Normalized defining polynomial
\( x^{20} - 10 x^{19} + 41 x^{18} - 84 x^{17} + 64 x^{16} + 100 x^{15} - 345 x^{14} + 461 x^{13} - 431 x^{12} + 701 x^{11} - 1487 x^{10} + 1995 x^{9} - 1210 x^{8} - 583 x^{7} + 1972 x^{6} - 2087 x^{5} + 1330 x^{4} - 551 x^{3} + 143 x^{2} - 20 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(52642662275826068288374921=149^{6}\cdot 1481^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $19.32$ | 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: | $149, 1481$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{21948076} a^{18} - \frac{9}{21948076} a^{17} - \frac{2257847}{21948076} a^{16} - \frac{971274}{5487019} a^{15} - \frac{4015267}{21948076} a^{14} - \frac{2671007}{21948076} a^{13} + \frac{2354401}{21948076} a^{12} + \frac{8438039}{21948076} a^{11} - \frac{10390291}{21948076} a^{10} - \frac{10180919}{21948076} a^{9} - \frac{3441029}{21948076} a^{8} + \frac{6269319}{21948076} a^{7} - \frac{2483448}{5487019} a^{6} - \frac{1482182}{5487019} a^{5} + \frac{1713528}{5487019} a^{4} + \frac{6266777}{21948076} a^{3} - \frac{7256209}{21948076} a^{2} + \frac{7829469}{21948076} a + \frac{3614375}{21948076}$, $\frac{1}{4060394060} a^{19} + \frac{83}{4060394060} a^{18} - \frac{451735}{812078812} a^{17} - \frac{47414736}{1015098515} a^{16} - \frac{229755643}{4060394060} a^{15} + \frac{1317926281}{4060394060} a^{14} + \frac{1753896673}{4060394060} a^{13} + \frac{62567047}{812078812} a^{12} - \frac{68117591}{4060394060} a^{11} + \frac{1974954493}{4060394060} a^{10} - \frac{413331753}{4060394060} a^{9} + \frac{1335800351}{4060394060} a^{8} + \frac{317295497}{1015098515} a^{7} - \frac{350673816}{1015098515} a^{6} - \frac{55461942}{203019703} a^{5} - \frac{416662567}{4060394060} a^{4} - \frac{1537728021}{4060394060} a^{3} - \frac{879222519}{4060394060} a^{2} + \frac{570288991}{4060394060} a - \frac{202194363}{1015098515}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 245186.265159 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 61440 |
| The 126 conjugacy class representatives for t20n669 are not computed |
| Character table for t20n669 is not computed |
Intermediate fields
| 5.5.220669.1, 10.6.7255526326589.1, 10.6.48694807561.1, 10.6.7255526326589.2 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $149$ | 149.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 149.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 149.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 149.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 149.8.6.2 | $x^{8} + 745 x^{4} + 199809$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 1481 | Data not computed | ||||||