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Properties

Label 27.27.4942064767...5929.2
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![-209910532282507790711, -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 - 209910532282507790711)
 
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 - 209910532282507790711, 1)
 

Normalized defining polynomial

\( x^{27} - 837 x^{25} + 311364 x^{23} - 67834107 x^{21} + 9600000795 x^{19} - 925265531169 x^{17} + 61919356816008 x^{15} - 2879250091944372 x^{13} + 91605614767388046 x^{11} - 1928241644177736030 x^{9} + 25316678528262981288 x^{7} - 187285883203400009301 x^{5} + 645095819922822254259 x^{3} - 659273750031016149957 x - 209910532282507790711 \)

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$$\chi_{2511}(1,·)$, $\chi_{2511}(67,·)$, $\chi_{2511}(838,·)$, $\chi_{2511}(583,·)$, $\chi_{2511}(904,·)$, $\chi_{2511}(1675,·)$, $\chi_{2511}(1420,·)$, $\chi_{2511}(1741,·)$, $\chi_{2511}(2257,·)$, $\chi_{2511}(280,·)$, $\chi_{2511}(25,·)$, $\chi_{2511}(346,·)$, $\chi_{2511}(1117,·)$, $\chi_{2511}(862,·)$, $\chi_{2511}(1183,·)$, $\chi_{2511}(1954,·)$, $\chi_{2511}(1699,·)$, $\chi_{2511}(2020,·)$, $\chi_{2511}(559,·)$, $\chi_{2511}(304,·)$, $\chi_{2511}(625,·)$, $\chi_{2511}(1396,·)$, $\chi_{2511}(1141,·)$, $\chi_{2511}(1462,·)$, $\chi_{2511}(2233,·)$, $\chi_{2511}(1978,·)$, $\chi_{2511}(2299,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{31} a^{3}$, $\frac{1}{31} a^{4}$, $\frac{1}{31} a^{5}$, $\frac{1}{961} a^{6}$, $\frac{1}{961} a^{7}$, $\frac{1}{961} a^{8}$, $\frac{1}{29791} a^{9}$, $\frac{1}{29791} a^{10}$, $\frac{1}{29791} a^{11}$, $\frac{1}{923521} a^{12}$, $\frac{1}{923521} a^{13}$, $\frac{1}{3485274978379} a^{14} - \frac{1069959}{3485274978379} a^{13} - \frac{14}{112428225109} a^{12} - \frac{1186129}{112428225109} a^{11} + \frac{77}{3626716939} a^{10} - \frac{1071056}{112428225109} a^{9} - \frac{210}{116990869} a^{8} + \frac{306195}{3626716939} a^{7} + \frac{294}{3773899} a^{6} + \frac{1529722}{116990869} a^{5} - \frac{6076}{3773899} a^{4} - \frac{1067297}{116990869} a^{3} + \frac{47089}{3773899} a^{2} + \frac{152471}{3773899} a - \frac{59582}{3773899}$, $\frac{1}{108043524329749} a^{15} - \frac{15}{3485274978379} a^{13} - \frac{1069959}{3485274978379} a^{12} + \frac{90}{112428225109} a^{11} + \frac{1517811}{112428225109} a^{10} - \frac{275}{3626716939} a^{9} - \frac{1169301}{3626716939} a^{8} + \frac{450}{116990869} a^{7} + \frac{1381032}{3626716939} a^{6} - \frac{378}{3773899} a^{5} + \frac{592232}{116990869} a^{4} + \frac{4340}{3773899} a^{3} + \frac{1522160}{3773899} a^{2} - \frac{14415}{3773899} a + \frac{313757}{3773899}$, $\frac{1}{108043524329749} a^{16} + \frac{1750151}{3485274978379} a^{13} - \frac{120}{112428225109} a^{12} - \frac{1178528}{112428225109} a^{11} + \frac{880}{3626716939} a^{10} + \frac{520415}{112428225109} a^{9} - \frac{2700}{116990869} a^{8} - \frac{1573841}{3626716939} a^{7} + \frac{100853}{3626716939} a^{6} + \frac{894668}{116990869} a^{5} + \frac{1083099}{116990869} a^{4} + \frac{986313}{116990869} a^{3} + \frac{691920}{3773899} a^{2} - \frac{1173077}{3773899} a - \frac{893730}{3773899}$, $\frac{1}{108043524329749} a^{17} - \frac{136}{112428225109} a^{13} - \frac{1557442}{3485274978379} a^{12} + \frac{1088}{3626716939} a^{11} + \frac{616171}{112428225109} a^{10} + \frac{179759}{112428225109} a^{9} - \frac{1491912}{3626716939} a^{8} - \frac{1274390}{3626716939} a^{7} - \frac{728744}{3626716939} a^{6} - \frac{1715333}{116990869} a^{5} - \frac{1423480}{116990869} a^{4} + \frac{668933}{116990869} a^{3} + \frac{372846}{3773899} a^{2} + \frac{457540}{3773899} a + \frac{893613}{3773899}$, $\frac{1}{3349349254222219} a^{18} - \frac{1426070}{3485274978379} a^{13} - \frac{816}{112428225109} a^{12} - \frac{720352}{112428225109} a^{11} + \frac{6732}{3626716939} a^{10} + \frac{26533}{112428225109} a^{9} - \frac{22032}{116990869} a^{8} - \frac{1354741}{3626716939} a^{7} - \frac{1029699}{3626716939} a^{6} + \frac{918783}{116990869} a^{5} - \frac{881886}{116990869} a^{4} - \frac{1371384}{116990869} a^{3} - \frac{1372412}{3773899} a^{2} - \frac{661125}{3773899} a - \frac{555354}{3773899}$, $\frac{1}{3349349254222219} a^{19} - \frac{969}{112428225109} a^{13} + \frac{317538}{3485274978379} a^{12} + \frac{8721}{3626716939} a^{11} - \frac{1275}{112428225109} a^{10} - \frac{538705}{112428225109} a^{9} - \frac{1278901}{3626716939} a^{8} - \frac{735945}{3626716939} a^{7} - \frac{1226476}{3626716939} a^{6} - \frac{1224599}{116990869} a^{5} + \frac{822920}{116990869} a^{4} - \frac{1380680}{116990869} a^{3} - \frac{1209701}{3773899} a^{2} + \frac{572731}{3773899} a + \frac{1233245}{3773899}$, $\frac{1}{3349349254222219} a^{20} - \frac{1656979}{3485274978379} a^{13} - \frac{882146}{3485274978379} a^{12} - \frac{749847}{112428225109} a^{11} - \frac{539693}{112428225109} a^{10} + \frac{904749}{112428225109} a^{9} - \frac{47087}{3626716939} a^{8} - \frac{426734}{3626716939} a^{7} - \frac{683604}{3626716939} a^{6} + \frac{1147854}{116990869} a^{5} + \frac{1441936}{116990869} a^{4} - \frac{1024319}{116990869} a^{3} - \frac{132923}{3773899} a^{2} - \frac{203772}{3773899} a - \frac{955572}{3773899}$, $\frac{1}{103829826880888789} a^{21} - \frac{5985}{112428225109} a^{13} - \frac{1304159}{3485274978379} a^{12} + \frac{57456}{3626716939} a^{11} + \frac{1738776}{112428225109} a^{10} + \frac{446894}{112428225109} a^{9} - \frac{1389341}{3626716939} a^{8} - \frac{1864995}{3626716939} a^{7} - \frac{289338}{3626716939} a^{6} - \frac{55551}{116990869} a^{5} - \frac{414040}{116990869} a^{4} - \frac{437864}{116990869} a^{3} - \frac{733060}{3773899} a^{2} - \frac{1082910}{3773899} a + \frac{457118}{3773899}$, $\frac{1}{103829826880888789} a^{22} - \frac{1512026}{3485274978379} a^{13} + \frac{1110319}{3485274978379} a^{12} + \frac{667148}{112428225109} a^{11} + \frac{1772756}{112428225109} a^{10} - \frac{1505898}{112428225109} a^{9} + \frac{1723734}{3626716939} a^{8} + \frac{1098340}{3626716939} a^{7} - \frac{1233501}{3626716939} a^{6} + \frac{482935}{116990869} a^{5} - \frac{763584}{116990869} a^{4} - \frac{775932}{116990869} a^{3} - \frac{1001480}{3773899} a^{2} + \frac{17199}{3773899} a - \frac{796199}{3773899}$, $\frac{1}{103829826880888789} a^{23} + \frac{1628402}{3485274978379} a^{13} - \frac{1522664}{3485274978379} a^{12} - \frac{1188424}{112428225109} a^{11} - \frac{147280}{112428225109} a^{10} - \frac{831610}{112428225109} a^{9} + \frac{137672}{3626716939} a^{8} - \frac{813953}{3626716939} a^{7} + \frac{810271}{3626716939} a^{6} + \frac{1262876}{116990869} a^{5} + \frac{1116746}{116990869} a^{4} + \frac{1700273}{116990869} a^{3} + \frac{1430979}{3773899} a^{2} - \frac{622065}{3773899} a + \frac{983796}{3773899}$, $\frac{1}{3218724633307552459} a^{24} + \frac{470317}{3485274978379} a^{13} - \frac{1034190}{3485274978379} a^{12} - \frac{1279168}{112428225109} a^{11} + \frac{585751}{112428225109} a^{10} + \frac{1429564}{112428225109} a^{9} + \frac{948128}{3626716939} a^{8} + \frac{236889}{3626716939} a^{7} - \frac{1022084}{3626716939} a^{6} + \frac{1413330}{116990869} a^{5} + \frac{766563}{116990869} a^{4} - \frac{1047768}{116990869} a^{3} + \frac{1506346}{3773899} a^{2} + \frac{1759750}{3773899} a + \frac{1226373}{3773899}$, $\frac{1}{3218724633307552459} a^{25} - \frac{367645}{3485274978379} a^{13} - \frac{1588186}{3485274978379} a^{12} - \frac{531536}{112428225109} a^{11} - \frac{369112}{112428225109} a^{10} - \frac{1219093}{112428225109} a^{9} + \frac{1368470}{3626716939} a^{8} - \frac{1523958}{3626716939} a^{7} + \frac{740853}{3626716939} a^{6} - \frac{1163850}{116990869} a^{5} + \frac{1249857}{116990869} a^{4} - \frac{746802}{116990869} a^{3} + \frac{241869}{3773899} a^{2} - \frac{622035}{3773899} a + \frac{1227419}{3773899}$, $\frac{1}{3218724633307552459} a^{26} - \frac{1850274}{3485274978379} a^{13} + \frac{1337707}{3485274978379} a^{12} - \frac{735867}{112428225109} a^{11} + \frac{804954}{112428225109} a^{10} + \frac{1148221}{112428225109} a^{9} + \frac{1532957}{3626716939} a^{8} + \frac{168357}{3626716939} a^{7} + \frac{1075994}{3626716939} a^{6} - \frac{1856130}{116990869} a^{5} - \frac{1615671}{116990869} a^{4} + \frac{919202}{116990869} a^{3} + \frac{538657}{3773899} a^{2} - \frac{1067532}{3773899} a - \frac{1314594}{3773899}$

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

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
31Data not computed