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Properties

Label 27.27.4942064767...5929.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 31^{18}$
Root discriminant $452.20$
Ramified primes $3, 31$
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![-254625961590055084699, -659273750031016149957, 0, 645095819922822254259, 0, -187285883203400009301, 0, 25316678528262981288, 0, -1928241644177736030, 0, 91605614767388046, 0, -2879250091944372, 0, 61919356816008, 0, -925265531169, 0, 9600000795, 0, -67834107, 0, 311364, 0, -837, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 837*x^25 + 311364*x^23 - 67834107*x^21 + 9600000795*x^19 - 925265531169*x^17 + 61919356816008*x^15 - 2879250091944372*x^13 + 91605614767388046*x^11 - 1928241644177736030*x^9 + 25316678528262981288*x^7 - 187285883203400009301*x^5 + 645095819922822254259*x^3 - 659273750031016149957*x - 254625961590055084699)
 
gp: K = bnfinit(x^27 - 837*x^25 + 311364*x^23 - 67834107*x^21 + 9600000795*x^19 - 925265531169*x^17 + 61919356816008*x^15 - 2879250091944372*x^13 + 91605614767388046*x^11 - 1928241644177736030*x^9 + 25316678528262981288*x^7 - 187285883203400009301*x^5 + 645095819922822254259*x^3 - 659273750031016149957*x - 254625961590055084699, 1)
 

Normalized defining polynomial

\( x^{27} - 837 x^{25} + 311364 x^{23} - 67834107 x^{21} + 9600000795 x^{19} - 925265531169 x^{17} + 61919356816008 x^{15} - 2879250091944372 x^{13} + 91605614767388046 x^{11} - 1928241644177736030 x^{9} + 25316678528262981288 x^{7} - 187285883203400009301 x^{5} + 645095819922822254259 x^{3} - 659273750031016149957 x - 254625961590055084699 \)

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:  \(494206476726811255379944367361178473261552448264840656699721419592855929=3^{94}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $452.20$
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:  $3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2511=3^{4}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{2511}(1885,·)$, $\chi_{2511}(1,·)$, $\chi_{2511}(1675,·)$, $\chi_{2511}(1606,·)$, $\chi_{2511}(769,·)$, $\chi_{2511}(1396,·)$, $\chi_{2511}(2443,·)$, $\chi_{2511}(718,·)$, $\chi_{2511}(1048,·)$, $\chi_{2511}(1555,·)$, $\chi_{2511}(280,·)$, $\chi_{2511}(559,·)$, $\chi_{2511}(1117,·)$, $\chi_{2511}(997,·)$, $\chi_{2511}(160,·)$, $\chi_{2511}(1954,·)$, $\chi_{2511}(2392,·)$, $\chi_{2511}(838,·)$, $\chi_{2511}(2113,·)$, $\chi_{2511}(1834,·)$, $\chi_{2511}(1327,·)$, $\chi_{2511}(211,·)$, $\chi_{2511}(2164,·)$, $\chi_{2511}(439,·)$, $\chi_{2511}(2233,·)$, $\chi_{2511}(1276,·)$, $\chi_{2511}(490,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{31} a^{3}$, $\frac{1}{31} a^{4}$, $\frac{1}{31} a^{5}$, $\frac{1}{961} a^{6}$, $\frac{1}{961} a^{7}$, $\frac{1}{961} a^{8}$, $\frac{1}{29791} a^{9}$, $\frac{1}{29791} a^{10}$, $\frac{1}{29791} a^{11}$, $\frac{1}{923521} a^{12}$, $\frac{1}{923521} a^{13}$, $\frac{1}{1923390404591} a^{14} + \frac{590462}{1923390404591} a^{13} - \frac{14}{62044851761} a^{12} + \frac{654678}{62044851761} a^{11} + \frac{77}{2001446831} a^{10} + \frac{575789}{62044851761} a^{9} - \frac{210}{64562801} a^{8} - \frac{132291}{2001446831} a^{7} + \frac{294}{2082671} a^{6} - \frac{886996}{64562801} a^{5} - \frac{6076}{2082671} a^{4} - \frac{830259}{64562801} a^{3} + \frac{47089}{2082671} a^{2} - \frac{178916}{2082671} a - \frac{59582}{2082671}$, $\frac{1}{59625102542321} a^{15} - \frac{15}{1923390404591} a^{13} + \frac{590462}{1923390404591} a^{12} + \frac{90}{62044851761} a^{11} - \frac{837531}{62044851761} a^{10} - \frac{275}{2001446831} a^{9} + \frac{644883}{2001446831} a^{8} + \frac{450}{64562801} a^{7} - \frac{735800}{2001446831} a^{6} - \frac{378}{2082671} a^{5} - \frac{351523}{64562801} a^{4} + \frac{4340}{2082671} a^{3} - \frac{831556}{2082671} a^{2} - \frac{14415}{2082671} a - \frac{187731}{2082671}$, $\frac{1}{59625102542321} a^{16} - \frac{965963}{1923390404591} a^{13} - \frac{120}{62044851761} a^{12} + \frac{651955}{62044851761} a^{11} + \frac{880}{2001446831} a^{10} - \frac{529186}{62044851761} a^{9} - \frac{2700}{64562801} a^{8} - \frac{637494}{2001446831} a^{7} - \frac{290590}{2001446831} a^{6} + \frac{922234}{64562801} a^{5} - \frac{608129}{64562801} a^{4} - \frac{744043}{64562801} a^{3} + \frac{691920}{2082671} a^{2} - \frac{788800}{2082671} a - \frac{893730}{2082671}$, $\frac{1}{59625102542321} a^{17} - \frac{136}{62044851761} a^{13} + \frac{855495}{1923390404591} a^{12} + \frac{1088}{2001446831} a^{11} - \frac{292302}{62044851761} a^{10} + \frac{571202}{62044851761} a^{9} + \frac{609796}{2001446831} a^{8} + \frac{25395}{2001446831} a^{7} - \frac{551080}{2001446831} a^{6} + \frac{758781}{64562801} a^{5} + \frac{633031}{64562801} a^{4} + \frac{8465}{2082671} a^{3} - \frac{91733}{2082671} a^{2} - \frac{842245}{2082671} a + \frac{605619}{2082671}$, $\frac{1}{1848378178811951} a^{18} + \frac{785833}{1923390404591} a^{13} - \frac{816}{62044851761} a^{12} + \frac{412487}{62044851761} a^{11} + \frac{6732}{2001446831} a^{10} - \frac{224398}{62044851761} a^{9} - \frac{22032}{64562801} a^{8} - \frac{71509}{2001446831} a^{7} - \frac{387344}{2001446831} a^{6} - \frac{487947}{64562801} a^{5} - \frac{615899}{64562801} a^{4} - \frac{542723}{64562801} a^{3} - \frac{72627}{2082671} a^{2} - \frac{261549}{2082671} a + \frac{227532}{2082671}$, $\frac{1}{1848378178811951} a^{19} - \frac{969}{62044851761} a^{13} - \frac{215451}{1923390404591} a^{12} + \frac{8721}{2001446831} a^{11} + \frac{478802}{62044851761} a^{10} + \frac{510168}{62044851761} a^{9} + \frac{661345}{2001446831} a^{8} - \frac{359577}{2001446831} a^{7} + \frac{136144}{2001446831} a^{6} - \frac{301182}{64562801} a^{5} + \frac{389855}{64562801} a^{4} + \frac{159798}{64562801} a^{3} + \frac{546642}{2082671} a^{2} - \frac{711979}{2082671} a + \frac{975055}{2082671}$, $\frac{1}{1848378178811951} a^{20} + \frac{646331}{1923390404591} a^{13} - \frac{490703}{1923390404591} a^{12} - \frac{311009}{62044851761} a^{11} - \frac{680224}{62044851761} a^{10} - \frac{781899}{62044851761} a^{9} - \frac{142393}{2001446831} a^{8} - \frac{16937}{2001446831} a^{7} - \frac{851457}{2001446831} a^{6} - \frac{472886}{64562801} a^{5} + \frac{751021}{64562801} a^{4} + \frac{119658}{64562801} a^{3} - \frac{339117}{2082671} a^{2} - \frac{191489}{2082671} a - \frac{769309}{2082671}$, $\frac{1}{57299723543170481} a^{21} - \frac{5985}{62044851761} a^{13} + \frac{406941}{1923390404591} a^{12} - \frac{301535}{62044851761} a^{11} + \frac{661675}{62044851761} a^{10} - \frac{541679}{62044851761} a^{9} + \frac{1027178}{2001446831} a^{8} + \frac{769581}{2001446831} a^{7} + \frac{742639}{2001446831} a^{6} + \frac{34899}{64562801} a^{5} - \frac{28183}{2082671} a^{4} - \frac{504541}{64562801} a^{3} - \frac{912108}{2082671} a^{2} + \frac{603462}{2082671} a + \frac{976266}{2082671}$, $\frac{1}{57299723543170481} a^{22} - \frac{885831}{1923390404591} a^{13} - \frac{314922}{1923390404591} a^{12} + \frac{806343}{62044851761} a^{11} + \frac{804114}{62044851761} a^{10} - \frac{994377}{62044851761} a^{9} + \frac{885911}{2001446831} a^{8} + \frac{409689}{2001446831} a^{7} + \frac{198489}{2001446831} a^{6} + \frac{29136}{2082671} a^{5} + \frac{84379}{64562801} a^{4} - \frac{626346}{64562801} a^{3} + \frac{456232}{2082671} a^{2} - \frac{593396}{2082671} a + \frac{271298}{2082671}$, $\frac{1}{57299723543170481} a^{23} + \frac{986047}{1923390404591} a^{13} + \frac{848062}{1923390404591} a^{12} + \frac{17835}{2001446831} a^{11} - \frac{426845}{62044851761} a^{10} - \frac{241736}{62044851761} a^{9} + \frac{565878}{2001446831} a^{8} + \frac{461496}{2001446831} a^{7} - \frac{438685}{2001446831} a^{6} + \frac{818873}{64562801} a^{5} + \frac{976983}{64562801} a^{4} + \frac{675864}{64562801} a^{3} + \frac{567775}{2082671} a^{2} + \frac{112531}{2082671} a - \frac{534160}{2082671}$, $\frac{1}{1776291429838284911} a^{24} + \frac{158186}{1923390404591} a^{13} - \frac{7134}{62044851761} a^{12} - \frac{975536}{62044851761} a^{11} - \frac{890078}{62044851761} a^{10} - \frac{312579}{62044851761} a^{9} - \frac{174599}{2001446831} a^{8} + \frac{293643}{2001446831} a^{7} + \frac{711880}{2001446831} a^{6} - \frac{573915}{64562801} a^{5} - \frac{533910}{64562801} a^{4} - \frac{461406}{64562801} a^{3} + \frac{713613}{2082671} a^{2} + \frac{609673}{2082671} a - \frac{48276}{2082671}$, $\frac{1}{1776291429838284911} a^{25} + \frac{585922}{1923390404591} a^{13} + \frac{923030}{1923390404591} a^{12} - \frac{968711}{62044851761} a^{11} - \frac{939110}{62044851761} a^{10} + \frac{527533}{62044851761} a^{9} - \frac{837642}{2001446831} a^{8} + \frac{617798}{2001446831} a^{7} + \frac{66339}{2001446831} a^{6} + \frac{270076}{64562801} a^{5} + \frac{129484}{64562801} a^{4} - \frac{753135}{64562801} a^{3} - \frac{579385}{2082671} a^{2} + \frac{541881}{2082671} a + \frac{951977}{2082671}$, $\frac{1}{1776291429838284911} a^{26} - \frac{859769}{1923390404591} a^{13} - \frac{668361}{1923390404591} a^{12} - \frac{672104}{62044851761} a^{11} - \frac{596040}{62044851761} a^{10} - \frac{707360}{62044851761} a^{9} - \frac{483254}{2001446831} a^{8} - \frac{575637}{2001446831} a^{7} + \frac{342430}{2001446831} a^{6} + \frac{878456}{64562801} a^{5} + \frac{434807}{64562801} a^{4} - \frac{412607}{64562801} a^{3} - \frac{796440}{2082671} a^{2} + \frac{527744}{2082671} a + \frac{673302}{2082671}$

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 $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ $27$ $27$ R ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ $27$

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
31Data not computed