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Properties

Label 21.21.3059439832...4401.1
Degree $21$
Signature $[21, 0]$
Discriminant $421^{20}$
Root discriminant $315.73$
Ramified prime $421$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{21}$ (as 21T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10850554247, -56191978934, -86032090549, -8634241360, 75239240850, 36814363818, -21119020890, -14625774577, 2906746078, 2686381285, -223777460, -280861686, 9863681, 17859943, -222713, -697221, 828, 16134, 67, -200, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 200*x^19 + 67*x^18 + 16134*x^17 + 828*x^16 - 697221*x^15 - 222713*x^14 + 17859943*x^13 + 9863681*x^12 - 280861686*x^11 - 223777460*x^10 + 2686381285*x^9 + 2906746078*x^8 - 14625774577*x^7 - 21119020890*x^6 + 36814363818*x^5 + 75239240850*x^4 - 8634241360*x^3 - 86032090549*x^2 - 56191978934*x - 10850554247)
 
gp: K = bnfinit(x^21 - x^20 - 200*x^19 + 67*x^18 + 16134*x^17 + 828*x^16 - 697221*x^15 - 222713*x^14 + 17859943*x^13 + 9863681*x^12 - 280861686*x^11 - 223777460*x^10 + 2686381285*x^9 + 2906746078*x^8 - 14625774577*x^7 - 21119020890*x^6 + 36814363818*x^5 + 75239240850*x^4 - 8634241360*x^3 - 86032090549*x^2 - 56191978934*x - 10850554247, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 200 x^{19} + 67 x^{18} + 16134 x^{17} + 828 x^{16} - 697221 x^{15} - 222713 x^{14} + 17859943 x^{13} + 9863681 x^{12} - 280861686 x^{11} - 223777460 x^{10} + 2686381285 x^{9} + 2906746078 x^{8} - 14625774577 x^{7} - 21119020890 x^{6} + 36814363818 x^{5} + 75239240850 x^{4} - 8634241360 x^{3} - 86032090549 x^{2} - 56191978934 x - 10850554247 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

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:  \(30594398325772002447442992111668517593745691297844401=421^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $315.73$
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:  $421$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(421\)
Dirichlet character group:    $\lbrace$$\chi_{421}(1,·)$, $\chi_{421}(385,·)$, $\chi_{421}(75,·)$, $\chi_{421}(237,·)$, $\chi_{421}(400,·)$, $\chi_{421}(20,·)$, $\chi_{421}(149,·)$, $\chi_{421}(152,·)$, $\chi_{421}(335,·)$, $\chi_{421}(93,·)$, $\chi_{421}(286,·)$, $\chi_{421}(33,·)$, $\chi_{421}(229,·)$, $\chi_{421}(109,·)$, $\chi_{421}(239,·)$, $\chi_{421}(176,·)$, $\chi_{421}(370,·)$, $\chi_{421}(243,·)$, $\chi_{421}(309,·)$, $\chi_{421}(247,·)$, $\chi_{421}(122,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} - \frac{3}{13} a^{11} - \frac{4}{13} a^{10} - \frac{1}{13} a^{9} + \frac{3}{13} a^{8} + \frac{4}{13} a^{7} + \frac{1}{13} a^{6} - \frac{3}{13} a^{5} - \frac{4}{13} a^{4} - \frac{1}{13} a^{3} + \frac{3}{13} a^{2} + \frac{4}{13} a$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{377} a^{14} + \frac{6}{377} a^{13} + \frac{3}{377} a^{12} - \frac{22}{377} a^{11} + \frac{92}{377} a^{10} + \frac{10}{377} a^{9} - \frac{30}{377} a^{8} - \frac{118}{377} a^{7} + \frac{172}{377} a^{6} - \frac{48}{377} a^{5} + \frac{144}{377} a^{4} - \frac{1}{13} a^{3} + \frac{86}{377} a^{2} + \frac{110}{377} a$, $\frac{1}{377} a^{15} - \frac{4}{377} a^{13} - \frac{11}{377} a^{12} + \frac{137}{377} a^{11} + \frac{96}{377} a^{10} - \frac{119}{377} a^{9} + \frac{149}{377} a^{8} - \frac{135}{377} a^{7} + \frac{80}{377} a^{6} - \frac{32}{377} a^{5} + \frac{122}{377} a^{4} - \frac{146}{377} a^{3} + \frac{2}{13} a^{2} + \frac{181}{377} a$, $\frac{1}{25259} a^{16} + \frac{3}{25259} a^{15} + \frac{16}{25259} a^{14} - \frac{164}{25259} a^{13} + \frac{193}{25259} a^{12} + \frac{9405}{25259} a^{11} - \frac{4516}{25259} a^{10} - \frac{7954}{25259} a^{9} - \frac{2463}{25259} a^{8} - \frac{5208}{25259} a^{7} + \frac{12348}{25259} a^{6} - \frac{4037}{25259} a^{5} + \frac{2230}{25259} a^{4} + \frac{8813}{25259} a^{3} - \frac{5755}{25259} a^{2} + \frac{762}{1943} a - \frac{8}{67}$, $\frac{1}{25259} a^{17} + \frac{7}{25259} a^{15} - \frac{11}{25259} a^{14} - \frac{4}{1943} a^{13} - \frac{22}{1943} a^{12} - \frac{616}{1943} a^{11} + \frac{956}{1943} a^{10} + \frac{605}{1943} a^{9} - \frac{595}{1943} a^{8} - \frac{719}{1943} a^{7} + \frac{695}{1943} a^{6} + \frac{8579}{25259} a^{5} - \frac{450}{1943} a^{4} - \frac{3049}{25259} a^{3} - \frac{343}{871} a^{2} + \frac{229}{1943} a + \frac{24}{67}$, $\frac{1}{25259} a^{18} - \frac{32}{25259} a^{15} - \frac{30}{25259} a^{14} - \frac{277}{25259} a^{13} + \frac{758}{25259} a^{12} - \frac{9723}{25259} a^{11} - \frac{12314}{25259} a^{10} - \frac{10950}{25259} a^{9} + \frac{7760}{25259} a^{8} - \frac{7238}{25259} a^{7} + \frac{5424}{25259} a^{6} + \frac{12091}{25259} a^{5} + \frac{12295}{25259} a^{4} - \frac{9462}{25259} a^{3} + \frac{8154}{25259} a^{2} - \frac{4751}{25259} a - \frac{11}{67}$, $\frac{1}{328367} a^{19} - \frac{2}{328367} a^{18} - \frac{4}{328367} a^{17} - \frac{5}{328367} a^{16} - \frac{315}{328367} a^{15} - \frac{9}{328367} a^{14} + \frac{10693}{328367} a^{13} + \frac{10392}{328367} a^{12} - \frac{18384}{328367} a^{11} + \frac{35262}{328367} a^{10} + \frac{19014}{328367} a^{9} + \frac{51092}{328367} a^{8} + \frac{99716}{328367} a^{7} - \frac{121725}{328367} a^{6} - \frac{63412}{328367} a^{5} - \frac{34192}{328367} a^{4} + \frac{79441}{328367} a^{3} + \frac{79285}{328367} a^{2} + \frac{8917}{25259} a - \frac{12}{67}$, $\frac{1}{25405742808078707299254621753686268524672856939997385174054198189} a^{20} + \frac{529956010785278785914550276570700183345690432342386202756}{1954287908313746715327278596437405271128681303076721936465707553} a^{19} - \frac{275027467013782674765160274005171279809516271843406959232204}{25405742808078707299254621753686268524672856939997385174054198189} a^{18} + \frac{165121444280260628685392212450686006548463073986902653135}{1954287908313746715327278596437405271128681303076721936465707553} a^{17} - \frac{338695077466345309949933362607810417471266628459519758708}{29168476243488756945183262633394108524308676165324208006950859} a^{16} + \frac{6466901030270047329344727401230076111877238730630459507654224}{25405742808078707299254621753686268524672856939997385174054198189} a^{15} - \frac{8034577425728579419285544099347516959294787509064533810658507}{25405742808078707299254621753686268524672856939997385174054198189} a^{14} - \frac{975953544248814269384781472029022530670873951233314083774148483}{25405742808078707299254621753686268524672856939997385174054198189} a^{13} + \frac{71376775078301282949809539799930184890002677118537292822273578}{25405742808078707299254621753686268524672856939997385174054198189} a^{12} + \frac{8703530380715382245239008859395201066439193410063169672773426185}{25405742808078707299254621753686268524672856939997385174054198189} a^{11} + \frac{4814590223457565386889746397856525590764718825233834563746488044}{25405742808078707299254621753686268524672856939997385174054198189} a^{10} - \frac{561453641419543699758397833905925939700901025502522991553688516}{25405742808078707299254621753686268524672856939997385174054198189} a^{9} + \frac{12211614774035065370797540987424562746718117267262104419245931153}{25405742808078707299254621753686268524672856939997385174054198189} a^{8} - \frac{8220414445124547602608625992133134866904835253760085934266522323}{25405742808078707299254621753686268524672856939997385174054198189} a^{7} - \frac{10961467574216897215811972934242118501141922529464795595790485603}{25405742808078707299254621753686268524672856939997385174054198189} a^{6} + \frac{9394785317150563392432758648197764712853750681370976800967494618}{25405742808078707299254621753686268524672856939997385174054198189} a^{5} + \frac{1673284702949620266632973028512842799624656200189902447126618391}{25405742808078707299254621753686268524672856939997385174054198189} a^{4} - \frac{2292200347195351914206537757356105514611953695595469539503316284}{25405742808078707299254621753686268524672856939997385174054198189} a^{3} + \frac{947402093940127231422059341765664067111010112613836090463901751}{25405742808078707299254621753686268524672856939997385174054198189} a^{2} - \frac{27682414234515431927422470757750571990280217614209312240038731}{67389238217715403976802710221979492107885562175059377119507157} a - \frac{57703116563021806334001086236614719527851350618180749370474}{178751295007202663068442202180316955193330403647372353102141}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.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:  \( 262575512314938100000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{21}$ (as 21T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

3.3.177241.1, 7.7.5567914722008521.1

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 $21$ $21$ $21$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{21}$ $21$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{21}$ $21$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
421Data not computed