↑

Properties

Label 27.27.1151231499...8681.1
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)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-315928389203, -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 - 315928389203)
 
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 - 315928389203, 1)
 

Normalized defining polynomial

\( x^{27} - 189 x^{25} + 15876 x^{23} - 781011 x^{21} + 24958395 x^{19} - 543185433 x^{17} + 8208135432 x^{15} - 86185422036 x^{13} + 619174216206 x^{11} - 2942988558510 x^{9} + 8725095491112 x^{7} - 14574875422653 x^{5} + 11336014217619 x^{3} - 2616003280989 x - 315928389203 \)

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$$\chi_{567}(64,·)$, $\chi_{567}(1,·)$, $\chi_{567}(529,·)$, $\chi_{567}(466,·)$, $\chi_{567}(403,·)$, $\chi_{567}(340,·)$, $\chi_{567}(277,·)$, $\chi_{567}(214,·)$, $\chi_{567}(151,·)$, $\chi_{567}(88,·)$, $\chi_{567}(25,·)$, $\chi_{567}(442,·)$, $\chi_{567}(121,·)$, $\chi_{567}(562,·)$, $\chi_{567}(499,·)$, $\chi_{567}(436,·)$, $\chi_{567}(373,·)$, $\chi_{567}(310,·)$, $\chi_{567}(247,·)$, $\chi_{567}(184,·)$, $\chi_{567}(505,·)$, $\chi_{567}(58,·)$, $\chi_{567}(379,·)$, $\chi_{567}(316,·)$, $\chi_{567}(253,·)$, $\chi_{567}(190,·)$, $\chi_{567}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{7} a^{4}$, $\frac{1}{7} a^{5}$, $\frac{1}{49} a^{6}$, $\frac{1}{49} a^{7}$, $\frac{1}{49} a^{8}$, $\frac{1}{343} a^{9}$, $\frac{1}{343} a^{10}$, $\frac{1}{343} a^{11}$, $\frac{1}{2401} a^{12}$, $\frac{1}{2401} a^{13}$, $\frac{1}{13870577} a^{14} - \frac{2522}{13870577} a^{13} - \frac{2}{283073} a^{12} - \frac{268}{283073} a^{11} + \frac{11}{40439} a^{10} + \frac{2113}{1981511} a^{9} - \frac{30}{5777} a^{8} - \frac{1605}{283073} a^{7} + \frac{2852}{283073} a^{6} - \frac{1016}{40439} a^{5} + \frac{1950}{40439} a^{4} - \frac{2221}{40439} a^{3} + \frac{2401}{5777} a^{2} - \frac{508}{5777} a - \frac{686}{5777}$, $\frac{1}{97094039} a^{15} - \frac{15}{13870577} a^{13} - \frac{2522}{13870577} a^{12} + \frac{90}{1981511} a^{11} + \frac{197}{283073} a^{10} - \frac{275}{283073} a^{9} + \frac{2460}{283073} a^{8} - \frac{2627}{283073} a^{7} + \frac{1514}{283073} a^{6} - \frac{378}{5777} a^{5} + \frac{747}{40439} a^{4} + \frac{1083}{40439} a^{3} + \frac{74}{5777} a^{2} - \frac{735}{5777} a + \frac{1255}{5777}$, $\frac{1}{97094039} a^{16} + \frac{87}{13870577} a^{13} - \frac{120}{1981511} a^{12} + \frac{2124}{1981511} a^{11} + \frac{383}{1981511} a^{10} + \frac{2699}{1981511} a^{9} - \frac{1569}{283073} a^{8} + \frac{547}{283073} a^{7} + \frac{1150}{283073} a^{6} + \frac{2838}{40439} a^{5} + \frac{1448}{40439} a^{4} + \frac{1865}{40439} a^{3} + \frac{618}{5777} a^{2} - \frac{588}{5777} a + \frac{1264}{5777}$, $\frac{1}{97094039} a^{17} - \frac{136}{1981511} a^{13} + \frac{286}{13870577} a^{12} + \frac{1839}{1981511} a^{11} + \frac{2022}{1981511} a^{10} + \frac{1604}{1981511} a^{9} + \frac{1343}{283073} a^{8} + \frac{2137}{283073} a^{7} + \frac{2822}{283073} a^{6} - \frac{2592}{40439} a^{5} - \frac{36}{5777} a^{4} + \frac{1135}{40439} a^{3} - \frac{1503}{5777} a^{2} - \frac{756}{5777} a + \frac{1912}{5777}$, $\frac{1}{679658273} a^{18} + \frac{1193}{13870577} a^{13} + \frac{65}{13870577} a^{12} + \frac{2642}{1981511} a^{11} + \frac{908}{1981511} a^{10} - \frac{139}{1981511} a^{9} + \frac{1755}{283073} a^{8} - \frac{236}{40439} a^{7} - \frac{1779}{283073} a^{6} + \frac{4}{371} a^{5} + \frac{2096}{40439} a^{4} + \frac{2622}{40439} a^{3} - \frac{2861}{5777} a^{2} - \frac{2792}{5777} a - \frac{864}{5777}$, $\frac{1}{679658273} a^{19} - \frac{1006}{13870577} a^{13} + \frac{2537}{13870577} a^{12} - \frac{2500}{1981511} a^{11} - \frac{1919}{1981511} a^{10} - \frac{1306}{1981511} a^{9} + \frac{1627}{283073} a^{8} + \frac{799}{283073} a^{7} - \frac{2508}{283073} a^{6} + \frac{1014}{40439} a^{5} - \frac{1374}{40439} a^{4} + \frac{1091}{40439} a^{3} - \frac{1793}{5777} a^{2} - \frac{1405}{5777} a - \frac{1936}{5777}$, $\frac{1}{679658273} a^{20} + \frac{1508}{13870577} a^{13} - \frac{548}{13870577} a^{12} - \frac{96}{1981511} a^{11} - \frac{2110}{1981511} a^{10} - \frac{423}{1981511} a^{9} + \frac{891}{283073} a^{8} + \frac{422}{283073} a^{7} - \frac{736}{283073} a^{6} - \frac{941}{40439} a^{5} - \frac{1389}{40439} a^{4} + \frac{376}{40439} a^{3} - \frac{785}{5777} a^{2} + \frac{1169}{5777} a - \frac{2653}{5777}$, $\frac{1}{4757607911} a^{21} - \frac{208}{1981511} a^{13} - \frac{2092}{13870577} a^{12} - \frac{314}{283073} a^{11} - \frac{2287}{1981511} a^{10} - \frac{2053}{1981511} a^{9} - \frac{260}{40439} a^{8} - \frac{137}{283073} a^{7} - \frac{44}{5777} a^{6} - \frac{849}{40439} a^{5} + \frac{1689}{40439} a^{4} + \frac{1492}{40439} a^{3} + \frac{2853}{5777} a^{2} + \frac{121}{5777} a - \frac{2418}{5777}$, $\frac{1}{4757607911} a^{22} + \frac{48}{13870577} a^{13} - \frac{2095}{13870577} a^{12} - \frac{1222}{1981511} a^{11} + \frac{2836}{1981511} a^{10} + \frac{1978}{1981511} a^{9} + \frac{2810}{283073} a^{8} + \frac{649}{283073} a^{7} - \frac{1317}{283073} a^{6} + \frac{1305}{40439} a^{5} - \frac{1592}{40439} a^{4} - \frac{1793}{40439} a^{3} + \frac{892}{5777} a^{2} - \frac{2610}{5777} a + \frac{605}{5777}$, $\frac{1}{4757607911} a^{23} - \frac{2356}{13870577} a^{13} + \frac{1927}{13870577} a^{12} + \frac{452}{1981511} a^{11} - \frac{786}{1981511} a^{10} - \frac{876}{1981511} a^{9} + \frac{1885}{283073} a^{8} + \frac{622}{283073} a^{7} - \frac{667}{283073} a^{6} + \frac{960}{40439} a^{5} + \frac{2816}{40439} a^{4} - \frac{384}{5777} a^{3} - \frac{2318}{5777} a^{2} + \frac{1881}{5777} a - \frac{1734}{5777}$, $\frac{1}{33303255377} a^{24} - \frac{2640}{13870577} a^{13} + \frac{2130}{13870577} a^{12} - \frac{1002}{1981511} a^{11} - \frac{276}{1981511} a^{10} - \frac{1637}{1981511} a^{9} - \frac{325}{283073} a^{8} + \frac{1095}{283073} a^{7} + \frac{2705}{283073} a^{6} + \frac{114}{40439} a^{5} + \frac{2304}{40439} a^{4} - \frac{187}{5777} a^{3} + \frac{422}{5777} a^{2} + \frac{2084}{5777} a + \frac{192}{5777}$, $\frac{1}{33303255377} a^{25} - \frac{846}{13870577} a^{13} + \frac{8}{13870577} a^{12} - \frac{2027}{1981511} a^{11} + \frac{181}{1981511} a^{10} + \frac{1240}{1981511} a^{9} + \frac{2439}{283073} a^{8} + \frac{46}{283073} a^{7} + \frac{2647}{283073} a^{6} + \frac{592}{40439} a^{5} - \frac{88}{5777} a^{4} - \frac{2608}{40439} a^{3} - \frac{2422}{5777} a^{2} - \frac{664}{5777} a - \frac{2839}{5777}$, $\frac{1}{33303255377} a^{26} - \frac{1891}{13870577} a^{13} + \frac{1112}{13870577} a^{12} + \frac{1760}{1981511} a^{11} + \frac{851}{1981511} a^{10} + \frac{321}{283073} a^{9} - \frac{31}{5777} a^{8} + \frac{2412}{283073} a^{7} + \frac{2150}{283073} a^{6} + \frac{621}{40439} a^{5} + \frac{647}{40439} a^{4} - \frac{152}{5777} a^{3} + \frac{2855}{5777} a^{2} + \frac{668}{5777} a - \frac{2656}{5777}$

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:  \( 1864771778993385600000 \) (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 $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{27}$ $27$

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
7Data not computed