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Properties

Label 27.27.8541091277...5009.1
Degree $27$
Signature $[27, 0]$
Discriminant $7^{18}\cdot 73^{24}$
Root discriminant $165.84$
Ramified primes $7, 73$
Class number $1$ (GRH)
Class group Trivial (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![2018477, -56615545, 314793135, -414810580, -940006807, 2333730233, 676032279, -4302987535, 641980245, 3935601510, -1390142439, -1975738003, 983894056, 550432787, -361313257, -79731532, 75852203, 4123822, -9451534, 356998, 701185, -65183, -29942, 3826, 670, -101, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 - 101*x^25 + 670*x^24 + 3826*x^23 - 29942*x^22 - 65183*x^21 + 701185*x^20 + 356998*x^19 - 9451534*x^18 + 4123822*x^17 + 75852203*x^16 - 79731532*x^15 - 361313257*x^14 + 550432787*x^13 + 983894056*x^12 - 1975738003*x^11 - 1390142439*x^10 + 3935601510*x^9 + 641980245*x^8 - 4302987535*x^7 + 676032279*x^6 + 2333730233*x^5 - 940006807*x^4 - 414810580*x^3 + 314793135*x^2 - 56615545*x + 2018477)
 
gp: K = bnfinit(x^27 - 6*x^26 - 101*x^25 + 670*x^24 + 3826*x^23 - 29942*x^22 - 65183*x^21 + 701185*x^20 + 356998*x^19 - 9451534*x^18 + 4123822*x^17 + 75852203*x^16 - 79731532*x^15 - 361313257*x^14 + 550432787*x^13 + 983894056*x^12 - 1975738003*x^11 - 1390142439*x^10 + 3935601510*x^9 + 641980245*x^8 - 4302987535*x^7 + 676032279*x^6 + 2333730233*x^5 - 940006807*x^4 - 414810580*x^3 + 314793135*x^2 - 56615545*x + 2018477, 1)
 

Normalized defining polynomial

\( x^{27} - 6 x^{26} - 101 x^{25} + 670 x^{24} + 3826 x^{23} - 29942 x^{22} - 65183 x^{21} + 701185 x^{20} + 356998 x^{19} - 9451534 x^{18} + 4123822 x^{17} + 75852203 x^{16} - 79731532 x^{15} - 361313257 x^{14} + 550432787 x^{13} + 983894056 x^{12} - 1975738003 x^{11} - 1390142439 x^{10} + 3935601510 x^{9} + 641980245 x^{8} - 4302987535 x^{7} + 676032279 x^{6} + 2333730233 x^{5} - 940006807 x^{4} - 414810580 x^{3} + 314793135 x^{2} - 56615545 x + 2018477 \)

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:  \(854109127767918484727194550995844259867162076193861770225009=7^{18}\cdot 73^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $165.84$
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:  $7, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(511=7\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{511}(128,·)$, $\chi_{511}(1,·)$, $\chi_{511}(2,·)$, $\chi_{511}(4,·)$, $\chi_{511}(8,·)$, $\chi_{511}(137,·)$, $\chi_{511}(74,·)$, $\chi_{511}(256,·)$, $\chi_{511}(16,·)$, $\chi_{511}(81,·)$, $\chi_{511}(274,·)$, $\chi_{511}(148,·)$, $\chi_{511}(470,·)$, $\chi_{511}(324,·)$, $\chi_{511}(347,·)$, $\chi_{511}(221,·)$, $\chi_{511}(32,·)$, $\chi_{511}(162,·)$, $\chi_{511}(37,·)$, $\chi_{511}(296,·)$, $\chi_{511}(64,·)$, $\chi_{511}(235,·)$, $\chi_{511}(429,·)$, $\chi_{511}(366,·)$, $\chi_{511}(373,·)$, $\chi_{511}(183,·)$, $\chi_{511}(442,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{249} a^{19} + \frac{41}{249} a^{18} - \frac{32}{249} a^{17} - \frac{38}{249} a^{16} - \frac{68}{249} a^{15} + \frac{106}{249} a^{14} - \frac{18}{83} a^{13} + \frac{95}{249} a^{12} - \frac{7}{249} a^{11} + \frac{18}{83} a^{10} + \frac{2}{83} a^{9} - \frac{122}{249} a^{8} + \frac{95}{249} a^{7} - \frac{26}{83} a^{6} + \frac{58}{249} a^{5} - \frac{22}{249} a^{4} - \frac{31}{83} a^{3} - \frac{47}{249} a^{2} + \frac{65}{249} a$, $\frac{1}{249} a^{20} + \frac{10}{83} a^{18} + \frac{29}{249} a^{17} - \frac{4}{249} a^{16} - \frac{94}{249} a^{15} + \frac{82}{249} a^{14} + \frac{68}{249} a^{13} + \frac{82}{249} a^{12} + \frac{92}{249} a^{11} + \frac{11}{83} a^{10} - \frac{119}{249} a^{9} + \frac{39}{83} a^{8} + \frac{11}{249} a^{7} + \frac{19}{249} a^{6} + \frac{30}{83} a^{5} + \frac{62}{249} a^{4} + \frac{31}{249} a^{3} + \frac{74}{249} a$, $\frac{1}{249} a^{21} - \frac{13}{83} a^{18} - \frac{40}{249} a^{17} - \frac{11}{83} a^{16} + \frac{47}{249} a^{15} + \frac{14}{83} a^{14} - \frac{124}{249} a^{13} - \frac{19}{249} a^{12} - \frac{89}{249} a^{11} + \frac{29}{83} a^{10} + \frac{20}{249} a^{9} + \frac{19}{249} a^{8} + \frac{74}{249} a^{7} + \frac{23}{249} a^{6} - \frac{6}{83} a^{5} + \frac{110}{249} a^{4} - \frac{32}{249} a^{3} + \frac{73}{249} a^{2} - \frac{124}{249} a - \frac{1}{3}$, $\frac{1}{249} a^{22} - \frac{6}{83} a^{18} - \frac{12}{83} a^{17} - \frac{8}{83} a^{16} + \frac{46}{249} a^{15} - \frac{19}{83} a^{14} + \frac{11}{83} a^{13} - \frac{119}{249} a^{12} - \frac{20}{249} a^{11} - \frac{32}{249} a^{10} + \frac{29}{83} a^{9} - \frac{119}{249} a^{8} - \frac{30}{83} a^{7} + \frac{11}{249} a^{6} + \frac{16}{83} a^{5} + \frac{23}{249} a^{4} + \frac{98}{249} a^{3} + \frac{118}{249} a^{2} + \frac{15}{83} a - \frac{1}{3}$, $\frac{1}{49053} a^{23} + \frac{70}{49053} a^{22} - \frac{68}{49053} a^{21} + \frac{85}{49053} a^{20} + \frac{53}{49053} a^{19} + \frac{391}{16351} a^{18} + \frac{3523}{49053} a^{17} + \frac{4378}{49053} a^{16} + \frac{475}{16351} a^{15} - \frac{9664}{49053} a^{14} - \frac{23702}{49053} a^{13} - \frac{2512}{16351} a^{12} - \frac{7064}{49053} a^{11} + \frac{17660}{49053} a^{10} - \frac{46}{249} a^{9} - \frac{20381}{49053} a^{8} + \frac{7197}{16351} a^{7} + \frac{7214}{16351} a^{6} - \frac{15248}{49053} a^{5} + \frac{6236}{49053} a^{4} + \frac{4274}{16351} a^{3} + \frac{5731}{49053} a^{2} - \frac{19345}{49053} a - \frac{47}{197}$, $\frac{1}{49053} a^{24} - \frac{43}{49053} a^{22} - \frac{80}{49053} a^{21} + \frac{13}{49053} a^{20} + \frac{8}{16351} a^{19} - \frac{3530}{49053} a^{18} - \frac{762}{16351} a^{17} - \frac{1865}{16351} a^{16} + \frac{272}{591} a^{15} + \frac{7328}{16351} a^{14} + \frac{7734}{16351} a^{13} + \frac{6692}{16351} a^{12} + \frac{4274}{49053} a^{11} - \frac{10072}{49053} a^{10} - \frac{5606}{49053} a^{9} + \frac{1023}{16351} a^{8} + \frac{5699}{49053} a^{7} + \frac{7397}{49053} a^{6} - \frac{13844}{49053} a^{5} - \frac{18863}{49053} a^{4} - \frac{4068}{16351} a^{3} + \frac{7763}{49053} a^{2} - \frac{17444}{49053} a - \frac{59}{197}$, $\frac{1}{49053} a^{25} - \frac{25}{49053} a^{22} + \frac{44}{49053} a^{21} - \frac{64}{49053} a^{20} - \frac{23}{16351} a^{19} + \frac{4025}{49053} a^{18} + \frac{4054}{49053} a^{17} + \frac{7132}{49053} a^{16} + \frac{14309}{49053} a^{15} + \frac{4293}{16351} a^{14} - \frac{5163}{16351} a^{13} - \frac{6347}{49053} a^{12} - \frac{3746}{49053} a^{11} - \frac{17087}{49053} a^{10} + \frac{15086}{49053} a^{9} + \frac{17786}{49053} a^{8} + \frac{19957}{49053} a^{7} - \frac{12290}{49053} a^{6} + \frac{2233}{16351} a^{5} - \frac{23008}{49053} a^{4} - \frac{649}{16351} a^{3} + \frac{2586}{16351} a^{2} + \frac{2424}{16351} a + \frac{241}{591}$, $\frac{1}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{26} + \frac{3169079459609457118178160612300206154457160205281345889167949539486711009}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{25} + \frac{5502481805898885769045281230579781434401650740944975145738685533088926564}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{24} + \frac{5513602004567658074316770922321189400984787055875584638905759657966990180}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{23} - \frac{603369911866773074913790191778713395012341403376533907756764037811112112481}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{22} + \frac{337125320892788510331351197756180449284785494965492805570436057587233000689}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{21} - \frac{748178750139256915529250797594461175920886827743553806564084975392922531514}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{20} + \frac{448838862568258640553095312431776962800434889500918688867287098840008755128}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{19} - \frac{2823682251349871045658869104993859532781026222673943960339932385737969775044}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{18} - \frac{2069549318654854488513171750727869752020359313927368176800736766107299411655}{181330668980342918625433216744488742194043855477159865736628437994704834357519} a^{17} + \frac{6787060825091533947822714273334750178228448404581895223694119698329670074426}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{16} + \frac{243806530296509232273001496208244305946079636913877586505164992685843051478256}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{15} + \frac{47335163139548730580225714920911492244387165611840012956480438346892860517946}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{14} - \frac{216495869659862587198521431136528436490437127907148665934927648756394138707361}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{13} - \frac{154771839424595238803231927232960881501764365255772135543639874707769487105677}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{12} + \frac{85273963394458093385740344230159452724044698380395011330523210280279826040634}{181330668980342918625433216744488742194043855477159865736628437994704834357519} a^{11} - \frac{105535415732424082850070896543599693309732665117511841767905435799825283105206}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{10} + \frac{116789797840534687133707103298252698491456025937545830709553073855334950685074}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{9} + \frac{100273029198047641869614862046289430585186776812306251827794179333785662106633}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{8} - \frac{171795184813385918124681914815010957625781223945699343340369152576816311986148}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{7} + \frac{161593334940372261313315569770619882443368500367927292235721630132959359019559}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{6} + \frac{100221747616188486963452011493637843807699997906223877204995316675245777442695}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{5} - \frac{17296088964061953350308048977105171190745137310667964747971412907126346136888}{181330668980342918625433216744488742194043855477159865736628437994704834357519} a^{4} + \frac{152238375073980614720901983431937430554668358291928421213149521232741226713839}{543992006941028755876299650233466226582131566431479597209885313984114503072557} a^{3} - \frac{19882951201058654581730037754579964344650722778834717946337980224555329592392}{60443556326780972875144405581496247398014618492386621912209479331568278119173} a^{2} + \frac{3175572407313300901614556670294849304166877819486680054227565840369593785540}{181330668980342918625433216744488742194043855477159865736628437994704834357519} a - \frac{2933514038740789207934890066389012076567988089225069354445004742260948749249}{6554120565554563323810839159439352127495561041343127677227533903423066302079}$

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:  $26$
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:  \( 2860207280794142500000 \) (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

3.3.261121.1, 3.3.5329.1, 3.3.261121.2, \(\Q(\zeta_{7})^+\), 9.9.17804320388674561.1, 9.9.806460091894081.1, 9.9.94879223351246735569.1, 9.9.94879223351246735569.2

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
73Data not computed