Normalized defining polynomial
\( x^{20} - x^{19} + 4 x^{18} - 7 x^{17} + 19 x^{16} - 40 x^{15} + 97 x^{14} - 217 x^{13} + 508 x^{12} - 1159 x^{11} + 2683 x^{10} + 3477 x^{9} + 4572 x^{8} + 5859 x^{7} + 7857 x^{6} + 9720 x^{5} + 13851 x^{4} + 15309 x^{3} + 26244 x^{2} + 19683 x + 59049 \)
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: | \(766481815643182771348259698369=11^{18}\cdot 13^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.21$ | 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: | $11, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(143=11\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{143}(64,·)$, $\chi_{143}(1,·)$, $\chi_{143}(131,·)$, $\chi_{143}(129,·)$, $\chi_{143}(12,·)$, $\chi_{143}(142,·)$, $\chi_{143}(79,·)$, $\chi_{143}(14,·)$, $\chi_{143}(25,·)$, $\chi_{143}(90,·)$, $\chi_{143}(27,·)$, $\chi_{143}(92,·)$, $\chi_{143}(38,·)$, $\chi_{143}(103,·)$, $\chi_{143}(40,·)$, $\chi_{143}(105,·)$, $\chi_{143}(51,·)$, $\chi_{143}(116,·)$, $\chi_{143}(53,·)$, $\chi_{143}(118,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{8049} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1159}{2683}$, $\frac{1}{24147} a^{12} - \frac{1}{24147} a^{11} + \frac{4}{9} a^{10} + \frac{2}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1159}{8049} a + \frac{508}{2683}$, $\frac{1}{72441} a^{13} - \frac{1}{72441} a^{12} + \frac{4}{72441} a^{11} + \frac{2}{27} a^{10} + \frac{10}{27} a^{9} - \frac{4}{27} a^{8} + \frac{7}{27} a^{7} + \frac{8}{27} a^{6} + \frac{13}{27} a^{5} + \frac{11}{27} a^{4} + \frac{1}{27} a^{3} + \frac{1159}{24147} a^{2} + \frac{508}{8049} a + \frac{217}{2683}$, $\frac{1}{217323} a^{14} - \frac{1}{217323} a^{13} + \frac{4}{217323} a^{12} - \frac{7}{217323} a^{11} + \frac{10}{81} a^{10} - \frac{4}{81} a^{9} + \frac{34}{81} a^{8} + \frac{35}{81} a^{7} - \frac{14}{81} a^{6} + \frac{38}{81} a^{5} + \frac{1}{81} a^{4} + \frac{1159}{72441} a^{3} + \frac{508}{24147} a^{2} + \frac{217}{8049} a + \frac{97}{2683}$, $\frac{1}{651969} a^{15} - \frac{1}{651969} a^{14} + \frac{4}{651969} a^{13} - \frac{7}{651969} a^{12} + \frac{19}{651969} a^{11} - \frac{4}{243} a^{10} + \frac{34}{243} a^{9} - \frac{46}{243} a^{8} - \frac{95}{243} a^{7} - \frac{43}{243} a^{6} + \frac{1}{243} a^{5} + \frac{1159}{217323} a^{4} + \frac{508}{72441} a^{3} + \frac{217}{24147} a^{2} + \frac{97}{8049} a + \frac{40}{2683}$, $\frac{1}{1955907} a^{16} - \frac{1}{1955907} a^{15} + \frac{4}{1955907} a^{14} - \frac{7}{1955907} a^{13} + \frac{19}{1955907} a^{12} - \frac{40}{1955907} a^{11} + \frac{34}{729} a^{10} - \frac{46}{729} a^{9} + \frac{148}{729} a^{8} - \frac{286}{729} a^{7} + \frac{1}{729} a^{6} + \frac{1159}{651969} a^{5} + \frac{508}{217323} a^{4} + \frac{217}{72441} a^{3} + \frac{97}{24147} a^{2} + \frac{40}{8049} a + \frac{19}{2683}$, $\frac{1}{5867721} a^{17} - \frac{1}{5867721} a^{16} + \frac{4}{5867721} a^{15} - \frac{7}{5867721} a^{14} + \frac{19}{5867721} a^{13} - \frac{40}{5867721} a^{12} + \frac{97}{5867721} a^{11} + \frac{683}{2187} a^{10} - \frac{581}{2187} a^{9} + \frac{443}{2187} a^{8} + \frac{1}{2187} a^{7} + \frac{1159}{1955907} a^{6} + \frac{508}{651969} a^{5} + \frac{217}{217323} a^{4} + \frac{97}{72441} a^{3} + \frac{40}{24147} a^{2} + \frac{19}{8049} a + \frac{7}{2683}$, $\frac{1}{17603163} a^{18} - \frac{1}{17603163} a^{17} + \frac{4}{17603163} a^{16} - \frac{7}{17603163} a^{15} + \frac{19}{17603163} a^{14} - \frac{40}{17603163} a^{13} + \frac{97}{17603163} a^{12} - \frac{217}{17603163} a^{11} - \frac{2768}{6561} a^{10} - \frac{1744}{6561} a^{9} + \frac{1}{6561} a^{8} + \frac{1159}{5867721} a^{7} + \frac{508}{1955907} a^{6} + \frac{217}{651969} a^{5} + \frac{97}{217323} a^{4} + \frac{40}{72441} a^{3} + \frac{19}{24147} a^{2} + \frac{7}{8049} a + \frac{4}{2683}$, $\frac{1}{52809489} a^{19} - \frac{1}{52809489} a^{18} + \frac{4}{52809489} a^{17} - \frac{7}{52809489} a^{16} + \frac{19}{52809489} a^{15} - \frac{40}{52809489} a^{14} + \frac{97}{52809489} a^{13} - \frac{217}{52809489} a^{12} + \frac{508}{52809489} a^{11} - \frac{8305}{19683} a^{10} + \frac{1}{19683} a^{9} + \frac{1159}{17603163} a^{8} + \frac{508}{5867721} a^{7} + \frac{217}{1955907} a^{6} + \frac{97}{651969} a^{5} + \frac{40}{217323} a^{4} + \frac{19}{72441} a^{3} + \frac{7}{24147} a^{2} + \frac{4}{8049} a + \frac{1}{2683}$
Class group and class number
$C_{25}$, which has order $25$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{7}{217323} a^{15} - \frac{75316}{217323} a^{4} \) (order $22$) | 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: | \( 2015201.7242 \) (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
| \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{13})\), \(\Q(\zeta_{11})^+\), 10.0.875489472034463.1, 10.10.79589952003133.1, \(\Q(\zeta_{11})\) |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | R | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| 13 | Data not computed | ||||||