Normalized defining polynomial
\( x^{22} - 2 x^{21} + 47 x^{20} - 82 x^{19} + 1234 x^{18} - 1928 x^{17} + 22201 x^{16} - 30962 x^{15} + 297640 x^{14} - 368132 x^{13} + 3078660 x^{12} - 3330816 x^{11} + 24869896 x^{10} - 23035280 x^{9} + 156123875 x^{8} - 119632870 x^{7} + 744424991 x^{6} - 445564762 x^{5} + 2565595585 x^{4} - 1074775766 x^{3} + 5757368622 x^{2} - 1278298632 x + 6391493137 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-2611441967281400084968119933496263205453824=-\,2^{33}\cdot 3^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $84.73$ | 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: | $2, 3, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(552=2^{3}\cdot 3\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{552}(1,·)$, $\chi_{552}(197,·)$, $\chi_{552}(193,·)$, $\chi_{552}(265,·)$, $\chi_{552}(269,·)$, $\chi_{552}(77,·)$, $\chi_{552}(533,·)$, $\chi_{552}(25,·)$, $\chi_{552}(409,·)$, $\chi_{552}(461,·)$, $\chi_{552}(361,·)$, $\chi_{552}(29,·)$, $\chi_{552}(485,·)$, $\chi_{552}(289,·)$, $\chi_{552}(101,·)$, $\chi_{552}(169,·)$, $\chi_{552}(173,·)$, $\chi_{552}(509,·)$, $\chi_{552}(49,·)$, $\chi_{552}(73,·)$, $\chi_{552}(121,·)$, $\chi_{552}(317,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{21} + \frac{5368482719810123978204546541866070538587862166740291192382636088228179293822982}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{20} - \frac{1273940848025776549986367000767747083627730917112142682469377765899147967953185}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{19} + \frac{4223564213305343908204329261684160104234651685558066594550250826976587223270768}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{18} + \frac{9504469766929994603866856009673948502872778345259818459906698206821975359975683}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{17} - \frac{8120968874425230365377489274251949572080401070466388850579170972926830315212483}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{16} - \frac{13098987681763158038975037759201587215387211397651740222386387434407995853576566}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{15} - \frac{6930095571634104153894722411123020181178232090780282058637002668806929165863616}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{14} - \frac{3796035123976950043803387642346261722820945219325134582310594479725731427775484}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{13} + \frac{13804981857043871707313552183160928942902920133710881174387906225437300947878642}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{12} + \frac{7936527747669074895243231711950910152830778715951880324126481665979278433666175}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{11} - \frac{7714647779915634432995959944878739940008197433959766277313053694226434942569246}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{10} + \frac{3789873698529516235073988189411747047568967194712465557596523186357398775500925}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{9} + \frac{6558731642089766829255003426749835037622557699627474462550531896459778187409125}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{8} - \frac{13287589391131311657760889778862978761895760374618110757620603705488697885864577}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{7} + \frac{3456075027872082745359870966589326120446753936682827895885652493326590116971728}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{6} - \frac{4638268530462187526164226576720915410439187368623107900072921766630399525378663}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{5} + \frac{6015195158110275864553453846688673101440598932807828117309027329891735655814990}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{4} + \frac{12838773864773513259446194842431843271187887823185437681003149517535073310255443}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{3} + \frac{9519504555567118813521111381137686435805339372672590059483210400431806031881776}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a^{2} + \frac{8238217722661693494387072336356077105184664257268525426830945446652463027626074}{30984097795921555015647072513886486852970884189157104437398008152367347156341497} a + \frac{12015826099734275700639321005424863541741700590013857044881385880189231807955526}{30984097795921555015647072513886486852970884189157104437398008152367347156341497}$
Class group and class number
$C_{495926}$, which has order $495926$ (assuming GRH)
Unit group
| Rank: | $10$ | 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: | \( 1038656.82438 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-6}) \), \(\Q(\zeta_{23})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | R | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 | ||||||
| 3 | Data not computed | ||||||
| 23 | Data not computed | ||||||