Normalized defining polynomial
\( x^{21} - 6 x^{20} - 109 x^{19} + 852 x^{18} + 4054 x^{17} - 49056 x^{16} - 19772 x^{15} + 1439643 x^{14} - 2914462 x^{13} - 20959603 x^{12} + 93601933 x^{11} + 73773944 x^{10} - 1209292466 x^{9} + 1909889766 x^{8} + 5378436465 x^{7} - 23689051810 x^{6} + 21160307927 x^{5} + 56749396678 x^{4} - 187732639665 x^{3} + 241813432964 x^{2} - 156262915354 x + 41965405847 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[5, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(133228076990589082766892787384406577621911901=3^{12}\cdot 19^{2}\cdot 271^{2}\cdot 2377^{2}\cdot 8689^{2}\cdot 280909^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $126.23$ | 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: | $3, 19, 271, 2377, 8689, 280909$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2775202536427652602480828400985862716418831334098437011731881} a^{20} - \frac{1289370387475139991815298068463936802573788930347068581037703}{2775202536427652602480828400985862716418831334098437011731881} a^{19} - \frac{259529275290176844418578291837814638815805227800366562319958}{925067512142550867493609466995287572139610444699479003910627} a^{18} - \frac{383726248983109544292731413694513349366779881240919878308200}{925067512142550867493609466995287572139610444699479003910627} a^{17} - \frac{841106714849870458804969603302958275328668588722712230130467}{2775202536427652602480828400985862716418831334098437011731881} a^{16} + \frac{461900856370796946456091630943684670019789951565149407857782}{2775202536427652602480828400985862716418831334098437011731881} a^{15} + \frac{518673863793910608360013020701615345651540213469196482880991}{2775202536427652602480828400985862716418831334098437011731881} a^{14} + \frac{892723886524344268449842382631819941029284631409231491697272}{2775202536427652602480828400985862716418831334098437011731881} a^{13} + \frac{810376746787313706537819841646994335055707419197624754441429}{2775202536427652602480828400985862716418831334098437011731881} a^{12} + \frac{5167848176728615972562555859698339339291871400779146591219}{13806977793172401007367305477541605554322543950738492595681} a^{11} + \frac{1322158613519648897045668011775308300109506825295514172595762}{2775202536427652602480828400985862716418831334098437011731881} a^{10} + \frac{155882562934885071597458131856006215118611069166432846847740}{925067512142550867493609466995287572139610444699479003910627} a^{9} - \frac{8011013411553884168727622388921068916152304546706222210479}{41420933379517203022101916432624816662967631852215477787043} a^{8} + \frac{824418121357872590997047420917682058763990742322013997495473}{2775202536427652602480828400985862716418831334098437011731881} a^{7} - \frac{413230884145990266695329402256290811145555654335573501032907}{2775202536427652602480828400985862716418831334098437011731881} a^{6} + \frac{771347508372076101232665811507611283892558616390351644270}{132152501734650123927658495285041081734230063528497000558661} a^{5} + \frac{4345501330843470281301262481596350635239512566813860153304}{2775202536427652602480828400985862716418831334098437011731881} a^{4} - \frac{23126322885368685357235966414676718283893036510832870067370}{132152501734650123927658495285041081734230063528497000558661} a^{3} - \frac{354594175111011847098873506157740435316985002563141644333637}{925067512142550867493609466995287572139610444699479003910627} a^{2} - \frac{1208617643053327334147119379472923512818630605223832118265889}{2775202536427652602480828400985862716418831334098437011731881} a + \frac{9083511761048506740447003700620935698179633237583939865920}{41420933379517203022101916432624816662967631852215477787043}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $12$ | 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: | \( 31233390232100 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 22044960 |
| The 261 conjugacy class representatives for t21n144 are not computed |
| Character table for t21n144 is not computed |
Intermediate fields
| 7.5.7584543.1 |
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 | ${\href{/LocalNumberField/2.14.0.1}{14} }{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | $15{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }$ | $15{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $21$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ | $18{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.9.0.1}{9} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }$ |
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 | ||||||
| $19$ | 19.3.2.2 | $x^{3} - 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 19.6.0.1 | $x^{6} - x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 19.6.0.1 | $x^{6} - x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 19.6.0.1 | $x^{6} - x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 271 | Data not computed | ||||||
| 2377 | Data not computed | ||||||
| 8689 | Data not computed | ||||||
| 280909 | Data not computed | ||||||