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Properties

Label 27.27.4512385497...8201.1
Degree $27$
Signature $[27, 0]$
Discriminant $19^{18}\cdot 37^{24}$
Root discriminant $176.38$
Ramified primes $19, 37$
Class number $4$ (GRH)
Class group $[2, 2]$ (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![-5332589, 9973258, 69450354, -139835954, -307934112, 712423287, 506463311, -1631686473, -109881561, 1780570826, -403762596, -1018443104, 406625862, 322170703, -176060310, -55948534, 42143562, 4517402, -5964855, 38939, 502442, -37459, -24256, 2854, 608, -89, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 - 89*x^25 + 608*x^24 + 2854*x^23 - 24256*x^22 - 37459*x^21 + 502442*x^20 + 38939*x^19 - 5964855*x^18 + 4517402*x^17 + 42143562*x^16 - 55948534*x^15 - 176060310*x^14 + 322170703*x^13 + 406625862*x^12 - 1018443104*x^11 - 403762596*x^10 + 1780570826*x^9 - 109881561*x^8 - 1631686473*x^7 + 506463311*x^6 + 712423287*x^5 - 307934112*x^4 - 139835954*x^3 + 69450354*x^2 + 9973258*x - 5332589)
 
gp: K = bnfinit(x^27 - 6*x^26 - 89*x^25 + 608*x^24 + 2854*x^23 - 24256*x^22 - 37459*x^21 + 502442*x^20 + 38939*x^19 - 5964855*x^18 + 4517402*x^17 + 42143562*x^16 - 55948534*x^15 - 176060310*x^14 + 322170703*x^13 + 406625862*x^12 - 1018443104*x^11 - 403762596*x^10 + 1780570826*x^9 - 109881561*x^8 - 1631686473*x^7 + 506463311*x^6 + 712423287*x^5 - 307934112*x^4 - 139835954*x^3 + 69450354*x^2 + 9973258*x - 5332589, 1)
 

Normalized defining polynomial

\( x^{27} - 6 x^{26} - 89 x^{25} + 608 x^{24} + 2854 x^{23} - 24256 x^{22} - 37459 x^{21} + 502442 x^{20} + 38939 x^{19} - 5964855 x^{18} + 4517402 x^{17} + 42143562 x^{16} - 55948534 x^{15} - 176060310 x^{14} + 322170703 x^{13} + 406625862 x^{12} - 1018443104 x^{11} - 403762596 x^{10} + 1780570826 x^{9} - 109881561 x^{8} - 1631686473 x^{7} + 506463311 x^{6} + 712423287 x^{5} - 307934112 x^{4} - 139835954 x^{3} + 69450354 x^{2} + 9973258 x - 5332589 \)

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:  \(4512385497467278486124161046693886108531644910659813162128201=19^{18}\cdot 37^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $176.38$
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:  $19, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(703=19\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{703}(1,·)$, $\chi_{703}(514,·)$, $\chi_{703}(581,·)$, $\chi_{703}(7,·)$, $\chi_{703}(201,·)$, $\chi_{703}(330,·)$, $\chi_{703}(334,·)$, $\chi_{703}(144,·)$, $\chi_{703}(83,·)$, $\chi_{703}(343,·)$, $\chi_{703}(292,·)$, $\chi_{703}(26,·)$, $\chi_{703}(349,·)$, $\chi_{703}(197,·)$, $\chi_{703}(482,·)$, $\chi_{703}(419,·)$, $\chi_{703}(676,·)$, $\chi_{703}(229,·)$, $\chi_{703}(305,·)$, $\chi_{703}(552,·)$, $\chi_{703}(49,·)$, $\chi_{703}(562,·)$, $\chi_{703}(182,·)$, $\chi_{703}(248,·)$, $\chi_{703}(121,·)$, $\chi_{703}(571,·)$, $\chi_{703}(638,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{31} a^{21} + \frac{15}{31} a^{20} + \frac{8}{31} a^{19} - \frac{13}{31} a^{18} - \frac{7}{31} a^{17} - \frac{5}{31} a^{16} + \frac{12}{31} a^{15} + \frac{3}{31} a^{14} - \frac{14}{31} a^{13} - \frac{10}{31} a^{12} - \frac{3}{31} a^{11} + \frac{14}{31} a^{10} - \frac{7}{31} a^{9} - \frac{1}{31} a^{8} + \frac{14}{31} a^{7} - \frac{11}{31} a^{6} - \frac{10}{31} a^{5} - \frac{4}{31} a^{4} - \frac{3}{31} a^{3} + \frac{3}{31} a^{2} - \frac{13}{31} a$, $\frac{1}{31} a^{22} - \frac{9}{31} a^{19} + \frac{2}{31} a^{18} + \frac{7}{31} a^{17} - \frac{6}{31} a^{16} + \frac{9}{31} a^{15} + \frac{3}{31} a^{14} + \frac{14}{31} a^{13} - \frac{8}{31} a^{12} - \frac{3}{31} a^{11} + \frac{11}{31} a^{9} - \frac{2}{31} a^{8} - \frac{4}{31} a^{7} - \frac{9}{31} a^{5} - \frac{5}{31} a^{4} - \frac{14}{31} a^{3} + \frac{4}{31} a^{2} + \frac{9}{31} a$, $\frac{1}{31} a^{23} - \frac{9}{31} a^{20} + \frac{2}{31} a^{19} + \frac{7}{31} a^{18} - \frac{6}{31} a^{17} + \frac{9}{31} a^{16} + \frac{3}{31} a^{15} + \frac{14}{31} a^{14} - \frac{8}{31} a^{13} - \frac{3}{31} a^{12} + \frac{11}{31} a^{10} - \frac{2}{31} a^{9} - \frac{4}{31} a^{8} - \frac{9}{31} a^{6} - \frac{5}{31} a^{5} - \frac{14}{31} a^{4} + \frac{4}{31} a^{3} + \frac{9}{31} a^{2}$, $\frac{1}{1333} a^{24} - \frac{1}{1333} a^{23} - \frac{10}{1333} a^{22} - \frac{21}{1333} a^{21} - \frac{262}{1333} a^{20} - \frac{63}{1333} a^{19} + \frac{278}{1333} a^{18} + \frac{5}{31} a^{17} + \frac{424}{1333} a^{16} + \frac{366}{1333} a^{15} - \frac{615}{1333} a^{14} + \frac{188}{1333} a^{13} - \frac{107}{1333} a^{12} - \frac{574}{1333} a^{11} - \frac{646}{1333} a^{10} + \frac{375}{1333} a^{9} - \frac{646}{1333} a^{8} + \frac{49}{1333} a^{7} - \frac{484}{1333} a^{6} - \frac{481}{1333} a^{5} + \frac{364}{1333} a^{4} + \frac{553}{1333} a^{3} + \frac{597}{1333} a^{2} - \frac{213}{1333} a + \frac{5}{43}$, $\frac{1}{6797674823} a^{25} - \frac{896436}{6797674823} a^{24} + \frac{535450}{219279833} a^{23} - \frac{45120512}{6797674823} a^{22} + \frac{32844507}{6797674823} a^{21} - \frac{3038942404}{6797674823} a^{20} - \frac{2354100877}{6797674823} a^{19} - \frac{41734317}{6797674823} a^{18} + \frac{1982768307}{6797674823} a^{17} - \frac{1729535402}{6797674823} a^{16} + \frac{2340700551}{6797674823} a^{15} + \frac{1853508060}{6797674823} a^{14} - \frac{691271857}{6797674823} a^{13} + \frac{3189849800}{6797674823} a^{12} - \frac{2859041669}{6797674823} a^{11} - \frac{2556255656}{6797674823} a^{10} - \frac{581860568}{6797674823} a^{9} - \frac{2458219637}{6797674823} a^{8} + \frac{2996564641}{6797674823} a^{7} + \frac{3039316600}{6797674823} a^{6} + \frac{1008648165}{6797674823} a^{5} + \frac{2435193946}{6797674823} a^{4} + \frac{442329641}{6797674823} a^{3} + \frac{2585776348}{6797674823} a^{2} + \frac{87684174}{219279833} a - \frac{16977}{39517}$, $\frac{1}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{26} - \frac{69734814305217910776529703506924855700698199776187494258966673000987438}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{25} + \frac{1076934055935299421481524990612778883183037467237365293512051491575079466480934}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{24} + \frac{56513410014182717917328242476103188417586566265468323625993784087069534082376837}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{23} + \frac{13001174999275539981014197292036722550123571882402863524368533868347917435660223}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{22} - \frac{59541715357644111579148888571925357188965005203765590904115430921299771433862611}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{21} + \frac{954564973443688525263889605151522818203951028533515462848393008739473949355430086}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{20} + \frac{508081220365123742524586700003182795299044007159152122649658593298014709459111458}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{19} + \frac{189925819366690574876744256407204327495526801810470893168218666176760736802423096}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{18} + \frac{1591558932612355324917388760713441517889594130606180679595589433198736768116127354}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{17} + \frac{445369073323612815252784187396576128259524246779048096778452366314890817500986161}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{16} + \frac{1071204945141226783814572707281390630614074209688921605875329004318855222351042224}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{15} + \frac{9462202605157783912400873043219152391094890077819248286048682447421973618741556}{126689268328766862300194409913706787609867382479919178969101187782664842697119441} a^{14} + \frac{1165099482714316879300216957574950702763680065138809632589746906480034823943451172}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{13} - \frac{659122862033814372611399934898243556515773322547462627363455900824538443474343274}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{12} - \frac{310471621167967809620860269870883702830059146112352166372715957489860408796594752}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{11} - \frac{1716555318794353709335948323602515035526613685178516887070150455636456363020961457}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{10} + \frac{70173772517675987275825281201877524224771067424530276368843372967407301752382719}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{9} + \frac{873192089527297926448516016612881257601442932114227306387495232585068089159973440}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{8} - \frac{49209973384391399719788670235376374324327453938078077546545062509902063713306875}{126689268328766862300194409913706787609867382479919178969101187782664842697119441} a^{7} + \frac{1513385833172339286707953211837462575260314908875700704682854199390982719104542943}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{6} - \frac{113908331212466997582171125881546888534045377351346202267007071191188177101811723}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{5} + \frac{865110939643958204736241186211532068004501620706014878025091229855912706503697895}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{4} + \frac{493272418790904582409247870199506372252213526197301512249205252716079735511303842}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{3} + \frac{169628197797506138983361639422500068298247075918617111860748263252808062580170852}{3927367318191772731306026707324910415905888856877494548042136821262610123610702671} a^{2} + \frac{59965881997542373401419137306609836053425694524952246476198756560205799799240934}{126689268328766862300194409913706787609867382479919178969101187782664842697119441} a - \frac{1224158872469913384396468829733359800008762609370726507102790772619404507547}{22831008889667843269092522961561864770204970711825406193746835066257855955509}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 2472314464659601600000 \) (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

3.3.494209.2, 3.3.1369.1, 3.3.494209.1, 3.3.361.1, 9.9.120706859316371329.1, 9.9.3512479453921.1, 9.9.165247690404112349401.1, 9.9.165247690404112349401.2

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{27}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
19Data not computed
$37$37.9.8.1$x^{9} - 37$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.1$x^{9} - 37$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.1$x^{9} - 37$$9$$1$$8$$C_9$$[\ ]_{9}$