32}{89} a^{4} - \frac{19}{89} a^{3} + \frac{21}{89} a^{2} + \frac{25}{89} a + \frac{22}{89}$, $\frac{1}{89} a^{16} + \frac{7}{89} a^{11} - \frac{21}{89} a^{10} + \frac{7}{89} a^{9} - \frac{26}{89} a^{8} - \frac{1}{89} a^{7} - \frac{10}{89} a^{6} - \frac{42}{89} a^{5} + \frac{6}{89} a^{4} - \frac{1}{89} a^{3} - \frac{12}{89} a^{2} + \frac{31}{89} a + \frac{20}{89}$, $\frac{1}{3827} a^{17} - \frac{12}{3827} a^{16} - \frac{13}{3827} a^{15} - \frac{13}{3827} a^{14} + \frac{10}{3827} a^{13} - \frac{7}{3827} a^{12} - \frac{638}{3827} a^{11} + \frac{1008}{3827} a^{10} + \frac{707}{3827} a^{9} + \frac{1736}{3827} a^{8} - \frac{1097}{3827} a^{7} - \frac{406}{3827} a^{6} + \frac{1045}{3827} a^{5} + \frac{923}{3827} a^{4} - \frac{366}{3827} a^{3} - \frac{1088}{3827} a^{2} - \frac{1213}{3827} a + \frac{238}{3827}$, $\frac{1}{3827} a^{18} + \frac{15}{3827} a^{16} + \frac{3}{3827} a^{15} - \frac{17}{3827} a^{14} - \frac{16}{3827} a^{13} + \frac{9}{3827} a^{12} + \frac{1393}{3827} a^{11} - \frac{656}{3827} a^{10} - \frac{14}{3827} a^{9} - \frac{561}{3827} a^{8} + \frac{1781}{3827} a^{7} + \frac{44}{89} a^{6} + \frac{1079}{3827} a^{5} - \frac{212}{3827} a^{4} + \frac{1099}{3827} a^{3} + \frac{1899}{3827} a^{2} + \frac{1635}{3827} a - \frac{799}{3827}$, $\frac{1}{34054663175532437375954755397} a^{19} - \frac{4294998099160415235013627}{34054663175532437375954755397} a^{18} - \frac{2419881960381393667845192}{34054663175532437375954755397} a^{17} - \frac{143840111638615490062581933}{34054663175532437375954755397} a^{16} + \frac{101472651951132298900697922}{34054663175532437375954755397} a^{15} - \frac{134516616592704378128870079}{34054663175532437375954755397} a^{14} + \frac{140017777589269302288932232}{34054663175532437375954755397} a^{13} - \frac{48216778282026782581185852}{34054663175532437375954755397} a^{12} + \frac{3410558970106953262624905773}{34054663175532437375954755397} a^{11} - \frac{405226024408215702595032224}{34054663175532437375954755397} a^{10} - \frac{367381516449699616921729667}{791968911058893892464064079} a^{9} + \frac{4588393099837690929490253742}{34054663175532437375954755397} a^{8} - \frac{78681258812779489614472062}{791968911058893892464064079} a^{7} - \frac{10448299870943388147844487061}{34054663175532437375954755397} a^{6} - \frac{4450756443937130304505158075}{34054663175532437375954755397} a^{5} + \frac{12099566439857476633339925579}{34054663175532437375954755397} a^{4} - \frac{1656107920618649593344511234}{34054663175532437375954755397} a^{3} - \frac{35502800948915501026247983}{382636664893622891864660173} a^{2} + \frac{10994621556969338456038237049}{34054663175532437375954755397} a - \frac{40119727098119999741221519}{171128960681067524502285203}$
Class group and class number
$C_{4}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{5450446576734732557919}{382636664893622891864660173} a^{19} - \frac{12760858735848176183373}{382636664893622891864660173} a^{18} + \frac{82955089180632474360445}{382636664893622891864660173} a^{17} + \frac{110344197394287202945456}{382636664893622891864660173} a^{16} - \frac{607494296181568648320204}{382636664893622891864660173} a^{15} - \frac{673043153011224117595484}{382636664893622891864660173} a^{14} + \frac{2969566953334063594678210}{382636664893622891864660173} a^{13} + \frac{2075035862535446958249014}{382636664893622891864660173} a^{12} - \frac{9243361835267626662955182}{382636664893622891864660173} a^{11} - \frac{4348110769522642391116419}{382636664893622891864660173} a^{10} + \frac{20894271280123993655079924}{382636664893622891864660173} a^{9} - \frac{706793935034519176890177}{382636664893622891864660173} a^{8} - \frac{27280821987949417065613259}{382636664893622891864660173} a^{7} + \frac{43434800389480446204509735}{382636664893622891864660173} a^{6} + \frac{67576111754189054710187241}{382636664893622891864660173} a^{5} - \frac{186909336001920118415349983}{382636664893622891864660173} a^{4} + \frac{56356266711451635366016568}{382636664893622891864660173} a^{3} - \frac{298024862461600890993355}{8898527090549369578247911} a^{2} - \frac{410333637697643852401163724}{382636664893622891864660173} a + \frac{3319818332989134310114055}{1922797311023230612385227} \) (order $6$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5362955.97373 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{10}$ (as 20T3):
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{165}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-3}, \sqrt{-55})\), \(\Q(\zeta_{11})^+\), 10.10.1790566527853125.1, 10.0.52089208083.1, 10.0.7368586534375.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | R | R | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| 11 | Data not computed | ||||||