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Properties

Label 27.27.1194067772...0801.2
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![-894239309602828075753, -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 - 894239309602828075753)
 
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 - 894239309602828075753, 1)
 

Normalized defining polynomial

\( x^{27} - 999 x^{25} + 443556 x^{23} - 115336881 x^{21} + 19481903595 x^{19} - 2241127346283 x^{17} + 179005600103112 x^{15} - 9934810805722716 x^{13} + 377261368227838926 x^{11} - 9478109683254965610 x^{9} + 148527554095242519912 x^{7} - 1311430790136402704223 x^{5} + 5391437692782988895139 x^{3} - 6576369053834195245719 x - 894239309602828075753 \)

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$$\chi_{2997}(1,·)$, $\chi_{2997}(898,·)$, $\chi_{2997}(2563,·)$, $\chi_{2997}(1543,·)$, $\chi_{2997}(1897,·)$, $\chi_{2997}(1666,·)$, $\chi_{2997}(334,·)$, $\chi_{2997}(1999,·)$, $\chi_{2997}(2896,·)$, $\chi_{2997}(211,·)$, $\chi_{2997}(1876,·)$, $\chi_{2997}(2542,·)$, $\chi_{2997}(667,·)$, $\chi_{2997}(2332,·)$, $\chi_{2997}(544,·)$, $\chi_{2997}(2209,·)$, $\chi_{2997}(232,·)$, $\chi_{2997}(1564,·)$, $\chi_{2997}(877,·)$, $\chi_{2997}(1231,·)$, $\chi_{2997}(1000,·)$, $\chi_{2997}(565,·)$, $\chi_{2997}(2230,·)$, $\chi_{2997}(2665,·)$, $\chi_{2997}(1210,·)$, $\chi_{2997}(2875,·)$, $\chi_{2997}(1333,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{37} a^{3}$, $\frac{1}{37} a^{4}$, $\frac{1}{37} a^{5}$, $\frac{1}{1369} a^{6}$, $\frac{1}{1369} a^{7}$, $\frac{1}{1369} a^{8}$, $\frac{1}{50653} a^{9}$, $\frac{1}{50653} a^{10}$, $\frac{1}{50653} a^{11}$, $\frac{1}{1874161} a^{12}$, $\frac{1}{1874161} a^{13}$, $\frac{1}{23520576239603} a^{14} - \frac{3487439}{23520576239603} a^{13} - \frac{14}{635691249719} a^{12} - \frac{4862985}{635691249719} a^{11} + \frac{77}{17180844587} a^{10} - \frac{3942231}{635691249719} a^{9} - \frac{210}{464347151} a^{8} - \frac{578584}{17180844587} a^{7} + \frac{294}{12549923} a^{6} - \frac{3508293}{464347151} a^{5} - \frac{7252}{12549923} a^{4} - \frac{4121156}{464347151} a^{3} + \frac{67081}{12549923} a^{2} + \frac{5967275}{12549923} a - \frac{101306}{12549923}$, $\frac{1}{870261320865311} a^{15} - \frac{15}{23520576239603} a^{13} - \frac{3487439}{23520576239603} a^{12} + \frac{90}{635691249719} a^{11} + \frac{4199499}{635691249719} a^{10} - \frac{275}{17180844587} a^{9} - \frac{72861}{17180844587} a^{8} + \frac{450}{464347151} a^{7} - \frac{5564080}{17180844587} a^{6} - \frac{378}{12549923} a^{5} + \frac{5216325}{464347151} a^{4} + \frac{5180}{12549923} a^{3} + \frac{1797238}{12549923} a^{2} - \frac{20535}{12549923} a + \frac{1883421}{12549923}$, $\frac{1}{870261320865311} a^{16} - \frac{5599332}{23520576239603} a^{13} - \frac{120}{635691249719} a^{12} - \frac{5995661}{635691249719} a^{11} + \frac{880}{17180844587} a^{10} + \frac{920293}{635691249719} a^{9} - \frac{2700}{464347151} a^{8} - \frac{1692917}{17180844587} a^{7} + \frac{4032}{12549923} a^{6} + \frac{2791622}{464347151} a^{5} - \frac{103600}{12549923} a^{4} + \frac{4680466}{464347151} a^{3} + \frac{985680}{12549923} a^{2} + \frac{3543085}{12549923} a - \frac{1519590}{12549923}$, $\frac{1}{870261320865311} a^{17} - \frac{136}{635691249719} a^{13} + \frac{2637394}{23520576239603} a^{12} + \frac{1088}{17180844587} a^{11} + \frac{2465028}{635691249719} a^{10} - \frac{3740}{464347151} a^{9} + \frac{2080484}{17180844587} a^{8} - \frac{3613091}{17180844587} a^{7} + \frac{3207257}{17180844587} a^{6} + \frac{4730195}{464347151} a^{5} - \frac{3897234}{464347151} a^{4} - \frac{2728379}{464347151} a^{3} + \frac{5687510}{12549923} a^{2} + \frac{494509}{12549923} a - \frac{1957915}{12549923}$, $\frac{1}{32199668872016507} a^{18} + \frac{6068522}{23520576239603} a^{13} - \frac{816}{635691249719} a^{12} - \frac{3615535}{635691249719} a^{11} + \frac{6732}{17180844587} a^{10} + \frac{5583757}{635691249719} a^{9} - \frac{22032}{464347151} a^{8} + \frac{4160913}{17180844587} a^{7} - \frac{3281324}{17180844587} a^{6} + \frac{1699018}{464347151} a^{5} + \frac{4136649}{464347151} a^{4} - \frac{2593094}{464347151} a^{3} - \frac{3752729}{12549923} a^{2} + \frac{839295}{12549923} a - \frac{1227693}{12549923}$, $\frac{1}{32199668872016507} a^{19} - \frac{969}{635691249719} a^{13} - \frac{2261919}{23520576239603} a^{12} + \frac{8721}{17180844587} a^{11} - \frac{22350}{5941039717} a^{10} - \frac{6126744}{635691249719} a^{9} - \frac{6033781}{17180844587} a^{8} + \frac{4294122}{17180844587} a^{7} - \frac{3917535}{17180844587} a^{6} + \frac{3665244}{464347151} a^{5} + \frac{94630}{464347151} a^{4} + \frac{1644211}{464347151} a^{3} + \frac{167364}{12549923} a^{2} - \frac{1577126}{12549923} a - \frac{5388269}{12549923}$, $\frac{1}{32199668872016507} a^{20} - \frac{2529537}{23520576239603} a^{13} + \frac{5917118}{23520576239603} a^{12} + \frac{1087584}{635691249719} a^{11} - \frac{4380931}{635691249719} a^{10} - \frac{926500}{635691249719} a^{9} + \frac{1814618}{17180844587} a^{8} - \frac{2866968}{17180844587} a^{7} - \frac{4516017}{17180844587} a^{6} + \frac{5143930}{464347151} a^{5} - \frac{5335143}{464347151} a^{4} + \frac{629879}{464347151} a^{3} - \frac{6107249}{12549923} a^{2} + \frac{784925}{12549923} a - \frac{5196271}{12549923}$, $\frac{1}{1191387748264610759} a^{21} - \frac{5985}{635691249719} a^{13} + \frac{3323835}{23520576239603} a^{12} + \frac{57456}{17180844587} a^{11} - \frac{4353524}{635691249719} a^{10} + \frac{771102}{635691249719} a^{9} + \frac{24441}{464347151} a^{8} - \frac{2908935}{17180844587} a^{7} - \frac{533723}{17180844587} a^{6} - \frac{1641452}{464347151} a^{5} + \frac{1427816}{464347151} a^{4} + \frac{2767157}{464347151} a^{3} + \frac{1618842}{12549923} a^{2} + \frac{4551790}{12549923} a + \frac{1685190}{12549923}$, $\frac{1}{1191387748264610759} a^{22} - \frac{543792}{23520576239603} a^{13} + \frac{1598523}{23520576239603} a^{12} - \frac{4274065}{635691249719} a^{11} + \frac{4171757}{635691249719} a^{10} + \frac{6209891}{635691249719} a^{9} - \frac{4197134}{17180844587} a^{8} - \frac{2903696}{17180844587} a^{7} + \frac{975015}{17180844587} a^{6} - \frac{2082177}{464347151} a^{5} - \frac{4905541}{464347151} a^{4} + \frac{5607756}{464347151} a^{3} + \frac{195003}{12549923} a^{2} + \frac{5855126}{12549923} a + \frac{5555154}{12549923}$, $\frac{1}{1191387748264610759} a^{23} + \frac{4134211}{23520576239603} a^{13} - \frac{577356}{23520576239603} a^{12} - \frac{99793}{17180844587} a^{11} - \frac{717153}{635691249719} a^{10} - \frac{5627820}{635691249719} a^{9} + \frac{1156515}{17180844587} a^{8} - \frac{1805903}{17180844587} a^{7} - \frac{3731219}{17180844587} a^{6} + \frac{2522171}{464347151} a^{5} - \frac{1432054}{464347151} a^{4} - \frac{4698331}{464347151} a^{3} + \frac{1340117}{12549923} a^{2} + \frac{3671382}{12549923} a + \frac{4769618}{12549923}$, $\frac{1}{44081346685790598083} a^{24} + \frac{3181296}{23520576239603} a^{13} + \frac{3986921}{23520576239603} a^{12} - \frac{2911755}{635691249719} a^{11} + \frac{5437338}{635691249719} a^{10} - \frac{3401264}{635691249719} a^{9} + \frac{5921873}{17180844587} a^{8} - \frac{1847772}{17180844587} a^{7} - \frac{3050978}{17180844587} a^{6} - \frac{996774}{464347151} a^{5} - \frac{3986728}{464347151} a^{4} - \frac{6035437}{464347151} a^{3} - \frac{4617221}{12549923} a^{2} - \frac{1579393}{12549923} a - \frac{560828}{12549923}$, $\frac{1}{44081346685790598083} a^{25} - \frac{4001363}{23520576239603} a^{13} - \frac{3464136}{23520576239603} a^{12} - \frac{3664277}{635691249719} a^{11} - \frac{158626}{17180844587} a^{10} - \frac{4680220}{635691249719} a^{9} + \frac{6023761}{17180844587} a^{8} - \frac{3092832}{17180844587} a^{7} + \frac{4661196}{17180844587} a^{6} + \frac{4428717}{464347151} a^{5} + \frac{3919776}{464347151} a^{4} + \frac{6050050}{464347151} a^{3} + \frac{5344246}{12549923} a^{2} - \frac{2523432}{12549923} a + \frac{2349936}{12549923}$, $\frac{1}{44081346685790598083} a^{26} - \frac{2461333}{23520576239603} a^{13} + \frac{502165}{23520576239603} a^{12} + \frac{6194245}{635691249719} a^{11} - \frac{127117}{635691249719} a^{10} - \frac{863458}{635691249719} a^{9} + \frac{5025852}{17180844587} a^{8} + \frac{1996783}{17180844587} a^{7} - \frac{666873}{17180844587} a^{6} + \frac{4936604}{464347151} a^{5} + \frac{2787011}{464347151} a^{4} + \frac{3226862}{464347151} a^{3} - \frac{4845153}{12549923} a^{2} + \frac{3244421}{12549923} a + \frac{432822}{12549923}$

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 $27$ $27$ $27$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{27}$ $27$

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
$37$37.9.6.3$x^{9} - 74 x^{6} + 1369 x^{3} - 202612$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
37.9.6.3$x^{9} - 74 x^{6} + 1369 x^{3} - 202612$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
37.9.6.3$x^{9} - 74 x^{6} + 1369 x^{3} - 202612$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$