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Properties

Label 27.27.7056772194...3489.1
Degree $27$
Signature $[27, 0]$
Discriminant $7^{18}\cdot 37^{24}$
Root discriminant $90.65$
Ramified primes $7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_9$ (as 27T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-223, -1124, 61524, -251138, -862152, 4979339, -453885, -17745073, 9025037, 28010042, -18615256, -24076806, 18475792, 12070299, -10646520, -3504440, 3777786, 521672, -834075, -14683, 112446, -6757, -8844, 982, 366, -53, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 - 53*x^25 + 366*x^24 + 982*x^23 - 8844*x^22 - 6757*x^21 + 112446*x^20 - 14683*x^19 - 834075*x^18 + 521672*x^17 + 3777786*x^16 - 3504440*x^15 - 10646520*x^14 + 12070299*x^13 + 18475792*x^12 - 24076806*x^11 - 18615256*x^10 + 28010042*x^9 + 9025037*x^8 - 17745073*x^7 - 453885*x^6 + 4979339*x^5 - 862152*x^4 - 251138*x^3 + 61524*x^2 - 1124*x - 223)
 
gp: K = bnfinit(x^27 - 6*x^26 - 53*x^25 + 366*x^24 + 982*x^23 - 8844*x^22 - 6757*x^21 + 112446*x^20 - 14683*x^19 - 834075*x^18 + 521672*x^17 + 3777786*x^16 - 3504440*x^15 - 10646520*x^14 + 12070299*x^13 + 18475792*x^12 - 24076806*x^11 - 18615256*x^10 + 28010042*x^9 + 9025037*x^8 - 17745073*x^7 - 453885*x^6 + 4979339*x^5 - 862152*x^4 - 251138*x^3 + 61524*x^2 - 1124*x - 223, 1)
 

Normalized defining polynomial

\( x^{27} - 6 x^{26} - 53 x^{25} + 366 x^{24} + 982 x^{23} - 8844 x^{22} - 6757 x^{21} + 112446 x^{20} - 14683 x^{19} - 834075 x^{18} + 521672 x^{17} + 3777786 x^{16} - 3504440 x^{15} - 10646520 x^{14} + 12070299 x^{13} + 18475792 x^{12} - 24076806 x^{11} - 18615256 x^{10} + 28010042 x^{9} + 9025037 x^{8} - 17745073 x^{7} - 453885 x^{6} + 4979339 x^{5} - 862152 x^{4} - 251138 x^{3} + 61524 x^{2} - 1124 x - 223 \)

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:  \(70567721948812723604880306782225092911730957083793489=7^{18}\cdot 37^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.65$
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:  $7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(259=7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{259}(256,·)$, $\chi_{259}(1,·)$, $\chi_{259}(197,·)$, $\chi_{259}(71,·)$, $\chi_{259}(9,·)$, $\chi_{259}(144,·)$, $\chi_{259}(81,·)$, $\chi_{259}(211,·)$, $\chi_{259}(149,·)$, $\chi_{259}(86,·)$, $\chi_{259}(218,·)$, $\chi_{259}(155,·)$, $\chi_{259}(158,·)$, $\chi_{259}(16,·)$, $\chi_{259}(219,·)$, $\chi_{259}(100,·)$, $\chi_{259}(232,·)$, $\chi_{259}(107,·)$, $\chi_{259}(44,·)$, $\chi_{259}(46,·)$, $\chi_{259}(53,·)$, $\chi_{259}(137,·)$, $\chi_{259}(120,·)$, $\chi_{259}(121,·)$, $\chi_{259}(186,·)$, $\chi_{259}(123,·)$, $\chi_{259}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{43} a^{21} - \frac{19}{43} a^{20} + \frac{11}{43} a^{19} - \frac{9}{43} a^{18} - \frac{3}{43} a^{17} + \frac{2}{43} a^{16} - \frac{2}{43} a^{15} + \frac{17}{43} a^{13} + \frac{21}{43} a^{12} + \frac{12}{43} a^{11} - \frac{15}{43} a^{10} + \frac{21}{43} a^{9} + \frac{20}{43} a^{8} - \frac{11}{43} a^{7} - \frac{9}{43} a^{6} + \frac{3}{43} a^{5} + \frac{1}{43} a^{4} + \frac{14}{43} a^{3} + \frac{5}{43} a^{2} + \frac{20}{43} a - \frac{18}{43}$, $\frac{1}{43} a^{22} - \frac{6}{43} a^{20} - \frac{15}{43} a^{19} - \frac{2}{43} a^{18} - \frac{12}{43} a^{17} - \frac{7}{43} a^{16} + \frac{5}{43} a^{15} + \frac{17}{43} a^{14} - \frac{19}{43} a^{12} - \frac{2}{43} a^{11} - \frac{6}{43} a^{10} - \frac{11}{43} a^{9} - \frac{18}{43} a^{8} - \frac{3}{43} a^{7} + \frac{4}{43} a^{6} + \frac{15}{43} a^{5} - \frac{10}{43} a^{4} + \frac{13}{43} a^{3} - \frac{14}{43} a^{2} + \frac{18}{43} a + \frac{2}{43}$, $\frac{1}{43} a^{23} + \frac{21}{43} a^{19} + \frac{20}{43} a^{18} + \frac{18}{43} a^{17} + \frac{17}{43} a^{16} + \frac{5}{43} a^{15} - \frac{3}{43} a^{13} - \frac{5}{43} a^{12} - \frac{20}{43} a^{11} - \frac{15}{43} a^{10} - \frac{21}{43} a^{9} - \frac{12}{43} a^{8} - \frac{19}{43} a^{7} + \frac{4}{43} a^{6} + \frac{8}{43} a^{5} + \frac{19}{43} a^{4} - \frac{16}{43} a^{3} + \frac{5}{43} a^{2} - \frac{7}{43} a + \frac{21}{43}$, $\frac{1}{1333} a^{24} - \frac{10}{1333} a^{23} + \frac{15}{1333} a^{22} + \frac{15}{1333} a^{21} + \frac{119}{1333} a^{20} + \frac{266}{1333} a^{19} - \frac{562}{1333} a^{18} + \frac{429}{1333} a^{17} + \frac{20}{43} a^{16} + \frac{597}{1333} a^{15} - \frac{221}{1333} a^{14} + \frac{366}{1333} a^{13} + \frac{275}{1333} a^{12} - \frac{267}{1333} a^{11} - \frac{358}{1333} a^{10} - \frac{39}{1333} a^{9} - \frac{643}{1333} a^{8} + \frac{113}{1333} a^{7} - \frac{365}{1333} a^{6} - \frac{479}{1333} a^{5} - \frac{83}{1333} a^{4} + \frac{97}{1333} a^{3} - \frac{579}{1333} a^{2} - \frac{328}{1333} a - \frac{536}{1333}$, $\frac{1}{1734233} a^{25} + \frac{367}{1734233} a^{24} + \frac{11776}{1734233} a^{23} - \frac{1739}{1734233} a^{22} + \frac{8626}{1734233} a^{21} + \frac{684566}{1734233} a^{20} - \frac{455366}{1734233} a^{19} + \frac{648185}{1734233} a^{18} + \frac{663561}{1734233} a^{17} + \frac{262671}{1734233} a^{16} + \frac{490363}{1734233} a^{15} - \frac{731440}{1734233} a^{14} + \frac{328101}{1734233} a^{13} + \frac{253076}{1734233} a^{12} - \frac{28012}{1734233} a^{11} + \frac{10286}{55943} a^{10} + \frac{844966}{1734233} a^{9} - \frac{824788}{1734233} a^{8} + \frac{251207}{1734233} a^{7} + \frac{828496}{1734233} a^{6} - \frac{592222}{1734233} a^{5} + \frac{684751}{1734233} a^{4} - \frac{652799}{1734233} a^{3} + \frac{10355}{1734233} a^{2} - \frac{49296}{1734233} a + \frac{667106}{1734233}$, $\frac{1}{1618365725208043775989176733101778566418398807413349157} a^{26} - \frac{294695514694584304696443758540314305533386838483}{1618365725208043775989176733101778566418398807413349157} a^{25} - \frac{247156418321751726838462204570797046256519835897170}{1618365725208043775989176733101778566418398807413349157} a^{24} + \frac{959262445536657235087035095327063256009542593756543}{1618365725208043775989176733101778566418398807413349157} a^{23} + \frac{2608214348551761633383345422759391213137588034538331}{1618365725208043775989176733101778566418398807413349157} a^{22} + \frac{6904640930052634662586683256521955508933107109253102}{1618365725208043775989176733101778566418398807413349157} a^{21} + \frac{540092325649894040489602205476198897569104598080128817}{1618365725208043775989176733101778566418398807413349157} a^{20} - \frac{669041488773471483557151725106562714018792715739740916}{1618365725208043775989176733101778566418398807413349157} a^{19} - \frac{733604479243151935949891616890747122799841580042817339}{1618365725208043775989176733101778566418398807413349157} a^{18} - \frac{709959713997777210738146593321165053464037477161358337}{1618365725208043775989176733101778566418398807413349157} a^{17} - \frac{104275676474123930474211173802831678889465150456977961}{1618365725208043775989176733101778566418398807413349157} a^{16} + \frac{84641654071632368355137230236396051131911414613325838}{1618365725208043775989176733101778566418398807413349157} a^{15} + \frac{478425161048917675358012550821155093909944548705223049}{1618365725208043775989176733101778566418398807413349157} a^{14} - \frac{23969249469038306155181588519764658900795769283446402}{1618365725208043775989176733101778566418398807413349157} a^{13} + \frac{559572642037783037466354477793169621699084430720255134}{1618365725208043775989176733101778566418398807413349157} a^{12} - \frac{787900339423935804258589846010087147793965431338989689}{1618365725208043775989176733101778566418398807413349157} a^{11} + \frac{310871121898926709675756675338663965026329200093142721}{1618365725208043775989176733101778566418398807413349157} a^{10} - \frac{765351462785027497465392849195689514288014504546460257}{1618365725208043775989176733101778566418398807413349157} a^{9} + \frac{705490283788395684743696427005297153838883598637870814}{1618365725208043775989176733101778566418398807413349157} a^{8} + \frac{576564060416948628774143822006241188200889432342692210}{1618365725208043775989176733101778566418398807413349157} a^{7} + \frac{25838788930471650371983349008921837102279176879357620}{52205345974453025031908926874250921497367703464946747} a^{6} + \frac{91307478637306507594106538213374888699145756660915621}{1618365725208043775989176733101778566418398807413349157} a^{5} - \frac{808813691192082343194041380024384833031595717073175491}{1618365725208043775989176733101778566418398807413349157} a^{4} - \frac{752656935165243332548947590227246396781909516207578731}{1618365725208043775989176733101778566418398807413349157} a^{3} - \frac{660428629467220522645591435716142363091568751748732911}{1618365725208043775989176733101778566418398807413349157} a^{2} - \frac{396115762852863692577411621279144280397494670024723660}{1618365725208043775989176733101778566418398807413349157} a + \frac{684627255431481189895760531710009110083835872957596}{7257245404520375677081510013909320925643043979432059}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 710242126153974400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_9$ (as 27T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3\times C_9$
Character table for $C_3\times C_9$ is not computed

Intermediate fields

3.3.67081.2, 3.3.1369.1, 3.3.67081.1, \(\Q(\zeta_{7})^+\), 9.9.301855146292441.1, 9.9.413239695274351729.1, 9.9.3512479453921.1, 9.9.413239695274351729.2

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{27}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
7Data not computed
37Data not computed