Normalized defining polynomial
\( x^{21} - 3 x^{20} - 180 x^{19} + 295 x^{18} + 13605 x^{17} - 5157 x^{16} - 551346 x^{15} - 398277 x^{14} + 12659553 x^{13} + 21281696 x^{12} - 159042903 x^{11} - 417305250 x^{10} + 922946239 x^{9} + 3784485999 x^{8} - 675851490 x^{7} - 14780825192 x^{6} - 12395275152 x^{5} + 16098793473 x^{4} + 29396147584 x^{3} + 14346185940 x^{2} + 2403226995 x + 72232579 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[21, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3265364406784528133678938964803591475967396373221129=3^{28}\cdot 7^{14}\cdot 29^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $283.82$ | 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: | $3, 7, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1827=3^{2}\cdot 7\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1827}(1,·)$, $\chi_{1827}(1474,·)$, $\chi_{1827}(1096,·)$, $\chi_{1827}(529,·)$, $\chi_{1827}(277,·)$, $\chi_{1827}(88,·)$, $\chi_{1827}(25,·)$, $\chi_{1827}(1822,·)$, $\chi_{1827}(1444,·)$, $\chi_{1827}(1765,·)$, $\chi_{1827}(1702,·)$, $\chi_{1827}(625,·)$, $\chi_{1827}(1387,·)$, $\chi_{1827}(877,·)$, $\chi_{1827}(1009,·)$, $\chi_{1827}(436,·)$, $\chi_{1827}(373,·)$, $\chi_{1827}(310,·)$, $\chi_{1827}(442,·)$, $\chi_{1827}(1789,·)$, $\chi_{1827}(190,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6494} a^{18} + \frac{1485}{6494} a^{17} + \frac{291}{3247} a^{16} - \frac{341}{3247} a^{15} - \frac{12}{191} a^{14} - \frac{1050}{3247} a^{13} - \frac{2617}{6494} a^{12} + \frac{609}{3247} a^{11} - \frac{907}{3247} a^{10} + \frac{693}{6494} a^{9} - \frac{2141}{6494} a^{8} + \frac{1079}{3247} a^{7} - \frac{523}{6494} a^{6} - \frac{30}{3247} a^{5} + \frac{351}{3247} a^{4} + \frac{1271}{3247} a^{3} + \frac{2563}{6494} a^{2} + \frac{1967}{6494} a + \frac{2525}{6494}$, $\frac{1}{383146} a^{19} + \frac{5}{383146} a^{18} + \frac{31347}{191573} a^{17} - \frac{15405}{191573} a^{16} - \frac{47514}{191573} a^{15} - \frac{5449}{383146} a^{14} - \frac{59444}{191573} a^{13} - \frac{142161}{383146} a^{12} + \frac{172969}{383146} a^{11} - \frac{171947}{383146} a^{10} + \frac{128151}{383146} a^{9} + \frac{72317}{191573} a^{8} + \frac{63659}{191573} a^{7} + \frac{62290}{191573} a^{6} - \frac{85129}{191573} a^{5} + \frac{129255}{383146} a^{4} - \frac{85831}{191573} a^{3} + \frac{157079}{383146} a^{2} + \frac{70149}{191573} a + \frac{3197}{6494}$, $\frac{1}{1544452060404108780683627922243920104632160522552923235765660119209546847642254} a^{20} + \frac{800332443477385197208090305658952037188711313393829221782550605823034113}{1544452060404108780683627922243920104632160522552923235765660119209546847642254} a^{19} - \frac{22572259698976076091179360338717677865149920079454874733810425576825707928}{772226030202054390341813961121960052316080261276461617882830059604773423821127} a^{18} + \frac{163149291891538165338666912141572051871488301425702616723616629699997352503603}{1544452060404108780683627922243920104632160522552923235765660119209546847642254} a^{17} + \frac{140977768718030125684291589517044342252951115639827040970613489899544002554236}{772226030202054390341813961121960052316080261276461617882830059604773423821127} a^{16} - \frac{78230815571806577723343821258014467165589675390413097627907880801342228862221}{772226030202054390341813961121960052316080261276461617882830059604773423821127} a^{15} + \frac{83105781848761853325012163911716419365845503977304990428605029839285302052447}{1544452060404108780683627922243920104632160522552923235765660119209546847642254} a^{14} - \frac{720780288207719224539524317387623432160588642366373605753670574983463497966233}{1544452060404108780683627922243920104632160522552923235765660119209546847642254} a^{13} - \frac{324843841073390356956288086258659442179136199782169166939379966476051984131565}{772226030202054390341813961121960052316080261276461617882830059604773423821127} a^{12} - \frac{599924717567822018617547370054673493269136404274546131279900662200925710975939}{1544452060404108780683627922243920104632160522552923235765660119209546847642254} a^{11} + \frac{23355786758450618273268703518849478449663580298020001180467678465956489895698}{772226030202054390341813961121960052316080261276461617882830059604773423821127} a^{10} - \frac{18961929367051044890220332315897132274758007834758956358672808238755660028919}{90850121200241692981389877779054123801891795444289602103862359953502755743662} a^{9} - \frac{707015561912975587939197341585732147751407596089978862099884623686356249629105}{1544452060404108780683627922243920104632160522552923235765660119209546847642254} a^{8} - \frac{451112986673567526337521120432804615839410974265226989381831155946259180443685}{1544452060404108780683627922243920104632160522552923235765660119209546847642254} a^{7} + \frac{186621750149193270231004463229740893406428103338446120384965489590756855413532}{772226030202054390341813961121960052316080261276461617882830059604773423821127} a^{6} + \frac{279884203617479726692280584922380471068033258483515673599885340104250416962399}{1544452060404108780683627922243920104632160522552923235765660119209546847642254} a^{5} - \frac{246378374598503884778561008087418615979665137261894029872059038774749356116463}{772226030202054390341813961121960052316080261276461617882830059604773423821127} a^{4} + \frac{231449792352017949103256391026006959491819577140731801589764250770659051228465}{1544452060404108780683627922243920104632160522552923235765660119209546847642254} a^{3} - \frac{223230368619724536488661105163059496473832107814703359811365651690262708760276}{772226030202054390341813961121960052316080261276461617882830059604773423821127} a^{2} - \frac{118944438954503230842732499953694629111878935074924339335331551825344004434405}{772226030202054390341813961121960052316080261276461617882830059604773423821127} a - \frac{5811283986489497978269808956311678851760806552610888102152717133911883500151}{13088576783085667632912101035965424615526784089431552845471695925504634302053}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $20$ | 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: | \( 4220636828185508400 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 21 |
| The 21 conjugacy class representatives for $C_{21}$ |
| Character table for $C_{21}$ is not computed |
Intermediate fields
| 3.3.3969.1, 7.7.594823321.1 |
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 | ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ | R | $21$ | R | $21$ | $21$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ | $21$ | $21$ | R | ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ | $21$ | ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/59.1.0.1}{1} }^{21}$ |
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 | ||||||
| 7 | Data not computed | ||||||
| 29 | Data not computed | ||||||