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Properties

Label 27.27.1151231499...8681.2
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 7^{18}$
Root discriminant $167.68$
Ramified primes $3, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
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![-191719986857, -2616003280989, 0, 11336014217619, 0, -14574875422653, 0, 8725095491112, 0, -2942988558510, 0, 619174216206, 0, -86185422036, 0, 8208135432, 0, -543185433, 0, 24958395, 0, -781011, 0, 15876, 0, -189, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 189*x^25 + 15876*x^23 - 781011*x^21 + 24958395*x^19 - 543185433*x^17 + 8208135432*x^15 - 86185422036*x^13 + 619174216206*x^11 - 2942988558510*x^9 + 8725095491112*x^7 - 14574875422653*x^5 + 11336014217619*x^3 - 2616003280989*x - 191719986857)
 
gp: K = bnfinit(x^27 - 189*x^25 + 15876*x^23 - 781011*x^21 + 24958395*x^19 - 543185433*x^17 + 8208135432*x^15 - 86185422036*x^13 + 619174216206*x^11 - 2942988558510*x^9 + 8725095491112*x^7 - 14574875422653*x^5 + 11336014217619*x^3 - 2616003280989*x - 191719986857, 1)
 

Normalized defining polynomial

\( x^{27} - 189 x^{25} + 15876 x^{23} - 781011 x^{21} + 24958395 x^{19} - 543185433 x^{17} + 8208135432 x^{15} - 86185422036 x^{13} + 619174216206 x^{11} - 2942988558510 x^{9} + 8725095491112 x^{7} - 14574875422653 x^{5} + 11336014217619 x^{3} - 2616003280989 x - 191719986857 \)

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:  \(1151231499063623584349492991499498083339556847040831466248681=3^{94}\cdot 7^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.68$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(567=3^{4}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{567}(256,·)$, $\chi_{567}(1,·)$, $\chi_{567}(130,·)$, $\chi_{567}(67,·)$, $\chi_{567}(4,·)$, $\chi_{567}(193,·)$, $\chi_{567}(520,·)$, $\chi_{567}(457,·)$, $\chi_{567}(394,·)$, $\chi_{567}(331,·)$, $\chi_{567}(268,·)$, $\chi_{567}(205,·)$, $\chi_{567}(142,·)$, $\chi_{567}(79,·)$, $\chi_{567}(16,·)$, $\chi_{567}(508,·)$, $\chi_{567}(64,·)$, $\chi_{567}(127,·)$, $\chi_{567}(253,·)$, $\chi_{567}(382,·)$, $\chi_{567}(505,·)$, $\chi_{567}(442,·)$, $\chi_{567}(379,·)$, $\chi_{567}(316,·)$, $\chi_{567}(445,·)$, $\chi_{567}(190,·)$, $\chi_{567}(319,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{7} a^{4}$, $\frac{1}{7} a^{5}$, $\frac{1}{49} a^{6}$, $\frac{1}{49} a^{7}$, $\frac{1}{49} a^{8}$, $\frac{1}{343} a^{9}$, $\frac{1}{343} a^{10}$, $\frac{1}{343} a^{11}$, $\frac{1}{2401} a^{12}$, $\frac{1}{2401} a^{13}$, $\frac{1}{16334003} a^{14} + \frac{2967}{16334003} a^{13} - \frac{2}{333347} a^{12} + \frac{321}{333347} a^{11} + \frac{11}{47621} a^{10} + \frac{2991}{2333429} a^{9} - \frac{30}{6803} a^{8} - \frac{248}{47621} a^{7} + \frac{800}{333347} a^{6} - \frac{2510}{47621} a^{5} - \frac{2801}{47621} a^{4} + \frac{1982}{47621} a^{3} + \frac{2401}{6803} a^{2} - \frac{1255}{6803} a - \frac{686}{6803}$, $\frac{1}{114338021} a^{15} - \frac{15}{16334003} a^{13} + \frac{2967}{16334003} a^{12} + \frac{90}{2333429} a^{11} - \frac{227}{333347} a^{10} - \frac{275}{333347} a^{9} - \frac{3054}{333347} a^{8} + \frac{450}{47621} a^{7} + \frac{498}{333347} a^{6} - \frac{378}{6803} a^{5} - \frac{3018}{47621} a^{4} + \frac{57}{47621} a^{3} + \frac{646}{6803} a^{2} - \frac{735}{6803} a - \frac{1763}{6803}$, $\frac{1}{114338021} a^{16} - \frac{149}{16334003} a^{13} - \frac{120}{2333429} a^{12} - \frac{1899}{2333429} a^{11} - \frac{643}{2333429} a^{10} + \frac{3078}{2333429} a^{9} + \frac{1509}{333347} a^{8} + \frac{1670}{333347} a^{7} + \frac{281}{333347} a^{6} + \frac{150}{47621} a^{5} - \frac{1140}{47621} a^{4} + \frac{237}{47621} a^{3} + \frac{1265}{6803} a^{2} - \frac{179}{6803} a + \frac{3316}{6803}$, $\frac{1}{114338021} a^{17} - \frac{136}{2333429} a^{13} - \frac{683}{16334003} a^{12} + \frac{813}{2333429} a^{11} + \frac{1753}{2333429} a^{10} + \frac{421}{2333429} a^{9} + \frac{48}{47621} a^{8} + \frac{131}{333347} a^{7} - \frac{2204}{333347} a^{6} - \frac{965}{47621} a^{5} - \frac{2129}{47621} a^{4} - \frac{1962}{47621} a^{3} - \frac{2989}{6803} a^{2} + \frac{2}{6803} a - \frac{169}{6803}$, $\frac{1}{800366147} a^{18} - \frac{1850}{16334003} a^{13} + \frac{1091}{16334003} a^{12} + \frac{2623}{2333429} a^{11} - \frac{71}{333347} a^{10} - \frac{1068}{2333429} a^{9} + \frac{2245}{333347} a^{8} + \frac{2666}{333347} a^{7} - \frac{1013}{333347} a^{6} - \frac{2486}{47621} a^{5} - \frac{2192}{47621} a^{4} + \frac{178}{6803} a^{3} + \frac{1936}{6803} a^{2} - \frac{1601}{6803} a + \frac{1946}{6803}$, $\frac{1}{800366147} a^{19} + \frac{20}{16334003} a^{13} + \frac{333}{16334003} a^{12} - \frac{180}{2333429} a^{11} + \frac{2844}{2333429} a^{10} - \frac{2183}{2333429} a^{9} - \frac{2437}{333347} a^{8} - \frac{1597}{333347} a^{7} - \frac{47}{333347} a^{6} + \frac{757}{47621} a^{5} + \frac{3282}{47621} a^{4} - \frac{171}{47621} a^{3} - \frac{2110}{6803} a^{2} + \frac{19}{6803} a + \frac{3061}{6803}$, $\frac{1}{800366147} a^{20} + \frac{2220}{16334003} a^{13} + \frac{100}{2333429} a^{12} - \frac{1278}{2333429} a^{11} + \frac{643}{2333429} a^{10} - \frac{2046}{2333429} a^{9} + \frac{591}{333347} a^{8} + \frac{94}{47621} a^{7} + \frac{415}{47621} a^{6} - \frac{942}{47621} a^{5} + \frac{1425}{47621} a^{4} + \frac{2}{6803} a^{3} - \frac{380}{6803} a^{2} + \frac{949}{6803} a + \frac{114}{6803}$, $\frac{1}{5602563029} a^{21} - \frac{1077}{16334003} a^{13} + \frac{370}{2333429} a^{12} + \frac{815}{2333429} a^{11} - \frac{443}{333347} a^{10} + \frac{2493}{2333429} a^{9} - \frac{3113}{333347} a^{8} - \frac{68}{333347} a^{7} + \frac{1914}{333347} a^{6} + \frac{179}{6803} a^{5} - \frac{1902}{47621} a^{4} + \frac{2750}{47621} a^{3} - \frac{2304}{6803} a^{2} + \frac{541}{6803} a - \frac{136}{6803}$, $\frac{1}{5602563029} a^{22} + \frac{639}{16334003} a^{13} + \frac{2204}{16334003} a^{12} + \frac{1853}{2333429} a^{11} - \frac{2062}{2333429} a^{10} + \frac{2106}{2333429} a^{9} + \frac{263}{47621} a^{8} + \frac{3067}{333347} a^{7} - \frac{59}{47621} a^{6} + \frac{346}{6803} a^{5} - \frac{198}{47621} a^{4} + \frac{2753}{47621} a^{3} + \frac{1278}{6803} a^{2} + \frac{2026}{6803} a + \frac{2705}{6803}$, $\frac{1}{5602563029} a^{23} - \frac{2475}{16334003} a^{13} + \frac{760}{16334003} a^{12} - \frac{2462}{2333429} a^{11} - \frac{2165}{2333429} a^{10} - \frac{325}{2333429} a^{9} - \frac{460}{47621} a^{8} + \frac{2}{333347} a^{7} + \frac{339}{47621} a^{6} - \frac{1816}{47621} a^{5} - \frac{3400}{47621} a^{4} + \frac{1003}{47621} a^{3} - \frac{1538}{6803} a^{2} + \frac{1896}{6803} a + \frac{2962}{6803}$, $\frac{1}{39217941203} a^{24} + \frac{1493}{16334003} a^{13} - \frac{3097}{16334003} a^{12} + \frac{3074}{2333429} a^{11} - \frac{2871}{2333429} a^{10} + \frac{117}{333347} a^{9} - \frac{778}{333347} a^{8} - \frac{1191}{333347} a^{7} + \frac{3090}{333347} a^{6} + \frac{3243}{47621} a^{5} + \frac{1084}{47621} a^{4} + \frac{2419}{47621} a^{3} - \frac{2151}{6803} a^{2} - \frac{1114}{6803} a + \frac{2358}{6803}$, $\frac{1}{39217941203} a^{25} + \frac{2728}{16334003} a^{13} - \frac{2243}{16334003} a^{12} + \frac{3040}{2333429} a^{11} - \frac{1154}{2333429} a^{10} - \frac{1438}{2333429} a^{9} + \frac{2953}{333347} a^{8} + \frac{2995}{333347} a^{7} - \frac{1583}{333347} a^{6} + \frac{61}{47621} a^{5} + \frac{467}{47621} a^{4} - \frac{1272}{47621} a^{3} - \frac{626}{6803} a^{2} - \frac{1555}{6803} a - \frac{3055}{6803}$, $\frac{1}{39217941203} a^{26} - \frac{649}{16334003} a^{13} + \frac{414}{2333429} a^{12} - \frac{1467}{2333429} a^{11} - \frac{2382}{2333429} a^{10} - \frac{2389}{2333429} a^{9} - \frac{615}{333347} a^{8} - \frac{663}{333347} a^{7} + \frac{1790}{333347} a^{6} - \frac{2874}{47621} a^{5} + \frac{87}{47621} a^{4} - \frac{2893}{47621} a^{3} - \frac{194}{6803} a^{2} - \frac{1324}{6803} a + \frac{583}{6803}$

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:  \( 880724655227693200000 \) (assuming GRH)
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 $27$ R $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ $27$

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
7Data not computed