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Properties

Label 27.27.9667962870...5089.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 61^{18}$
Root discriminant $710.08$
Ramified primes $3, 61$
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![-2525651376607972493539559, -4371712760668124402754687, 0, 2173911809949723063664899, 0, -320741086713893566770231, 0, 22033735855443117779688, 0, -852854074550393903130, 0, 20590515659785963086, 0, -328894323275269692, 0, 3594473478418248, 0, -27296507852019, 0, 143927517195, 0, -516835737, 0, 1205604, 0, -1647, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 1647*x^25 + 1205604*x^23 - 516835737*x^21 + 143927517195*x^19 - 27296507852019*x^17 + 3594473478418248*x^15 - 328894323275269692*x^13 + 20590515659785963086*x^11 - 852854074550393903130*x^9 + 22033735855443117779688*x^7 - 320741086713893566770231*x^5 + 2173911809949723063664899*x^3 - 4371712760668124402754687*x - 2525651376607972493539559)
 
gp: K = bnfinit(x^27 - 1647*x^25 + 1205604*x^23 - 516835737*x^21 + 143927517195*x^19 - 27296507852019*x^17 + 3594473478418248*x^15 - 328894323275269692*x^13 + 20590515659785963086*x^11 - 852854074550393903130*x^9 + 22033735855443117779688*x^7 - 320741086713893566770231*x^5 + 2173911809949723063664899*x^3 - 4371712760668124402754687*x - 2525651376607972493539559, 1)
 

Normalized defining polynomial

\( x^{27} - 1647 x^{25} + 1205604 x^{23} - 516835737 x^{21} + 143927517195 x^{19} - 27296507852019 x^{17} + 3594473478418248 x^{15} - 328894323275269692 x^{13} + 20590515659785963086 x^{11} - 852854074550393903130 x^{9} + 22033735855443117779688 x^{7} - 320741086713893566770231 x^{5} + 2173911809949723063664899 x^{3} - 4371712760668124402754687 x - 2525651376607972493539559 \)

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:  \(96679628704383224900411211844182831619287468040811424837360737394634888225089=3^{94}\cdot 61^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $710.08$
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:  $3, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4941=3^{4}\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{4941}(1,·)$, $\chi_{4941}(3844,·)$, $\chi_{4941}(2758,·)$, $\chi_{4941}(3463,·)$, $\chi_{4941}(1099,·)$, $\chi_{4941}(13,·)$, $\chi_{4941}(718,·)$, $\chi_{4941}(3856,·)$, $\chi_{4941}(4561,·)$, $\chi_{4941}(2197,·)$, $\chi_{4941}(1111,·)$, $\chi_{4941}(1816,·)$, $\chi_{4941}(3295,·)$, $\chi_{4941}(2209,·)$, $\chi_{4941}(2914,·)$, $\chi_{4941}(550,·)$, $\chi_{4941}(4393,·)$, $\chi_{4941}(3307,·)$, $\chi_{4941}(4012,·)$, $\chi_{4941}(1648,·)$, $\chi_{4941}(562,·)$, $\chi_{4941}(1267,·)$, $\chi_{4941}(4405,·)$, $\chi_{4941}(169,·)$, $\chi_{4941}(2746,·)$, $\chi_{4941}(1660,·)$, $\chi_{4941}(2365,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{61} a^{3}$, $\frac{1}{61} a^{4}$, $\frac{1}{61} a^{5}$, $\frac{1}{3721} a^{6}$, $\frac{1}{3721} a^{7}$, $\frac{1}{3721} a^{8}$, $\frac{1}{226981} a^{9}$, $\frac{1}{226981} a^{10}$, $\frac{1}{226981} a^{11}$, $\frac{1}{13845841} a^{12}$, $\frac{1}{13845841} a^{13}$, $\frac{1}{91522933581899} a^{14} + \frac{2962113}{91522933581899} a^{13} - \frac{14}{1500375960359} a^{12} + \frac{1153365}{1500375960359} a^{11} + \frac{77}{24596327219} a^{10} - \frac{1438958}{1500375960359} a^{9} - \frac{210}{403218479} a^{8} - \frac{1834612}{24596327219} a^{7} + \frac{294}{6610139} a^{6} - \frac{62999}{403218479} a^{5} - \frac{11956}{6610139} a^{4} - \frac{1383600}{403218479} a^{3} + \frac{182329}{6610139} a^{2} - \frac{1690954}{6610139} a - \frac{453962}{6610139}$, $\frac{1}{5582898948495839} a^{15} - \frac{15}{91522933581899} a^{13} + \frac{2962113}{91522933581899} a^{12} + \frac{90}{1500375960359} a^{11} - \frac{2494661}{1500375960359} a^{10} - \frac{275}{24596327219} a^{9} + \frac{1310766}{24596327219} a^{8} + \frac{450}{403218479} a^{7} + \frac{3089602}{24596327219} a^{6} - \frac{378}{6610139} a^{5} + \frac{1234835}{403218479} a^{4} + \frac{8540}{6610139} a^{3} - \frac{423372}{6610139} a^{2} - \frac{55815}{6610139} a - \frac{768619}{6610139}$, $\frac{1}{5582898948495839} a^{16} + \frac{1122835}{91522933581899} a^{13} - \frac{120}{1500375960359} a^{12} + \frac{1585536}{1500375960359} a^{11} + \frac{880}{24596327219} a^{10} - \frac{1118895}{1500375960359} a^{9} - \frac{2700}{403218479} a^{8} + \frac{2010978}{24596327219} a^{7} + \frac{1782794}{24596327219} a^{6} + \frac{289850}{403218479} a^{5} + \frac{2801478}{403218479} a^{4} - \frac{308719}{403218479} a^{3} + \frac{2679120}{6610139} a^{2} + \frac{307627}{6610139} a - \frac{199291}{6610139}$, $\frac{1}{5582898948495839} a^{17} - \frac{136}{1500375960359} a^{13} - \frac{2003454}{91522933581899} a^{12} + \frac{1088}{24596327219} a^{11} - \frac{183968}{1500375960359} a^{10} - \frac{696262}{1500375960359} a^{9} + \frac{1864864}{24596327219} a^{8} - \frac{2149868}{24596327219} a^{7} + \frac{1074374}{24596327219} a^{6} - \frac{1423935}{403218479} a^{5} - \frac{2458013}{403218479} a^{4} - \frac{1849769}{403218479} a^{3} - \frac{2460119}{6610139} a^{2} - \frac{1140366}{6610139} a - \frac{3226437}{6610139}$, $\frac{1}{340556835858246179} a^{18} - \frac{2896302}{91522933581899} a^{13} - \frac{816}{1500375960359} a^{12} - \frac{1571986}{1500375960359} a^{11} + \frac{6732}{24596327219} a^{10} - \frac{2139393}{1500375960359} a^{9} - \frac{22032}{403218479} a^{8} - \frac{471596}{24596327219} a^{7} + \frac{1933471}{24596327219} a^{6} + \frac{819416}{403218479} a^{5} + \frac{1451866}{403218479} a^{4} + \frac{1064312}{403218479} a^{3} - \frac{2529410}{6610139} a^{2} + \frac{2307495}{6610139} a - \frac{2247581}{6610139}$, $\frac{1}{340556835858246179} a^{19} - \frac{969}{1500375960359} a^{13} + \frac{2011017}{91522933581899} a^{12} + \frac{8721}{24596327219} a^{11} - \frac{17523}{14022205237} a^{10} - \frac{3915}{1500375960359} a^{9} + \frac{609991}{24596327219} a^{8} - \frac{1795647}{24596327219} a^{7} - \frac{1400292}{24596327219} a^{6} - \frac{3011015}{403218479} a^{5} - \frac{2136697}{403218479} a^{4} - \frac{1650792}{403218479} a^{3} - \frac{1849857}{6610139} a^{2} + \frac{2386801}{6610139} a - \frac{1520312}{6610139}$, $\frac{1}{340556835858246179} a^{20} + \frac{186502}{91522933581899} a^{13} + \frac{1802172}{91522933581899} a^{12} + \frac{2013317}{1500375960359} a^{11} + \frac{5220}{1500375960359} a^{10} + \frac{1448847}{1500375960359} a^{9} + \frac{1184048}{24596327219} a^{8} + \frac{2459434}{24596327219} a^{7} - \frac{1868694}{24596327219} a^{6} + \frac{2173808}{403218479} a^{5} + \frac{536322}{403218479} a^{4} - \frac{3041606}{403218479} a^{3} - \frac{1465047}{6610139} a^{2} - \frac{208479}{6610139} a - \frac{2685657}{6610139}$, $\frac{1}{20773966987353016919} a^{21} - \frac{5985}{1500375960359} a^{13} - \frac{32554}{1500375960359} a^{12} - \frac{3105323}{1500375960359} a^{11} - \frac{141358}{1500375960359} a^{10} + \frac{3083786}{1500375960359} a^{9} + \frac{520171}{24596327219} a^{8} + \frac{156291}{24596327219} a^{7} + \frac{2177274}{24596327219} a^{6} - \frac{1672187}{403218479} a^{5} + \frac{1501029}{403218479} a^{4} + \frac{228775}{403218479} a^{3} - \frac{38961}{6610139} a^{2} - \frac{2917673}{6610139} a - \frac{181306}{6610139}$, $\frac{1}{20773966987353016919} a^{22} + \frac{2298411}{91522933581899} a^{13} + \frac{1163271}{91522933581899} a^{12} - \frac{2954911}{1500375960359} a^{11} - \frac{748109}{1500375960359} a^{10} - \frac{2504669}{1500375960359} a^{9} - \frac{3214286}{24596327219} a^{8} - \frac{1590293}{24596327219} a^{7} - \frac{562051}{24596327219} a^{6} - \frac{1815305}{403218479} a^{5} + \frac{905974}{403218479} a^{4} - \frac{380519}{403218479} a^{3} - \frac{1434438}{6610139} a^{2} - \frac{1410769}{6610139} a + \frac{1298377}{6610139}$, $\frac{1}{20773966987353016919} a^{23} + \frac{384712}{91522933581899} a^{13} - \frac{2144107}{91522933581899} a^{12} + \frac{2763019}{1500375960359} a^{11} + \frac{2825990}{1500375960359} a^{10} + \frac{2181202}{1500375960359} a^{9} - \frac{504489}{24596327219} a^{8} - \frac{1760426}{24596327219} a^{7} - \frac{2513041}{24596327219} a^{6} - \frac{3204371}{403218479} a^{5} + \frac{2387347}{403218479} a^{4} + \frac{1509040}{403218479} a^{3} + \frac{1202334}{6610139} a^{2} - \frac{1974247}{6610139} a + \frac{643649}{6610139}$, $\frac{1}{1267211986228534032059} a^{24} - \frac{2509356}{91522933581899} a^{13} + \frac{1538848}{91522933581899} a^{12} - \frac{1056406}{1500375960359} a^{11} + \frac{2488362}{1500375960359} a^{10} - \frac{1287798}{1500375960359} a^{9} - \frac{944992}{24596327219} a^{8} - \frac{1455489}{24596327219} a^{7} - \frac{1644263}{24596327219} a^{6} - \frac{441994}{403218479} a^{5} + \frac{2885732}{403218479} a^{4} - \frac{648523}{403218479} a^{3} - \frac{2589783}{6610139} a^{2} + \frac{2720866}{6610139} a + \frac{836517}{6610139}$, $\frac{1}{1267211986228534032059} a^{25} + \frac{2024800}{91522933581899} a^{13} + \frac{355636}{91522933581899} a^{12} + \frac{2781125}{1500375960359} a^{11} - \frac{720503}{1500375960359} a^{10} - \frac{1514169}{1500375960359} a^{9} - \frac{1199892}{24596327219} a^{8} + \frac{1133805}{24596327219} a^{7} - \frac{1196617}{24596327219} a^{6} - \frac{2558727}{403218479} a^{5} + \frac{2005216}{403218479} a^{4} - \frac{1835972}{403218479} a^{3} - \frac{2900173}{6610139} a^{2} + \frac{2138329}{6610139} a - \frac{574046}{6610139}$, $\frac{1}{1267211986228534032059} a^{26} - \frac{2865670}{91522933581899} a^{13} + \frac{1717932}{91522933581899} a^{12} + \frac{1495641}{1500375960359} a^{11} - \frac{9748}{1500375960359} a^{10} - \frac{2171625}{1500375960359} a^{9} + \frac{636369}{24596327219} a^{8} + \frac{1536736}{24596327219} a^{7} + \frac{1415106}{24596327219} a^{6} - \frac{82006}{403218479} a^{5} - \frac{1072050}{403218479} a^{4} - \frac{877919}{403218479} a^{3} - \frac{1357721}{6610139} a^{2} + \frac{2607602}{6610139} a + \frac{2768816}{6610139}$

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

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
61Data not computed