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Properties

Label 27.27.1194067772...0801.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 37^{18}$
Root discriminant $508.81$
Ramified primes $3, 37$
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![-2893634395862792231483, -6576369053834195245719, 0, 5391437692782988895139, 0, -1311430790136402704223, 0, 148527554095242519912, 0, -9478109683254965610, 0, 377261368227838926, 0, -9934810805722716, 0, 179005600103112, 0, -2241127346283, 0, 19481903595, 0, -115336881, 0, 443556, 0, -999, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 999*x^25 + 443556*x^23 - 115336881*x^21 + 19481903595*x^19 - 2241127346283*x^17 + 179005600103112*x^15 - 9934810805722716*x^13 + 377261368227838926*x^11 - 9478109683254965610*x^9 + 148527554095242519912*x^7 - 1311430790136402704223*x^5 + 5391437692782988895139*x^3 - 6576369053834195245719*x - 2893634395862792231483)
 
gp: K = bnfinit(x^27 - 999*x^25 + 443556*x^23 - 115336881*x^21 + 19481903595*x^19 - 2241127346283*x^17 + 179005600103112*x^15 - 9934810805722716*x^13 + 377261368227838926*x^11 - 9478109683254965610*x^9 + 148527554095242519912*x^7 - 1311430790136402704223*x^5 + 5391437692782988895139*x^3 - 6576369053834195245719*x - 2893634395862792231483, 1)
 

Normalized defining polynomial

\( x^{27} - 999 x^{25} + 443556 x^{23} - 115336881 x^{21} + 19481903595 x^{19} - 2241127346283 x^{17} + 179005600103112 x^{15} - 9934810805722716 x^{13} + 377261368227838926 x^{11} - 9478109683254965610 x^{9} + 148527554095242519912 x^{7} - 1311430790136402704223 x^{5} + 5391437692782988895139 x^{3} - 6576369053834195245719 x - 2893634395862792231483 \)

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:  \(11940677720052639937823094663962830546187812969131075398124270924891510801=3^{94}\cdot 37^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $508.81$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2997=3^{4}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2997}(1,·)$, $\chi_{2997}(1666,·)$, $\chi_{2997}(1987,·)$, $\chi_{2997}(454,·)$, $\chi_{2997}(2119,·)$, $\chi_{2997}(1120,·)$, $\chi_{2997}(322,·)$, $\chi_{2997}(334,·)$, $\chi_{2997}(655,·)$, $\chi_{2997}(2320,·)$, $\chi_{2997}(787,·)$, $\chi_{2997}(2452,·)$, $\chi_{2997}(667,·)$, $\chi_{2997}(2332,·)$, $\chi_{2997}(2653,·)$, $\chi_{2997}(1999,·)$, $\chi_{2997}(2785,·)$, $\chi_{2997}(1000,·)$, $\chi_{2997}(1321,·)$, $\chi_{2997}(2986,·)$, $\chi_{2997}(988,·)$, $\chi_{2997}(1453,·)$, $\chi_{2997}(1333,·)$, $\chi_{2997}(1654,·)$, $\chi_{2997}(2665,·)$, $\chi_{2997}(121,·)$, $\chi_{2997}(1786,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{37} a^{3}$, $\frac{1}{37} a^{4}$, $\frac{1}{37} a^{5}$, $\frac{1}{1369} a^{6}$, $\frac{1}{1369} a^{7}$, $\frac{1}{1369} a^{8}$, $\frac{1}{50653} a^{9}$, $\frac{1}{50653} a^{10}$, $\frac{1}{50653} a^{11}$, $\frac{1}{1874161} a^{12}$, $\frac{1}{1874161} a^{13}$, $\frac{1}{5312434923287} a^{14} - \frac{787690}{5312434923287} a^{13} - \frac{14}{143579322251} a^{12} - \frac{1098298}{143579322251} a^{11} + \frac{77}{3880522223} a^{10} - \frac{903694}{143579322251} a^{9} - \frac{210}{104878979} a^{8} - \frac{98788}{3880522223} a^{7} + \frac{294}{2834567} a^{6} - \frac{829603}{104878979} a^{5} - \frac{7252}{2834567} a^{4} - \frac{242463}{104878979} a^{3} + \frac{67081}{2834567} a^{2} + \frac{1249452}{2834567} a - \frac{101306}{2834567}$, $\frac{1}{196560092161619} a^{15} - \frac{15}{5312434923287} a^{13} - \frac{787690}{5312434923287} a^{12} + \frac{90}{143579322251} a^{11} + \frac{948579}{143579322251} a^{10} - \frac{275}{3880522223} a^{9} - \frac{16755}{3880522223} a^{8} + \frac{450}{104878979} a^{7} - \frac{1233824}{3880522223} a^{6} - \frac{378}{2834567} a^{5} + \frac{1156710}{104878979} a^{4} + \frac{5180}{2834567} a^{3} + \frac{413290}{2834567} a^{2} - \frac{20535}{2834567} a + \frac{410267}{2834567}$, $\frac{1}{196560092161619} a^{16} - \frac{1264772}{5312434923287} a^{13} - \frac{120}{143579322251} a^{12} - \frac{1353056}{143579322251} a^{11} + \frac{880}{3880522223} a^{10} - \frac{2510}{143579322251} a^{9} - \frac{2700}{104878979} a^{8} + \frac{118923}{3880522223} a^{7} - \frac{149326}{3880522223} a^{6} + \frac{50933}{104878979} a^{5} - \frac{998633}{104878979} a^{4} + \frac{316517}{104878979} a^{3} + \frac{985680}{2834567} a^{2} - \frac{689922}{2834567} a + \frac{1314977}{2834567}$, $\frac{1}{196560092161619} a^{17} - \frac{136}{143579322251} a^{13} + \frac{592215}{5312434923287} a^{12} + \frac{1088}{3880522223} a^{11} + \frac{598261}{143579322251} a^{10} + \frac{549074}{143579322251} a^{9} + \frac{284272}{3880522223} a^{8} + \frac{433131}{3880522223} a^{7} + \frac{689317}{3880522223} a^{6} + \frac{683973}{104878979} a^{5} + \frac{168464}{104878979} a^{4} - \frac{327185}{104878979} a^{3} + \frac{55733}{2834567} a^{2} - \frac{717146}{2834567} a - \frac{894698}{2834567}$, $\frac{1}{7272723409979903} a^{18} + \frac{1369811}{5312434923287} a^{13} - \frac{816}{143579322251} a^{12} - \frac{805726}{143579322251} a^{11} + \frac{6732}{3880522223} a^{10} - \frac{731731}{143579322251} a^{9} - \frac{22032}{104878979} a^{8} - \frac{9802}{3880522223} a^{7} - \frac{1269271}{3880522223} a^{6} + \frac{637835}{104878979} a^{5} + \frac{501684}{104878979} a^{4} + \frac{1095569}{104878979} a^{3} + \frac{293493}{2834567} a^{2} + \frac{746590}{2834567} a + \frac{395219}{2834567}$, $\frac{1}{7272723409979903} a^{19} - \frac{969}{143579322251} a^{13} - \frac{545844}{5312434923287} a^{12} + \frac{8721}{3880522223} a^{11} - \frac{124511}{143579322251} a^{10} - \frac{1258008}{143579322251} a^{9} - \frac{377417}{3880522223} a^{8} + \frac{225784}{3880522223} a^{7} + \frac{123847}{3880522223} a^{6} - \frac{770119}{104878979} a^{5} - \frac{1205990}{104878979} a^{4} - \frac{1044491}{104878979} a^{3} + \frac{613338}{2834567} a^{2} - \frac{1143753}{2834567} a + \frac{1011114}{2834567}$, $\frac{1}{7272723409979903} a^{20} - \frac{804393}{5312434923287} a^{13} - \frac{963671}{5312434923287} a^{12} + \frac{402059}{143579322251} a^{11} - \frac{1157223}{143579322251} a^{10} - \frac{831766}{143579322251} a^{9} - \frac{564460}{3880522223} a^{8} - \frac{1348134}{3880522223} a^{7} - \frac{598772}{3880522223} a^{6} + \frac{983749}{104878979} a^{5} - \frac{744465}{104878979} a^{4} + \frac{608020}{104878979} a^{3} + \frac{198524}{2834567} a^{2} + \frac{116802}{2834567} a - \frac{1043691}{2834567}$, $\frac{1}{269090766169256411} a^{21} - \frac{5985}{143579322251} a^{13} + \frac{478825}{5312434923287} a^{12} - \frac{708695}{143579322251} a^{11} - \frac{597913}{143579322251} a^{10} + \frac{37052}{143579322251} a^{9} + \frac{119125}{3880522223} a^{8} + \frac{593334}{3880522223} a^{7} + \frac{862474}{3880522223} a^{6} - \frac{297101}{104878979} a^{5} + \frac{1016602}{104878979} a^{4} - \frac{346474}{104878979} a^{3} - \frac{668240}{2834567} a^{2} - \frac{1307441}{2834567} a + \frac{30525}{2834567}$, $\frac{1}{269090766169256411} a^{22} + \frac{1216254}{5312434923287} a^{13} + \frac{798125}{5312434923287} a^{12} + \frac{1153778}{143579322251} a^{11} - \frac{1174584}{143579322251} a^{10} + \frac{650861}{143579322251} a^{9} + \frac{547853}{3880522223} a^{8} - \frac{892647}{3880522223} a^{7} - \frac{1266947}{3880522223} a^{6} + \frac{702104}{104878979} a^{5} - \frac{1161200}{104878979} a^{4} + \frac{935302}{104878979} a^{3} + \frac{313524}{2834567} a^{2} + \frac{9228}{2834567} a - \frac{943932}{2834567}$, $\frac{1}{269090766169256411} a^{23} - \frac{712409}{5312434923287} a^{13} + \frac{916979}{5312434923287} a^{12} - \frac{1379611}{143579322251} a^{11} - \frac{615911}{143579322251} a^{10} - \frac{1325351}{143579322251} a^{9} - \frac{1045445}{3880522223} a^{8} + \frac{1241776}{3880522223} a^{7} - \frac{688933}{3880522223} a^{6} + \frac{163807}{104878979} a^{5} - \frac{1128813}{104878979} a^{4} - \frac{156690}{104878979} a^{3} - \frac{183385}{2834567} a^{2} - \frac{884102}{2834567} a + \frac{869368}{2834567}$, $\frac{1}{9956358348262487207} a^{24} + \frac{532653}{5312434923287} a^{13} - \frac{15069}{5312434923287} a^{12} - \frac{413550}{143579322251} a^{11} - \frac{262859}{143579322251} a^{10} - \frac{1316992}{143579322251} a^{9} + \frac{200063}{3880522223} a^{8} - \frac{1266833}{3880522223} a^{7} + \frac{42731}{3880522223} a^{6} + \frac{97835}{104878979} a^{5} - \frac{740690}{104878979} a^{4} + \frac{276828}{104878979} a^{3} + \frac{1232758}{2834567} a^{2} - \frac{1388394}{2834567} a - \frac{393746}{2834567}$, $\frac{1}{9956358348262487207} a^{25} + \frac{1322862}{5312434923287} a^{13} - \frac{167260}{5312434923287} a^{12} - \frac{648560}{143579322251} a^{11} + \frac{482523}{143579322251} a^{10} - \frac{610860}{143579322251} a^{9} - \frac{1020843}{3880522223} a^{8} - \frac{1134493}{3880522223} a^{7} - \frac{618686}{3880522223} a^{6} + \frac{632738}{104878979} a^{5} - \frac{676874}{104878979} a^{4} + \frac{361659}{104878979} a^{3} + \frac{267315}{2834567} a^{2} + \frac{401461}{2834567} a - \frac{707161}{2834567}$, $\frac{1}{9956358348262487207} a^{26} + \frac{1164918}{5312434923287} a^{13} + \frac{791685}{5312434923287} a^{12} + \frac{171611}{143579322251} a^{11} + \frac{529412}{143579322251} a^{10} - \frac{94440}{143579322251} a^{9} - \frac{636695}{3880522223} a^{8} + \frac{230169}{3880522223} a^{7} + \frac{1026850}{3880522223} a^{6} - \frac{1194777}{104878979} a^{5} + \frac{166939}{104878979} a^{4} + \frac{1047175}{104878979} a^{3} + \frac{450141}{2834567} a^{2} - \frac{253683}{2834567} a + \frac{1199146}{2834567}$

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

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
$37$37.9.6.2$x^{9} - 1369 x^{3} + 101306$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
37.9.6.2$x^{9} - 1369 x^{3} + 101306$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
37.9.6.2$x^{9} - 1369 x^{3} + 101306$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$