Normalized defining polynomial
\( x^{20} - x^{19} + 3 x^{18} - 3 x^{17} + 8 x^{16} - 5 x^{15} + 12 x^{14} - 10 x^{13} + 16 x^{12} - 6 x^{11} + 8 x^{10} + x^{8} - 5 x^{7} + 7 x^{6} + 2 x^{5} - 8 x^{4} + 6 x^{3} - 2 x + 1 \)
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: | \(10595546867344882662901=11^{4}\cdot 19^{4}\cdot 199^{4}\cdot 3541\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.63$ | 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: | $11, 19, 199, 3541$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10} a^{15} - \frac{3}{10} a^{14} + \frac{1}{5} a^{13} - \frac{3}{10} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{2} a^{6} - \frac{1}{10} a^{4} - \frac{1}{5} a^{3} + \frac{3}{10} a - \frac{1}{10}$, $\frac{1}{10} a^{16} + \frac{3}{10} a^{14} + \frac{3}{10} a^{13} + \frac{3}{10} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{2} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} + \frac{2}{5} a^{3} + \frac{3}{10} a^{2} - \frac{1}{5} a - \frac{3}{10}$, $\frac{1}{10} a^{17} + \frac{1}{5} a^{14} - \frac{3}{10} a^{13} + \frac{1}{10} a^{12} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} + \frac{2}{5} a^{9} - \frac{1}{10} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{2} a^{5} - \frac{3}{10} a^{4} - \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{3}{10}$, $\frac{1}{50} a^{18} + \frac{1}{25} a^{17} + \frac{1}{25} a^{16} - \frac{1}{50} a^{15} - \frac{12}{25} a^{14} - \frac{3}{10} a^{13} - \frac{1}{10} a^{12} + \frac{3}{10} a^{11} - \frac{2}{25} a^{10} - \frac{13}{50} a^{9} - \frac{23}{50} a^{8} + \frac{7}{50} a^{7} - \frac{1}{25} a^{6} - \frac{3}{10} a^{5} + \frac{8}{25} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{12}{25} a + \frac{3}{50}$, $\frac{1}{2150} a^{19} + \frac{11}{2150} a^{18} - \frac{8}{215} a^{17} - \frac{103}{2150} a^{16} + \frac{31}{1075} a^{15} + \frac{262}{1075} a^{14} - \frac{101}{215} a^{13} + \frac{5}{86} a^{12} - \frac{419}{2150} a^{11} + \frac{341}{2150} a^{10} - \frac{4}{43} a^{9} - \frac{179}{430} a^{8} + \frac{113}{1075} a^{7} + \frac{171}{1075} a^{6} - \frac{1049}{2150} a^{5} - \frac{331}{2150} a^{4} - \frac{11}{215} a^{3} + \frac{418}{1075} a^{2} - \frac{144}{1075} a + \frac{421}{1075}$
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: | \( -1 \) (order $2$) | 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: | \( 395.600477658 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 1966080 |
| The 280 conjugacy class representatives for t20n992 are not computed |
| Character table for t20n992 is not computed |
Intermediate fields
| 5.3.41591.1, 10.2.1729811281.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ | $16{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $11$ | 11.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 11.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.6.0.1 | $x^{6} + x^{2} - 2 x + 8$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $19$ | 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 19.8.0.1 | $x^{8} - x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 19.8.4.1 | $x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $199$ | $\Q_{199}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{199}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{199}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{199}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 199.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 199.4.2.1 | $x^{4} + 2189 x^{2} + 1425636$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 199.4.2.1 | $x^{4} + 2189 x^{2} + 1425636$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3541 | Data not computed | ||||||