Normalized defining polynomial
\( x^{21} - x^{20} - 278 x^{19} - 325 x^{18} + 30572 x^{17} + 90706 x^{16} - 1619293 x^{15} - 7475214 x^{14} + 39793032 x^{13} + 274015207 x^{12} - 273602661 x^{11} - 4607226493 x^{10} - 4412589549 x^{9} + 30914338952 x^{8} + 60233241889 x^{7} - 74040616576 x^{6} - 218112157253 x^{5} + 55980136111 x^{4} + 294993180914 x^{3} - 10861258704 x^{2} - 116080052977 x - 14284976423 \)
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: | \(373181000690273382911499008901975594970451729658121=19^{14}\cdot 43^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $255.97$ | 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: | $19, 43$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(817=19\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{817}(704,·)$, $\chi_{817}(1,·)$, $\chi_{817}(514,·)$, $\chi_{817}(771,·)$, $\chi_{817}(197,·)$, $\chi_{817}(273,·)$, $\chi_{817}(723,·)$, $\chi_{817}(666,·)$, $\chi_{817}(410,·)$, $\chi_{817}(482,·)$, $\chi_{817}(676,·)$, $\chi_{817}(742,·)$, $\chi_{817}(305,·)$, $\chi_{817}(296,·)$, $\chi_{817}(615,·)$, $\chi_{817}(748,·)$, $\chi_{817}(239,·)$, $\chi_{817}(49,·)$, $\chi_{817}(182,·)$, $\chi_{817}(444,·)$, $\chi_{817}(767,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{79} a^{19} - \frac{20}{79} a^{18} - \frac{23}{79} a^{17} + \frac{5}{79} a^{16} + \frac{14}{79} a^{15} - \frac{8}{79} a^{14} + \frac{31}{79} a^{13} + \frac{19}{79} a^{12} - \frac{28}{79} a^{11} - \frac{32}{79} a^{10} + \frac{14}{79} a^{9} - \frac{34}{79} a^{8} + \frac{9}{79} a^{7} - \frac{11}{79} a^{6} + \frac{21}{79} a^{5} - \frac{22}{79} a^{4} + \frac{9}{79} a^{3} - \frac{39}{79} a^{2} - \frac{17}{79} a + \frac{32}{79}$, $\frac{1}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{20} - \frac{884303246215051719602736258504015536797150169308012549655500060884510383669853530493256837758661022}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{19} + \frac{556685765854728699758103594234636586529632634905756168093105403638501715706339286338079475463360064339}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{18} + \frac{113254824982439162072150318772628150320951975454402064683107742720581507451063150657236747277458041008}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{17} - \frac{441916278457719216092782850489345113843223931240930566765114843840297933983267369650783315682043269733}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{16} - \frac{54395272480930252707531524225277309442207612150733833965572669415585261979383798697765588170142561386}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{15} - \frac{79892562153798669781160377798098018961921903733307908727140997827580650396527288079262327793439723842}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{14} + \frac{66305731478098534253052429037477106732626076298304772237054187975963030763470678007346268200362353505}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{13} + \frac{571058293752862838669068710519731181206040347983260074898654214646176433207507141059195965432249249455}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{12} - \frac{366514859164044860718006096415101114545238685949872313241063727817610882331599868452821071474653470820}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{11} + \frac{536175361315361995797201484011158898440889955898473869796880490151653397882215526994965923476875836831}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{10} + \frac{327236986721851243056315573253701585288449704007889058109773383288142578241206847624647556380143981375}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{9} + \frac{144712179249661039277938072661134334307787818535239405177881931164269486913528134975467912528494502469}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{8} + \frac{323621975927643109883642126963727742856421110085139850058301873951697427181487062744819746284047996798}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{7} + \frac{278080990079824875846389766020399538195789648720557537792332264747438879952573140744065580603122767821}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{6} + \frac{29357483077503511407932507927496368151186285736747591012278293008320358015148420513251250022258177601}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{5} - \frac{381106154055462928007976600702060424676182310551807293568067126103743998353973175174275795624110892213}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{4} - \frac{531245559323647399234034347110644965701720234248373875876047924831249202024450353054394818296806926998}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{3} - \frac{192951216015819757376126412077218336839611766703216507855174812896622836982195728128564625402563277681}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a^{2} + \frac{219823533579650964138412092858184592497354014485308968966982367098221180338562986152205212725651616488}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893} a + \frac{575956338380050099793195445938736983086012775585440110847718020995450866735365282409394416799091103019}{1153795010937951052187554670463543300547700065999769628837805912139866159159549295326414663326523269893}$
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: | \( 864898073802425300 \) (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
| 3.3.667489.2, 7.7.6321363049.1 |
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 | $21$ | $21$ | $21$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ | R | ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ | $21$ | R | $21$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ | $21$ |
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 |
|---|---|---|---|---|---|---|---|
| 19 | Data not computed | ||||||
| 43 | Data not computed | ||||||