Normalized defining polynomial
\( x^{22} - 2 x^{21} + 36 x^{20} - 62 x^{19} + 779 x^{18} - 1208 x^{17} + 11785 x^{16} - 16330 x^{15} + 135402 x^{14} - 166994 x^{13} + 1216660 x^{12} - 1317110 x^{11} + 8641614 x^{10} - 8043038 x^{9} + 48218481 x^{8} - 37307012 x^{7} + 206484136 x^{6} - 125495212 x^{5} + 645715650 x^{4} - 276536200 x^{3} + 1328901156 x^{2} - 304156912 x + 1368706081 \)
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: | \(-351468714257323283030813737164800000000000=-\,2^{22}\cdot 5^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $77.35$ | 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: | $2, 5, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(460=2^{2}\cdot 5\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{460}(1,·)$, $\chi_{460}(259,·)$, $\chi_{460}(261,·)$, $\chi_{460}(139,·)$, $\chi_{460}(141,·)$, $\chi_{460}(399,·)$, $\chi_{460}(81,·)$, $\chi_{460}(121,·)$, $\chi_{460}(279,·)$, $\chi_{460}(219,·)$, $\chi_{460}(361,·)$, $\chi_{460}(101,·)$, $\chi_{460}(39,·)$, $\chi_{460}(119,·)$, $\chi_{460}(41,·)$, $\chi_{460}(301,·)$, $\chi_{460}(239,·)$, $\chi_{460}(179,·)$, $\chi_{460}(439,·)$, $\chi_{460}(441,·)$, $\chi_{460}(59,·)$, $\chi_{460}(381,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{47} a^{19} + \frac{5}{47} a^{18} + \frac{3}{47} a^{17} + \frac{13}{47} a^{16} + \frac{4}{47} a^{15} + \frac{11}{47} a^{14} - \frac{20}{47} a^{13} + \frac{9}{47} a^{12} + \frac{18}{47} a^{11} - \frac{3}{47} a^{10} + \frac{16}{47} a^{9} - \frac{8}{47} a^{7} + \frac{22}{47} a^{6} - \frac{1}{47} a^{5} - \frac{12}{47} a^{4} - \frac{20}{47} a^{3} + \frac{5}{47} a^{2} - \frac{3}{47} a - \frac{13}{47}$, $\frac{1}{47} a^{20} - \frac{22}{47} a^{18} - \frac{2}{47} a^{17} - \frac{14}{47} a^{16} - \frac{9}{47} a^{15} + \frac{19}{47} a^{14} + \frac{15}{47} a^{13} + \frac{20}{47} a^{12} + \frac{1}{47} a^{11} - \frac{16}{47} a^{10} + \frac{14}{47} a^{9} - \frac{8}{47} a^{8} + \frac{15}{47} a^{7} - \frac{17}{47} a^{6} - \frac{7}{47} a^{5} - \frac{7}{47} a^{4} + \frac{11}{47} a^{3} + \frac{19}{47} a^{2} + \frac{2}{47} a + \frac{18}{47}$, $\frac{1}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{21} - \frac{3971133917411194426979979954135685139411046549509626468379652474950976}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{20} + \frac{5090255405117689755713611914946730683256152744379276253080296344195808}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{19} + \frac{1248241006208431257453449491490075673850879250747885450877362404155678358}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{18} - \frac{551051924389160082779179227485407716061986630485480412236465648926844740}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{17} + \frac{816281095324676253389639717082463639025619409953648998164231774602339315}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{16} - \frac{605412515675865731101638900754777466695750187542569137507302856308325176}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{15} - \frac{171796892232371985476156744791190876243895833904521737350404274985762933}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{14} + \frac{396230866578563015584287190039393003439196951757472741325439697613070778}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{13} - \frac{668446076965952903255604086214775296250013995591853534743896280034901918}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{12} - \frac{1232080757920173607816531302874362689891541825061576716018463030840984435}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{11} - \frac{508325573988273104069204334834709117505549348797369066100064654716171766}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{10} + \frac{491220447465025000404263305998034465619897595998871916145912546023725793}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{9} - \frac{242899949589158050925515186690652259606503437182618419788816196442490705}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{8} + \frac{85160525419446042993052565982625954978487398723630792116405612555921020}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{7} + \frac{449234446591243832086450093777050114155839787015421722793240223233518997}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{6} + \frac{945453211785246244454850262347573769193299367931544661819153103235264080}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{5} - \frac{775782408009472621868982562867609786592751885971653411586291637537155483}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{4} + \frac{549194122930160845533525786042389281044445309158439963528670519543740541}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{3} + \frac{740610956207869364509020155798787650002057016237516697873919890741576566}{2520374299582000455108234351615157322139260762675464516201276876183751821} a^{2} - \frac{1112030377777828615615848896832527047181083936919912091682398440891644847}{2520374299582000455108234351615157322139260762675464516201276876183751821} a - \frac{817315073215622505682145594251758103280208450579260262280124392072207425}{2520374299582000455108234351615157322139260762675464516201276876183751821}$
Class group and class number
$C_{374638}$, which has order $374638$ (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
| \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{23})^+\) |
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 | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | R | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | $22$ | R | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | $22$ | $22$ |
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 | ||||||
| 5 | Data not computed | ||||||
| $23$ | 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |