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Properties

Label 27.27.7361439730...5329.2
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 19^{18}$
Root discriminant $326.28$
Ramified primes $3, 19$
Class number Not computed
Class group Not computed
Galois group $C_{27}$ (as 27T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4600558507235203, -1135430553480940593, 0, 1812704918715185859, 0, -858649698338772249, 0, 189376374570957288, 0, -23533613799022470, 0, 1824136571981646, 0, -93545465229828, 0, 3282297025608, 0, -80025043581, 0, 1354686795, 0, -15617943, 0, 116964, 0, -513, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 513*x^25 + 116964*x^23 - 15617943*x^21 + 1354686795*x^19 - 80025043581*x^17 + 3282297025608*x^15 - 93545465229828*x^13 + 1824136571981646*x^11 - 23533613799022470*x^9 + 189376374570957288*x^7 - 858649698338772249*x^5 + 1812704918715185859*x^3 - 1135430553480940593*x - 4600558507235203)
 
gp: K = bnfinit(x^27 - 513*x^25 + 116964*x^23 - 15617943*x^21 + 1354686795*x^19 - 80025043581*x^17 + 3282297025608*x^15 - 93545465229828*x^13 + 1824136571981646*x^11 - 23533613799022470*x^9 + 189376374570957288*x^7 - 858649698338772249*x^5 + 1812704918715185859*x^3 - 1135430553480940593*x - 4600558507235203, 1)
 

Normalized defining polynomial

\( x^{27} - 513 x^{25} + 116964 x^{23} - 15617943 x^{21} + 1354686795 x^{19} - 80025043581 x^{17} + 3282297025608 x^{15} - 93545465229828 x^{13} + 1824136571981646 x^{11} - 23533613799022470 x^{9} + 189376374570957288 x^{7} - 858649698338772249 x^{5} + 1812704918715185859 x^{3} - 1135430553480940593 x - 4600558507235203 \)

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:  \(73614397307175798532497185733881845387702404681973260785482012855329=3^{94}\cdot 19^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $326.28$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1539=3^{4}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1539}(562,·)$, $\chi_{1539}(1,·)$, $\chi_{1539}(514,·)$, $\chi_{1539}(1027,·)$, $\chi_{1539}(7,·)$, $\chi_{1539}(520,·)$, $\chi_{1539}(1033,·)$, $\chi_{1539}(343,·)$, $\chi_{1539}(856,·)$, $\chi_{1539}(1369,·)$, $\chi_{1539}(220,·)$, $\chi_{1539}(733,·)$, $\chi_{1539}(1246,·)$, $\chi_{1539}(1375,·)$, $\chi_{1539}(1075,·)$, $\chi_{1539}(49,·)$, $\chi_{1539}(391,·)$, $\chi_{1539}(172,·)$, $\chi_{1539}(685,·)$, $\chi_{1539}(1198,·)$, $\chi_{1539}(349,·)$, $\chi_{1539}(904,·)$, $\chi_{1539}(178,·)$, $\chi_{1539}(691,·)$, $\chi_{1539}(1204,·)$, $\chi_{1539}(862,·)$, $\chi_{1539}(1417,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{19} a^{3}$, $\frac{1}{19} a^{4}$, $\frac{1}{19} a^{5}$, $\frac{1}{361} a^{6}$, $\frac{1}{361} a^{7}$, $\frac{1}{361} a^{8}$, $\frac{1}{6859} a^{9}$, $\frac{1}{6859} a^{10}$, $\frac{1}{6859} a^{11}$, $\frac{1}{130321} a^{12}$, $\frac{1}{130321} a^{13}$, $\frac{1}{85475328443} a^{14} + \frac{14190}{85475328443} a^{13} - \frac{14}{4498701497} a^{12} - \frac{184470}{4498701497} a^{11} + \frac{77}{236773763} a^{10} - \frac{184191}{4498701497} a^{9} - \frac{210}{12461777} a^{8} - \frac{82648}{236773763} a^{7} + \frac{294}{655883} a^{6} - \frac{122205}{12461777} a^{5} - \frac{3724}{655883} a^{4} + \frac{177123}{12461777} a^{3} + \frac{17689}{655883} a^{2} - \frac{306396}{655883} a - \frac{13718}{655883}$, $\frac{1}{1624031240417} a^{15} - \frac{15}{85475328443} a^{13} + \frac{14190}{85475328443} a^{12} + \frac{90}{4498701497} a^{11} - \frac{170280}{4498701497} a^{10} - \frac{275}{236773763} a^{9} + \frac{110377}{236773763} a^{8} + \frac{450}{12461777} a^{7} - \frac{25702}{236773763} a^{6} - \frac{378}{655883} a^{5} + \frac{106081}{12461777} a^{4} + \frac{2660}{655883} a^{3} + \frac{132285}{655883} a^{2} - \frac{5415}{655883} a - \frac{248948}{655883}$, $\frac{1}{1624031240417} a^{16} + \frac{227040}{85475328443} a^{13} - \frac{120}{4498701497} a^{12} - \frac{313798}{4498701497} a^{11} + \frac{880}{236773763} a^{10} - \frac{9819}{4498701497} a^{9} - \frac{2700}{12461777} a^{8} + \frac{46344}{236773763} a^{7} + \frac{143786}{236773763} a^{6} + \frac{240655}{12461777} a^{5} + \frac{300966}{12461777} a^{4} - \frac{76804}{12461777} a^{3} + \frac{259920}{655883} a^{2} - \frac{253707}{655883} a - \frac{205770}{655883}$, $\frac{1}{1624031240417} a^{17} - \frac{136}{4498701497} a^{13} - \frac{7811}{85475328443} a^{12} + \frac{1088}{236773763} a^{11} - \frac{292541}{4498701497} a^{10} - \frac{38374}{4498701497} a^{9} + \frac{161521}{236773763} a^{8} - \frac{266924}{236773763} a^{7} - \frac{196559}{236773763} a^{6} - \frac{94383}{12461777} a^{5} - \frac{176883}{12461777} a^{4} - \frac{160125}{12461777} a^{3} + \frac{263225}{655883} a^{2} - \frac{320676}{655883} a - \frac{253647}{655883}$, $\frac{1}{30856593567923} a^{18} + \frac{203421}{85475328443} a^{13} - \frac{816}{4498701497} a^{12} + \frac{12341}{236773763} a^{11} + \frac{6732}{236773763} a^{10} + \frac{35099}{4498701497} a^{9} + \frac{237275}{236773763} a^{8} - \frac{273063}{236773763} a^{7} - \frac{4715}{12461777} a^{6} - \frac{94034}{12461777} a^{5} - \frac{310801}{12461777} a^{4} + \frac{84282}{12461777} a^{3} - \frac{303746}{655883} a^{2} + \frac{258786}{655883} a + \frac{102001}{655883}$, $\frac{1}{30856593567923} a^{19} - \frac{51}{236773763} a^{13} + \frac{191500}{85475328443} a^{12} + \frac{459}{12461777} a^{11} + \frac{10582}{236773763} a^{10} + \frac{262197}{4498701497} a^{9} + \frac{49456}{236773763} a^{8} + \frac{284}{236773763} a^{7} - \frac{2700}{236773763} a^{6} + \frac{130921}{12461777} a^{5} - \frac{11877}{12461777} a^{4} - \frac{129288}{12461777} a^{3} + \frac{118855}{655883} a^{2} + \frac{232993}{655883} a - \frac{252887}{655883}$, $\frac{1}{30856593567923} a^{20} - \frac{253727}{85475328443} a^{13} + \frac{218604}{85475328443} a^{12} + \frac{86062}{4498701497} a^{11} + \frac{306287}{4498701497} a^{10} + \frac{58390}{4498701497} a^{9} - \frac{710}{236773763} a^{8} + \frac{13532}{236773763} a^{7} + \frac{1136}{12461777} a^{6} - \frac{249442}{12461777} a^{5} - \frac{234366}{12461777} a^{4} + \frac{251873}{12461777} a^{3} - \frac{68679}{655883} a^{2} - \frac{59960}{655883} a - \frac{47143}{655883}$, $\frac{1}{586275277790537} a^{21} - \frac{315}{236773763} a^{13} - \frac{186701}{85475328443} a^{12} - \frac{220102}{4498701497} a^{11} - \frac{170958}{4498701497} a^{10} + \frac{140393}{4498701497} a^{9} + \frac{17166}{236773763} a^{8} - \frac{75158}{236773763} a^{7} - \frac{293583}{236773763} a^{6} + \frac{306809}{12461777} a^{5} + \frac{157751}{12461777} a^{4} - \frac{7846}{655883} a^{3} + \frac{271402}{655883} a^{2} - \frac{163597}{655883} a - \frac{199537}{655883}$, $\frac{1}{586275277790537} a^{22} - \frac{43031}{85475328443} a^{13} - \frac{324212}{85475328443} a^{12} - \frac{71019}{4498701497} a^{11} - \frac{4676}{236773763} a^{10} + \frac{14311}{4498701497} a^{9} + \frac{73028}{236773763} a^{8} + \frac{192487}{236773763} a^{7} + \frac{51151}{236773763} a^{6} - \frac{10037}{655883} a^{5} + \frac{205030}{12461777} a^{4} - \frac{59528}{12461777} a^{3} - \frac{252123}{655883} a^{2} - \frac{203951}{655883} a - \frac{252596}{655883}$, $\frac{1}{586275277790537} a^{23} + \frac{16552}{4498701497} a^{13} + \frac{322053}{85475328443} a^{12} + \frac{134535}{4498701497} a^{11} + \frac{3896}{4498701497} a^{10} - \frac{156983}{4498701497} a^{9} - \frac{315740}{236773763} a^{8} - \frac{177311}{236773763} a^{7} - \frac{205117}{236773763} a^{6} - \frac{184314}{12461777} a^{5} - \frac{152078}{12461777} a^{4} + \frac{220197}{12461777} a^{3} + \frac{147128}{655883} a^{2} - \frac{218806}{655883} a - \frac{4558}{655883}$, $\frac{1}{11139230278020203} a^{24} - \frac{84336}{85475328443} a^{13} - \frac{53814}{85475328443} a^{12} - \frac{236482}{4498701497} a^{11} + \frac{319994}{4498701497} a^{10} - \frac{6868}{236773763} a^{9} - \frac{176515}{236773763} a^{8} - \frac{54957}{236773763} a^{7} + \frac{196275}{236773763} a^{6} + \frac{227625}{12461777} a^{5} - \frac{311177}{12461777} a^{4} + \frac{204242}{12461777} a^{3} - \frac{34385}{655883} a^{2} - \frac{62645}{655883} a + \frac{124818}{655883}$, $\frac{1}{11139230278020203} a^{25} - \frac{312449}{85475328443} a^{13} - \frac{35331}{85475328443} a^{12} - \frac{253049}{4498701497} a^{11} - \frac{52928}{4498701497} a^{10} - \frac{73574}{4498701497} a^{9} - \frac{87618}{236773763} a^{8} + \frac{63188}{236773763} a^{7} - \frac{184583}{236773763} a^{6} - \frac{46595}{12461777} a^{5} + \frac{149760}{12461777} a^{4} + \frac{112571}{12461777} a^{3} + \frac{278917}{655883} a^{2} - \frac{265687}{655883} a + \frac{56364}{655883}$, $\frac{1}{11139230278020203} a^{26} - \frac{153101}{85475328443} a^{13} - \frac{31043}{85475328443} a^{12} + \frac{166316}{4498701497} a^{11} - \frac{111138}{4498701497} a^{10} - \frac{192900}{4498701497} a^{9} + \frac{225261}{236773763} a^{8} - \frac{44059}{236773763} a^{7} - \frac{11144}{12461777} a^{6} + \frac{204443}{12461777} a^{5} + \frac{319408}{12461777} a^{4} - \frac{139188}{12461777} a^{3} + \frac{174516}{655883} a^{2} + \frac{271123}{655883} a + \frac{20023}{655883}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{27}$ (as 27T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 27
The 27 conjugacy class representatives for $C_{27}$
Character table for $C_{27}$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), \(\Q(\zeta_{27})^+\)

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 $27$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R $27$ $27$ $27$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{27}$ $27$

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
3Data not computed
$19$19.9.6.3$x^{9} - 361 x^{3} + 27436$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
19.9.6.3$x^{9} - 361 x^{3} + 27436$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
19.9.6.3$x^{9} - 361 x^{3} + 27436$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$