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Properties

Label 27.27.5898328880...6889.1
Degree $27$
Signature $[27, 0]$
Discriminant $13^{18}\cdot 73^{24}$
Root discriminant $250.56$
Ramified primes $13, 73$
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![241567, 11769907, 197825781, 1318748708, 2520335249, -2479823783, -8633121087, 854116693, 11059085461, 860974092, -7601774021, -727188729, 3185085572, 177434127, -849961797, 1202790, 144737679, -8000144, -15429798, 1505470, 999299, -127355, -37564, 5530, 746, -119, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 - 119*x^25 + 746*x^24 + 5530*x^23 - 37564*x^22 - 127355*x^21 + 999299*x^20 + 1505470*x^19 - 15429798*x^18 - 8000144*x^17 + 144737679*x^16 + 1202790*x^15 - 849961797*x^14 + 177434127*x^13 + 3185085572*x^12 - 727188729*x^11 - 7601774021*x^10 + 860974092*x^9 + 11059085461*x^8 + 854116693*x^7 - 8633121087*x^6 - 2479823783*x^5 + 2520335249*x^4 + 1318748708*x^3 + 197825781*x^2 + 11769907*x + 241567)
 
gp: K = bnfinit(x^27 - 6*x^26 - 119*x^25 + 746*x^24 + 5530*x^23 - 37564*x^22 - 127355*x^21 + 999299*x^20 + 1505470*x^19 - 15429798*x^18 - 8000144*x^17 + 144737679*x^16 + 1202790*x^15 - 849961797*x^14 + 177434127*x^13 + 3185085572*x^12 - 727188729*x^11 - 7601774021*x^10 + 860974092*x^9 + 11059085461*x^8 + 854116693*x^7 - 8633121087*x^6 - 2479823783*x^5 + 2520335249*x^4 + 1318748708*x^3 + 197825781*x^2 + 11769907*x + 241567, 1)
 

Normalized defining polynomial

\( x^{27} - 6 x^{26} - 119 x^{25} + 746 x^{24} + 5530 x^{23} - 37564 x^{22} - 127355 x^{21} + 999299 x^{20} + 1505470 x^{19} - 15429798 x^{18} - 8000144 x^{17} + 144737679 x^{16} + 1202790 x^{15} - 849961797 x^{14} + 177434127 x^{13} + 3185085572 x^{12} - 727188729 x^{11} - 7601774021 x^{10} + 860974092 x^{9} + 11059085461 x^{8} + 854116693 x^{7} - 8633121087 x^{6} - 2479823783 x^{5} + 2520335249 x^{4} + 1318748708 x^{3} + 197825781 x^{2} + 11769907 x + 241567 \)

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:  \(58983288808059104765952482916808157759827216339270579472138526889=13^{18}\cdot 73^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $250.56$
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:  $13, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(949=13\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{949}(256,·)$, $\chi_{949}(1,·)$, $\chi_{949}(835,·)$, $\chi_{949}(81,·)$, $\chi_{949}(586,·)$, $\chi_{949}(183,·)$, $\chi_{949}(588,·)$, $\chi_{949}(16,·)$, $\chi_{949}(913,·)$, $\chi_{949}(274,·)$, $\chi_{949}(659,·)$, $\chi_{949}(794,·)$, $\chi_{949}(731,·)$, $\chi_{949}(347,·)$, $\chi_{949}(807,·)$, $\chi_{949}(105,·)$, $\chi_{949}(235,·)$, $\chi_{949}(300,·)$, $\chi_{949}(367,·)$, $\chi_{949}(880,·)$, $\chi_{949}(178,·)$, $\chi_{949}(308,·)$, $\chi_{949}(373,·)$, $\chi_{949}(55,·)$, $\chi_{949}(867,·)$, $\chi_{949}(74,·)$, $\chi_{949}(575,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{249} a^{19} - \frac{25}{249} a^{18} - \frac{11}{249} a^{17} + \frac{17}{249} a^{16} - \frac{89}{249} a^{15} + \frac{116}{249} a^{14} - \frac{14}{83} a^{13} - \frac{31}{83} a^{12} + \frac{53}{249} a^{11} + \frac{12}{83} a^{10} + \frac{15}{83} a^{9} + \frac{7}{83} a^{8} - \frac{41}{249} a^{7} - \frac{37}{249} a^{6} + \frac{15}{83} a^{5} + \frac{80}{249} a^{4} + \frac{53}{249} a^{3} - \frac{6}{83} a^{2} - \frac{4}{83} a - \frac{13}{83}$, $\frac{1}{249} a^{20} + \frac{28}{249} a^{18} - \frac{3}{83} a^{17} + \frac{4}{249} a^{16} - \frac{34}{249} a^{15} + \frac{12}{83} a^{14} - \frac{64}{249} a^{13} - \frac{31}{249} a^{12} - \frac{50}{249} a^{11} + \frac{32}{249} a^{10} - \frac{16}{249} a^{9} + \frac{23}{83} a^{8} - \frac{22}{83} a^{7} + \frac{11}{83} a^{6} - \frac{40}{249} a^{5} + \frac{61}{249} a^{4} - \frac{7}{83} a^{3} - \frac{47}{249} a^{2} + \frac{76}{249} a + \frac{104}{249}$, $\frac{1}{249} a^{21} + \frac{9}{83} a^{18} - \frac{20}{249} a^{17} - \frac{4}{83} a^{16} + \frac{38}{249} a^{15} - \frac{25}{83} a^{14} - \frac{100}{249} a^{13} - \frac{19}{249} a^{12} - \frac{124}{249} a^{11} - \frac{37}{83} a^{10} + \frac{18}{83} a^{9} - \frac{73}{249} a^{8} - \frac{64}{249} a^{7} + \frac{1}{3} a^{6} - \frac{37}{249} a^{5} - \frac{103}{249} a^{4} - \frac{37}{249} a^{3} - \frac{1}{249} a^{2} + \frac{25}{249} a - \frac{70}{249}$, $\frac{1}{249} a^{22} - \frac{3}{83} a^{18} + \frac{12}{83} a^{17} - \frac{2}{83} a^{16} - \frac{79}{249} a^{15} - \frac{26}{83} a^{14} - \frac{47}{249} a^{13} - \frac{20}{249} a^{12} + \frac{35}{249} a^{11} - \frac{5}{249} a^{10} + \frac{41}{83} a^{9} - \frac{50}{249} a^{8} - \frac{55}{249} a^{7} + \frac{49}{249} a^{6} - \frac{73}{249} a^{5} - \frac{122}{249} a^{4} + \frac{62}{249} a^{3} + \frac{32}{83} a^{2} + \frac{5}{249} a - \frac{109}{249}$, $\frac{1}{249} a^{23} - \frac{23}{249} a^{18} - \frac{22}{249} a^{17} - \frac{3}{83} a^{16} - \frac{49}{249} a^{15} - \frac{82}{249} a^{14} - \frac{22}{83} a^{13} - \frac{55}{249} a^{12} - \frac{26}{249} a^{11} + \frac{32}{249} a^{10} + \frac{106}{249} a^{9} - \frac{115}{249} a^{8} - \frac{71}{249} a^{7} + \frac{3}{83} a^{6} + \frac{39}{83} a^{5} + \frac{35}{249} a^{4} - \frac{91}{249} a^{3} - \frac{74}{249} a^{2} + \frac{32}{249} a - \frac{34}{83}$, $\frac{1}{343371} a^{24} + \frac{283}{343371} a^{23} + \frac{17}{114457} a^{22} + \frac{341}{343371} a^{21} + \frac{1}{343371} a^{20} - \frac{58}{343371} a^{19} + \frac{8930}{343371} a^{18} - \frac{20056}{343371} a^{17} + \frac{43543}{343371} a^{16} + \frac{4362}{16351} a^{15} + \frac{42913}{114457} a^{14} + \frac{9587}{114457} a^{13} + \frac{267}{114457} a^{12} + \frac{82157}{343371} a^{11} - \frac{46439}{114457} a^{10} + \frac{12569}{114457} a^{9} + \frac{111002}{343371} a^{8} - \frac{49474}{114457} a^{7} + \frac{126758}{343371} a^{6} + \frac{2741}{343371} a^{5} - \frac{134119}{343371} a^{4} - \frac{125}{1743} a^{3} + \frac{162844}{343371} a^{2} + \frac{98989}{343371} a + \frac{42691}{343371}$, $\frac{1}{28499793} a^{25} + \frac{11}{28499793} a^{24} + \frac{7914}{9499931} a^{23} + \frac{1851}{1357133} a^{22} - \frac{579}{9499931} a^{21} - \frac{56869}{28499793} a^{20} + \frac{48149}{28499793} a^{19} - \frac{1344036}{9499931} a^{18} + \frac{1769959}{28499793} a^{17} - \frac{1530946}{28499793} a^{16} - \frac{2408593}{28499793} a^{15} - \frac{289865}{1357133} a^{14} + \frac{1326095}{28499793} a^{13} + \frac{627254}{9499931} a^{12} - \frac{497866}{28499793} a^{11} + \frac{5901898}{28499793} a^{10} - \frac{8931782}{28499793} a^{9} + \frac{3553475}{28499793} a^{8} - \frac{10013849}{28499793} a^{7} - \frac{2355477}{9499931} a^{6} + \frac{1844686}{9499931} a^{5} - \frac{5403802}{28499793} a^{4} + \frac{13631734}{28499793} a^{3} + \frac{4749147}{9499931} a^{2} - \frac{1120943}{4071399} a + \frac{10862990}{28499793}$, $\frac{1}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{26} + \frac{55831477906891652208978669455195107073380847419826769561328722281682396981487}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{25} - \frac{2158899283131779945744282316919161068008340027681022243770334997804047072442312}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{24} - \frac{4097492283497094686390011708631783187129989150632255552118704593623735743291320638}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{23} + \frac{5628903225596482771287598420862711980474123928002976741818974321607566285585367772}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{22} - \frac{4576829872572896365424398922774331350217586288205476501226583914299731778385509094}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{21} - \frac{2821118359291827648377555310836315667668164322765516975282332079122507708479452728}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{20} + \frac{4371167153096359127061839492143138753206656223566990822477381799643313624809043716}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{19} + \frac{362911703219225434338742463062729847985959005703657526841395212158350209887051643201}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{18} - \frac{471889408266585193732620977652231520654234463385926007848891549651208682667490281977}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{17} + \frac{144023695750287384212411470180609865110923555354613396939151623065740227165932958855}{1102990122660622143365444909443303685424445846016239244888204776886817368312452358241} a^{16} - \frac{341775033794496152848446938015189796514726061711107213729098856827833846751665241731}{1102990122660622143365444909443303685424445846016239244888204776886817368312452358241} a^{15} + \frac{495544559876279520776414856619432071478950567443897924106652113270958666631936143036}{1102990122660622143365444909443303685424445846016239244888204776886817368312452358241} a^{14} - \frac{5214924689920063667507856822097539984376776879755298384875037320571391185870254310}{52523339174315340160259281402062080258306945048392344994676417946991303252973921821} a^{13} + \frac{230474482014484155526781372603897062658957068385418979065794676857176095935252323016}{1102990122660622143365444909443303685424445846016239244888204776886817368312452358241} a^{12} + \frac{896267368518251191426530046672706694820938405148213978248589018996185323760792482759}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{11} - \frac{1596818444616819515381832579522808493260935709347193925410015714744071383586965251}{472710052568838061442333532618558722324762505435531104952087761522921729276765296389} a^{10} - \frac{97752132177289352572041228957964538582475905908411031135569878793374603267552256808}{367663374220207381121814969814434561808148615338746414962734925628939122770817452747} a^{9} + \frac{59474168640374748581818224110613073596068201119913806789147204562385025725626650996}{367663374220207381121814969814434561808148615338746414962734925628939122770817452747} a^{8} + \frac{1307023654260784938226703801390761307351434169294807009156560058404715638729348249813}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{7} + \frac{1383553469306883962158210038596933995311731932730461177810893564451761193930557004736}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{6} - \frac{793891946407724387989229785400011553514331662221771049141746630322936383746040904634}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{5} + \frac{18156569153603565821281565779524764504401019662135106754517683626752465619206888613}{52523339174315340160259281402062080258306945048392344994676417946991303252973921821} a^{4} - \frac{541414410711229127375692138303025182056774380503626985548614537459523568058298902059}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{3} - \frac{1010960563427254871159583776872613357029518302658187142535359494900180638856771397825}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723} a^{2} + \frac{234365989638087208065470084131180383778317410404374008455151310890342373716198505022}{472710052568838061442333532618558722324762505435531104952087761522921729276765296389} a + \frac{534910848273794344680221698810354134824433522607096413839589340519538813915822537003}{3308970367981866430096334728329911056273337538048717734664614330660452104937357074723}$

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:  \( 185722068650970870000000 \) (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

3.3.169.1, 3.3.5329.1, 3.3.900601.1, 3.3.900601.2, 9.9.730461405459781801.1, 9.9.3892628829695177217529.1, 9.9.806460091894081.1, 9.9.3892628829695177217529.2

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
13Data not computed
$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}$