Normalized defining polynomial
\( x^{20} - 5 x^{19} + 5 x^{18} + 14 x^{17} - 55 x^{16} + 81 x^{15} + 28 x^{14} - 281 x^{13} + 368 x^{12} + x^{11} - 699 x^{10} + 872 x^{9} + 242 x^{8} - 1190 x^{7} + 324 x^{6} + 650 x^{5} - 277 x^{4} - 157 x^{3} + 67 x^{2} + 14 x - 4 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[14, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-2854696149767119301830443008=-\,2^{16}\cdot 83^{4}\cdot 983^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.59$ | 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: | $2, 83, 983$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{403} a^{18} - \frac{59}{403} a^{17} + \frac{134}{403} a^{16} - \frac{149}{403} a^{15} + \frac{144}{403} a^{14} + \frac{5}{31} a^{13} + \frac{1}{31} a^{12} + \frac{200}{403} a^{11} - \frac{201}{403} a^{10} - \frac{75}{403} a^{9} + \frac{9}{403} a^{8} - \frac{49}{403} a^{7} - \frac{42}{403} a^{6} + \frac{149}{403} a^{5} + \frac{175}{403} a^{4} - \frac{37}{403} a^{3} - \frac{85}{403} a^{2} - \frac{134}{403} a - \frac{41}{403}$, $\frac{1}{53987188595002} a^{19} + \frac{54612054975}{53987188595002} a^{18} - \frac{3866056764763}{53987188595002} a^{17} - \frac{2635338285692}{26993594297501} a^{16} - \frac{12124835892647}{53987188595002} a^{15} - \frac{337723075775}{1741522212742} a^{14} - \frac{566725260492}{2076430330577} a^{13} - \frac{25317027520663}{53987188595002} a^{12} + \frac{7111892942122}{26993594297501} a^{11} + \frac{12197707754263}{53987188595002} a^{10} - \frac{9717306378421}{53987188595002} a^{9} - \frac{68542446077}{2076430330577} a^{8} + \frac{5239936681601}{26993594297501} a^{7} + \frac{7165007646872}{26993594297501} a^{6} - \frac{10612526344457}{26993594297501} a^{5} - \frac{785405272907}{2076430330577} a^{4} + \frac{23502196061215}{53987188595002} a^{3} - \frac{615421284133}{53987188595002} a^{2} - \frac{26519737901883}{53987188595002} a + \frac{11065675201694}{26993594297501}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $16$ | 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: | \( 6043091.49938 \) (assuming GRH) | 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
| 5.5.81589.1, 10.10.1704131819776.1 |
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 | R | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | $16{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $83$ | 83.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 83.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 83.4.2.1 | $x^{4} + 249 x^{2} + 27556$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 83.4.2.1 | $x^{4} + 249 x^{2} + 27556$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 83.6.0.1 | $x^{6} - x + 34$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 983 | Data not computed | ||||||