Normalized defining polynomial
\( x^{20} - x^{19} + 4 x^{18} - 3 x^{17} + 9 x^{16} + 9 x^{14} + 9 x^{13} + 6 x^{12} + 23 x^{11} + 7 x^{10} + 23 x^{9} + 6 x^{8} + 9 x^{7} + 9 x^{6} + 9 x^{4} - 3 x^{3} + 4 x^{2} - x + 1 \)
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: | \(47644389152686853402121=3^{18}\cdot 223^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.61$ | 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: | $3, 223$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} + \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{33} a^{19} + \frac{4}{33} a^{17} + \frac{1}{33} a^{16} - \frac{1}{33} a^{15} - \frac{1}{33} a^{14} - \frac{1}{11} a^{13} - \frac{5}{33} a^{12} + \frac{1}{33} a^{11} + \frac{2}{33} a^{10} + \frac{3}{11} a^{9} + \frac{10}{33} a^{8} + \frac{16}{33} a^{7} + \frac{14}{33} a^{6} - \frac{10}{33} a^{5} + \frac{4}{11} a^{4} + \frac{10}{33} a^{3} + \frac{7}{33} a^{2} - \frac{1}{3} a - \frac{4}{11}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{8}{33} a^{19} + \frac{2}{3} a^{18} + \frac{10}{33} a^{17} + \frac{74}{33} a^{16} + \frac{14}{33} a^{15} + \frac{212}{33} a^{14} + \frac{130}{33} a^{13} + \frac{268}{33} a^{12} + \frac{283}{33} a^{11} + \frac{108}{11} a^{10} + \frac{622}{33} a^{9} + \frac{388}{33} a^{8} + \frac{458}{33} a^{7} + \frac{134}{33} a^{6} + \frac{74}{33} a^{5} + \frac{118}{33} a^{4} - \frac{8}{33} a^{3} + \frac{100}{33} a^{2} - \frac{4}{3} a + \frac{12}{11} \) (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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 2771.70187261 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times S_5$ (as 20T62):
| A non-solvable group of order 240 |
| The 14 conjugacy class representatives for $C_2\times S_5$ |
| Character table for $C_2\times S_5$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 10.4.72758649087.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
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 | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 | ||||||
| 223 | Data not computed | ||||||