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Properties

Label 27.27.7872520728...5361.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 73^{24}$
Root discriminant $196.09$
Ramified primes $3, 73$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times C_9$ (as 27T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6213203, 119304222, 491101713, 53422363, -2591621310, -2585377374, 4106020983, 5691128145, -2877753387, -5490461910, 928156266, 2973052770, -61951104, -994582143, -50437221, 214692790, 17664747, -30400659, -2737311, 2816421, 234648, -167001, -11376, 6036, 290, -120, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 3*x^26 - 120*x^25 + 290*x^24 + 6036*x^23 - 11376*x^22 - 167001*x^21 + 234648*x^20 + 2816421*x^19 - 2737311*x^18 - 30400659*x^17 + 17664747*x^16 + 214692790*x^15 - 50437221*x^14 - 994582143*x^13 - 61951104*x^12 + 2973052770*x^11 + 928156266*x^10 - 5490461910*x^9 - 2877753387*x^8 + 5691128145*x^7 + 4106020983*x^6 - 2585377374*x^5 - 2591621310*x^4 + 53422363*x^3 + 491101713*x^2 + 119304222*x + 6213203)
 
gp: K = bnfinit(x^27 - 3*x^26 - 120*x^25 + 290*x^24 + 6036*x^23 - 11376*x^22 - 167001*x^21 + 234648*x^20 + 2816421*x^19 - 2737311*x^18 - 30400659*x^17 + 17664747*x^16 + 214692790*x^15 - 50437221*x^14 - 994582143*x^13 - 61951104*x^12 + 2973052770*x^11 + 928156266*x^10 - 5490461910*x^9 - 2877753387*x^8 + 5691128145*x^7 + 4106020983*x^6 - 2585377374*x^5 - 2591621310*x^4 + 53422363*x^3 + 491101713*x^2 + 119304222*x + 6213203, 1)
 

Normalized defining polynomial

\( x^{27} - 3 x^{26} - 120 x^{25} + 290 x^{24} + 6036 x^{23} - 11376 x^{22} - 167001 x^{21} + 234648 x^{20} + 2816421 x^{19} - 2737311 x^{18} - 30400659 x^{17} + 17664747 x^{16} + 214692790 x^{15} - 50437221 x^{14} - 994582143 x^{13} - 61951104 x^{12} + 2973052770 x^{11} + 928156266 x^{10} - 5490461910 x^{9} - 2877753387 x^{8} + 5691128145 x^{7} + 4106020983 x^{6} - 2585377374 x^{5} - 2591621310 x^{4} + 53422363 x^{3} + 491101713 x^{2} + 119304222 x + 6213203 \)

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

Invariants

Degree:  $27$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[27, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(78725207281899442169440296957686289952130469188765531717705361=3^{36}\cdot 73^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $196.09$
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:  $3, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(657=3^{2}\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{657}(256,·)$, $\chi_{657}(1,·)$, $\chi_{657}(4,·)$, $\chi_{657}(454,·)$, $\chi_{657}(64,·)$, $\chi_{657}(586,·)$, $\chi_{657}(439,·)$, $\chi_{657}(397,·)$, $\chi_{657}(16,·)$, $\chi_{657}(274,·)$, $\chi_{657}(148,·)$, $\chi_{657}(154,·)$, $\chi_{657}(283,·)$, $\chi_{657}(220,·)$, $\chi_{657}(223,·)$, $\chi_{657}(592,·)$, $\chi_{657}(475,·)$, $\chi_{657}(37,·)$, $\chi_{657}(616,·)$, $\chi_{657}(235,·)$, $\chi_{657}(493,·)$, $\chi_{657}(367,·)$, $\chi_{657}(178,·)$, $\chi_{657}(373,·)$, $\chi_{657}(502,·)$, $\chi_{657}(55,·)$, $\chi_{657}(442,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{9} a^{18} - \frac{1}{3} a^{17} + \frac{2}{9} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{2}{9} a^{12} - \frac{1}{3} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9}$, $\frac{1}{9} a^{19} + \frac{2}{9} a^{16} - \frac{1}{3} a^{14} - \frac{2}{9} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{9} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{20} + \frac{2}{9} a^{17} - \frac{1}{3} a^{15} - \frac{2}{9} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{9} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{21} - \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{4}{9} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{9} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} + \frac{2}{9}$, $\frac{1}{9} a^{22} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{4}{9} a^{13} - \frac{1}{3} a^{11} + \frac{1}{9} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3} + \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{1773} a^{23} + \frac{10}{591} a^{22} + \frac{8}{197} a^{21} + \frac{68}{1773} a^{20} + \frac{4}{197} a^{19} - \frac{70}{1773} a^{18} + \frac{781}{1773} a^{17} + \frac{130}{591} a^{16} - \frac{785}{1773} a^{15} + \frac{874}{1773} a^{14} - \frac{292}{591} a^{13} + \frac{323}{1773} a^{12} + \frac{182}{591} a^{11} + \frac{271}{591} a^{10} - \frac{826}{1773} a^{9} + \frac{130}{591} a^{8} - \frac{87}{197} a^{7} - \frac{43}{591} a^{6} + \frac{794}{1773} a^{5} + \frac{65}{591} a^{4} - \frac{113}{591} a^{3} + \frac{188}{591} a^{2} + \frac{101}{591} a - \frac{410}{1773}$, $\frac{1}{441477} a^{24} - \frac{8}{49053} a^{23} + \frac{19864}{441477} a^{22} + \frac{20698}{441477} a^{21} - \frac{2369}{441477} a^{20} - \frac{7814}{147159} a^{19} - \frac{5962}{147159} a^{18} - \frac{17611}{441477} a^{17} + \frac{20822}{147159} a^{16} + \frac{104387}{441477} a^{15} - \frac{146957}{441477} a^{14} - \frac{136678}{441477} a^{13} - \frac{23497}{49053} a^{12} - \frac{19616}{441477} a^{11} + \frac{149299}{441477} a^{10} - \frac{141908}{441477} a^{9} + \frac{13862}{147159} a^{8} - \frac{6853}{16351} a^{7} + \frac{154610}{441477} a^{6} - \frac{13141}{147159} a^{5} - \frac{176056}{441477} a^{4} + \frac{148417}{441477} a^{3} - \frac{190003}{441477} a^{2} - \frac{156805}{441477} a + \frac{215180}{441477}$, $\frac{1}{441477} a^{25} - \frac{11}{441477} a^{23} - \frac{19937}{441477} a^{22} - \frac{11342}{441477} a^{21} - \frac{5242}{147159} a^{20} + \frac{2428}{49053} a^{19} + \frac{17285}{441477} a^{18} - \frac{56894}{147159} a^{17} - \frac{146491}{441477} a^{16} + \frac{64990}{441477} a^{15} - \frac{12076}{441477} a^{14} + \frac{72406}{147159} a^{13} - \frac{124400}{441477} a^{12} + \frac{133837}{441477} a^{11} - \frac{109838}{441477} a^{10} - \frac{3750}{16351} a^{9} + \frac{18890}{49053} a^{8} - \frac{192979}{441477} a^{7} + \frac{12548}{147159} a^{6} + \frac{183893}{441477} a^{5} + \frac{59335}{441477} a^{4} + \frac{24575}{441477} a^{3} + \frac{98264}{441477} a^{2} + \frac{23648}{441477} a - \frac{38701}{147159}$, $\frac{1}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{26} + \frac{1470143366948490408389843212950330164939779532938445262256616138738069450055586}{2892187954641570285616023718903234059400893170020004779764358386737801006342365441779} a^{25} + \frac{2277922883502357656249545427841606349778328262883772165207424633307237193773761}{2892187954641570285616023718903234059400893170020004779764358386737801006342365441779} a^{24} - \frac{1081608421171952244117900598920997740905640874859130094067733240095263683007402898}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{23} + \frac{113665713532465053408556914462269817575068875544115788348447359751598806431142314587}{2892187954641570285616023718903234059400893170020004779764358386737801006342365441779} a^{22} - \frac{473293655406067985256848026575146048596046417194120421836918816825286899473188388436}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{21} - \frac{213189109380295946847883896320755043021945724745764566599314724197603933323954824093}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{20} + \frac{132612001730473525643875675420416103920198346452590782522450905238668800734400846864}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{19} + \frac{7819464686058158272155940420868392329514907613854930482505835321592155216017957700}{321354217182396698401780413211470451044543685557778308862706487415311222926929493531} a^{18} - \frac{268196872345504758807349722019978319086806218588414282052572314924962182531178374860}{2892187954641570285616023718903234059400893170020004779764358386737801006342365441779} a^{17} - \frac{1437419330451498039554381162602147651451045078899944810256952317807774303182232591799}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{16} + \frac{1104465596804380367479023798383063232434242562228982920793559459185417371694961971974}{2892187954641570285616023718903234059400893170020004779764358386737801006342365441779} a^{15} + \frac{667114954432561644928956270598744559399772802882228837154814878502028242933281996039}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{14} - \frac{1620858648823092780618887261248909611234933384852423813598858163472743955898393276591}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{13} + \frac{312506184532057082991439682709496671681883629403688986814526821260709537771228346652}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{12} + \frac{236152638878459698145016137905852874994116328863011786877892527310189301136939838451}{2892187954641570285616023718903234059400893170020004779764358386737801006342365441779} a^{11} + \frac{1295482773144718757972568977866573232207559208273814698377909601494289231935633865202}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{10} + \frac{1375391311400598162974170767093793225532089203839359747091725197638865302162691924479}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{9} + \frac{1273542568818825951101603970089131857789043075202923307722394261439684869783147772983}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{8} + \frac{766939135421331563133716611494173225749074138106597849824815999675354194013652441346}{2892187954641570285616023718903234059400893170020004779764358386737801006342365441779} a^{7} + \frac{372961077610238950158004155128730554198581120693620543014460750671889350794759863910}{964062651547190095205341239634411353133631056673334926588119462245933668780788480593} a^{6} - \frac{1008274398598846028725606938175458577422600429167773566301427243744082425207093887580}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{5} + \frac{272110134035725849415658202860921385012957793567833860515522206210050770548837671815}{964062651547190095205341239634411353133631056673334926588119462245933668780788480593} a^{4} + \frac{2351470671353589389968731629472364660789364341564234328071673366235168592428623969050}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a^{3} - \frac{45183961914536173454514385851916315626919566178203638738172228519094892114533488368}{964062651547190095205341239634411353133631056673334926588119462245933668780788480593} a^{2} + \frac{4165872316889529452932134309968757205037870856977806180619179931872023323438281996996}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337} a + \frac{433082360113613797882280646568524140725804053834960946099441306563597836290768570636}{8676563863924710856848071156709702178202679510060014339293075160213403019027096325337}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $26$
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:  \( 5754131910241353000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_9$ (as 27T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3\times C_9$
Character table for $C_3\times C_9$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.5329.1, 3.3.431649.2, 3.3.431649.1, 9.9.80425212553252449.3, 9.9.428585957696282300721.1, 9.9.428585957696282300721.2, 9.9.806460091894081.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

In the table, R denotes a ramified prime. 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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$73$73.9.8.1$x^{9} - 73$$9$$1$$8$$C_9$$[\ ]_{9}$
73.9.8.1$x^{9} - 73$$9$$1$$8$$C_9$$[\ ]_{9}$
73.9.8.1$x^{9} - 73$$9$$1$$8$$C_9$$[\ ]_{9}$