Normalized defining polynomial
\( x^{20} - 5 x^{19} + 16 x^{18} - 46 x^{17} + 108 x^{16} - 214 x^{15} + 395 x^{14} - 698 x^{13} + 1123 x^{12} - 1588 x^{11} + 2057 x^{10} - 2588 x^{9} + 3128 x^{8} - 3348 x^{7} + 2932 x^{6} - 1996 x^{5} + 1016 x^{4} - 371 x^{3} + 92 x^{2} - 14 x + 1 \)
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: | \(14861658978964734748713=3^{5}\cdot 11^{19}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.84$ | 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: | $3, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} - \frac{7}{23} a^{17} - \frac{9}{23} a^{16} - \frac{8}{23} a^{15} - \frac{8}{23} a^{14} - \frac{1}{23} a^{13} - \frac{4}{23} a^{12} - \frac{7}{23} a^{11} + \frac{5}{23} a^{10} + \frac{9}{23} a^{9} + \frac{4}{23} a^{8} - \frac{3}{23} a^{7} + \frac{11}{23} a^{6} - \frac{10}{23} a^{5} - \frac{7}{23} a^{4} - \frac{5}{23} a^{3} + \frac{11}{23} a^{2} + \frac{9}{23} a - \frac{10}{23}$, $\frac{1}{22332092443} a^{19} - \frac{405185842}{22332092443} a^{18} + \frac{6248587414}{22332092443} a^{17} + \frac{1361723408}{22332092443} a^{16} - \frac{7284615703}{22332092443} a^{15} + \frac{9816693260}{22332092443} a^{14} + \frac{10365062637}{22332092443} a^{13} + \frac{4605014631}{22332092443} a^{12} + \frac{3057372871}{22332092443} a^{11} - \frac{4392053315}{22332092443} a^{10} - \frac{4222339189}{22332092443} a^{9} + \frac{7908870301}{22332092443} a^{8} + \frac{9089785145}{22332092443} a^{7} + \frac{82091825}{22332092443} a^{6} + \frac{10646982395}{22332092443} a^{5} + \frac{11052225531}{22332092443} a^{4} + \frac{4781870836}{22332092443} a^{3} - \frac{9098316457}{22332092443} a^{2} + \frac{5425134077}{22332092443} a + \frac{10685951026}{22332092443}$
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: | \( -\frac{498258201233}{22332092443} a^{19} + \frac{2299914328811}{22332092443} a^{18} - \frac{7089012985276}{22332092443} a^{17} + \frac{20192473954921}{22332092443} a^{16} - \frac{46041149717855}{22332092443} a^{15} + \frac{88899555703222}{22332092443} a^{14} - \frac{7066771082192}{970960541} a^{13} + \frac{285094971179593}{22332092443} a^{12} - \frac{449558441453469}{22332092443} a^{11} + \frac{617669905806245}{22332092443} a^{10} - \frac{786154168476411}{22332092443} a^{9} + \frac{985289583822582}{22332092443} a^{8} - \frac{51184177137416}{970960541} a^{7} + \frac{1212360392856302}{22332092443} a^{6} - \frac{990686242950518}{22332092443} a^{5} + \frac{26471731661149}{970960541} a^{4} - \frac{267689714276496}{22332092443} a^{3} + \frac{78856206641197}{22332092443} a^{2} - \frac{611680183799}{970960541} a + \frac{1163961873210}{22332092443} \) (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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5805.73215178 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_5\times D_4$ (as 20T12):
| A solvable group of order 40 |
| The 25 conjugacy class representatives for $C_5\times D_4$ |
| Character table for $C_5\times D_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-11}) \), 4.0.3993.1, \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{11})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 20 sibling: | 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 | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ | $20$ | R | $20$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.10.5.1 | $x^{10} - 18 x^{6} + 81 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 3.10.0.1 | $x^{10} - x^{3} - x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 11 | Data not computed | ||||||