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Properties

Label 27.27.1785789002...6481.2
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 43^{18}$
Root discriminant $562.43$
Ramified primes $3, 43$
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![-13967167800173896001791, -46393131355621851201561, 0, 32726937622958050072419, 0, -6849824153642382573297, 0, 667535465692723883112, 0, -36654079188553959990, 0, 1255382839016013006, 0, -28446361289569764, 0, 441028857202632, 0, -4751165869317, 0, 35538436395, 0, -181037439, 0, 599076, 0, -1161, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 1161*x^25 + 599076*x^23 - 181037439*x^21 + 35538436395*x^19 - 4751165869317*x^17 + 441028857202632*x^15 - 28446361289569764*x^13 + 1255382839016013006*x^11 - 36654079188553959990*x^9 + 667535465692723883112*x^7 - 6849824153642382573297*x^5 + 32726937622958050072419*x^3 - 46393131355621851201561*x - 13967167800173896001791)
 
gp: K = bnfinit(x^27 - 1161*x^25 + 599076*x^23 - 181037439*x^21 + 35538436395*x^19 - 4751165869317*x^17 + 441028857202632*x^15 - 28446361289569764*x^13 + 1255382839016013006*x^11 - 36654079188553959990*x^9 + 667535465692723883112*x^7 - 6849824153642382573297*x^5 + 32726937622958050072419*x^3 - 46393131355621851201561*x - 13967167800173896001791, 1)
 

Normalized defining polynomial

\( x^{27} - 1161 x^{25} + 599076 x^{23} - 181037439 x^{21} + 35538436395 x^{19} - 4751165869317 x^{17} + 441028857202632 x^{15} - 28446361289569764 x^{13} + 1255382839016013006 x^{11} - 36654079188553959990 x^{9} + 667535465692723883112 x^{7} - 6849824153642382573297 x^{5} + 32726937622958050072419 x^{3} - 46393131355621851201561 x - 13967167800173896001791 \)

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:  \(178578900240970840458250689871157457776932189215069373149025720786258346481=3^{94}\cdot 43^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $562.43$
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:  $3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3483=3^{4}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{3483}(1984,·)$, $\chi_{3483}(1,·)$, $\chi_{3483}(2371,·)$, $\chi_{3483}(388,·)$, $\chi_{3483}(2758,·)$, $\chi_{3483}(775,·)$, $\chi_{3483}(3145,·)$, $\chi_{3483}(1162,·)$, $\chi_{3483}(1549,·)$, $\chi_{3483}(79,·)$, $\chi_{3483}(1936,·)$, $\chi_{3483}(466,·)$, $\chi_{3483}(2323,·)$, $\chi_{3483}(853,·)$, $\chi_{3483}(2710,·)$, $\chi_{3483}(1240,·)$, $\chi_{3483}(3097,·)$, $\chi_{3483}(1627,·)$, $\chi_{3483}(2014,·)$, $\chi_{3483}(2401,·)$, $\chi_{3483}(2788,·)$, $\chi_{3483}(3175,·)$, $\chi_{3483}(49,·)$, $\chi_{3483}(436,·)$, $\chi_{3483}(823,·)$, $\chi_{3483}(1210,·)$, $\chi_{3483}(1597,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{43} a^{3}$, $\frac{1}{43} a^{4}$, $\frac{1}{43} a^{5}$, $\frac{1}{1849} a^{6}$, $\frac{1}{1849} a^{7}$, $\frac{1}{1849} a^{8}$, $\frac{1}{79507} a^{9}$, $\frac{1}{79507} a^{10}$, $\frac{1}{79507} a^{11}$, $\frac{1}{3418801} a^{12}$, $\frac{1}{3418801} a^{13}$, $\frac{1}{69452255135999} a^{14} + \frac{9714942}{69452255135999} a^{13} - \frac{14}{1615168724093} a^{12} - \frac{4405452}{1615168724093} a^{11} + \frac{77}{37562063351} a^{10} - \frac{7623373}{1615168724093} a^{9} - \frac{210}{873536357} a^{8} + \frac{2044256}{37562063351} a^{7} + \frac{294}{20314799} a^{6} - \frac{9156565}{873536357} a^{5} - \frac{8428}{20314799} a^{4} + \frac{3875557}{873536357} a^{3} + \frac{90601}{20314799} a^{2} - \frac{553651}{20314799} a - \frac{159014}{20314799}$, $\frac{1}{2986446970847957} a^{15} - \frac{15}{69452255135999} a^{13} + \frac{9714942}{69452255135999} a^{12} + \frac{90}{1615168724093} a^{11} + \frac{5309490}{1615168724093} a^{10} - \frac{275}{37562063351} a^{9} - \frac{3577906}{37562063351} a^{8} + \frac{450}{873536357} a^{7} - \frac{2178575}{37562063351} a^{6} - \frac{378}{20314799} a^{5} + \frac{3312089}{873536357} a^{4} + \frac{6020}{20314799} a^{3} - \frac{5778956}{20314799} a^{2} - \frac{27735}{20314799} a + \frac{9290884}{20314799}$, $\frac{1}{2986446970847957} a^{16} - \frac{7079320}{69452255135999} a^{13} - \frac{120}{1615168724093} a^{12} + \frac{172107}{1615168724093} a^{11} + \frac{880}{37562063351} a^{10} - \frac{4108166}{1615168724093} a^{9} - \frac{2700}{873536357} a^{8} + \frac{8170466}{37562063351} a^{7} + \frac{4032}{20314799} a^{6} + \frac{8167207}{873536357} a^{5} - \frac{120400}{20314799} a^{4} - \frac{7528562}{873536357} a^{3} + \frac{1331280}{20314799} a^{2} + \frac{986119}{20314799} a - \frac{2385210}{20314799}$, $\frac{1}{2986446970847957} a^{17} - \frac{136}{1615168724093} a^{13} - \frac{8557048}{69452255135999} a^{12} + \frac{1088}{37562063351} a^{11} - \frac{7757692}{1615168724093} a^{10} - \frac{3740}{873536357} a^{9} - \frac{7731480}{37562063351} a^{8} - \frac{8244527}{37562063351} a^{7} - \frac{7911925}{37562063351} a^{6} + \frac{9753311}{873536357} a^{5} - \frac{5133333}{873536357} a^{4} - \frac{9825959}{873536357} a^{3} - \frac{4691388}{20314799} a^{2} + \frac{1392133}{20314799} a - \frac{7033493}{20314799}$, $\frac{1}{128417219746462151} a^{18} + \frac{3405799}{69452255135999} a^{13} - \frac{816}{1615168724093} a^{12} + \frac{8704775}{1615168724093} a^{11} + \frac{6732}{37562063351} a^{10} - \frac{8455459}{1615168724093} a^{9} - \frac{22032}{873536357} a^{8} - \frac{8462117}{37562063351} a^{7} + \frac{2424531}{37562063351} a^{6} + \frac{6073886}{873536357} a^{5} - \frac{4633922}{873536357} a^{4} - \frac{5800410}{873536357} a^{3} - \frac{8433125}{20314799} a^{2} - \frac{7901588}{20314799} a - \frac{1311105}{20314799}$, $\frac{1}{128417219746462151} a^{19} - \frac{969}{1615168724093} a^{13} + \frac{7135242}{69452255135999} a^{12} + \frac{8721}{37562063351} a^{11} + \frac{9972296}{1615168724093} a^{10} + \frac{1818924}{1615168724093} a^{9} + \frac{9611966}{37562063351} a^{8} + \frac{5926865}{37562063351} a^{7} - \frac{6164819}{37562063351} a^{6} + \frac{4823221}{873536357} a^{5} + \frac{3137543}{873536357} a^{4} - \frac{7295977}{873536357} a^{3} + \frac{5100023}{20314799} a^{2} + \frac{3067864}{20314799} a - \frac{2504355}{20314799}$, $\frac{1}{128417219746462151} a^{20} + \frac{6938682}{69452255135999} a^{13} - \frac{4845}{37562063351} a^{12} - \frac{7787223}{1615168724093} a^{11} - \frac{2425232}{1615168724093} a^{10} + \frac{6132931}{1615168724093} a^{9} - \frac{4659763}{37562063351} a^{8} - \frac{8102274}{37562063351} a^{7} + \frac{3680830}{37562063351} a^{6} + \frac{8783707}{873536357} a^{5} + \frac{6627011}{873536357} a^{4} - \frac{3665532}{873536357} a^{3} - \frac{412883}{20314799} a^{2} + \frac{6131092}{20314799} a - \frac{3011864}{20314799}$, $\frac{1}{5521940449097872493} a^{21} - \frac{5985}{1615168724093} a^{13} + \frac{8095129}{69452255135999} a^{12} + \frac{57456}{37562063351} a^{11} - \frac{9258174}{1615168724093} a^{10} + \frac{532930}{1615168724093} a^{9} + \frac{1828262}{37562063351} a^{8} + \frac{7182037}{37562063351} a^{7} + \frac{9267945}{37562063351} a^{6} - \frac{6471529}{873536357} a^{5} - \frac{4817484}{873536357} a^{4} - \frac{165097}{873536357} a^{3} - \frac{1981417}{20314799} a^{2} + \frac{8398428}{20314799} a + \frac{1654899}{20314799}$, $\frac{1}{5521940449097872493} a^{22} - \frac{6263788}{69452255135999} a^{13} - \frac{8061968}{69452255135999} a^{12} - \frac{5425444}{1615168724093} a^{11} - \frac{586223}{1615168724093} a^{10} + \frac{6225879}{1615168724093} a^{9} - \frac{846527}{37562063351} a^{8} - \frac{3893677}{37562063351} a^{7} - \frac{2167144}{37562063351} a^{6} + \frac{3766142}{873536357} a^{5} - \frac{1404308}{873536357} a^{4} - \frac{5656503}{873536357} a^{3} + \frac{3629531}{20314799} a^{2} + \frac{4801980}{20314799} a - \frac{9042784}{20314799}$, $\frac{1}{5521940449097872493} a^{23} - \frac{1272604}{69452255135999} a^{13} - \frac{2079065}{69452255135999} a^{12} - \frac{7588759}{1615168724093} a^{11} + \frac{4218168}{1615168724093} a^{10} + \frac{6479051}{1615168724093} a^{9} - \frac{9498901}{37562063351} a^{8} + \frac{558502}{37562063351} a^{7} + \frac{7760455}{37562063351} a^{6} + \frac{8958971}{873536357} a^{5} + \frac{9101802}{873536357} a^{4} + \frac{4361131}{873536357} a^{3} - \frac{3966296}{20314799} a^{2} - \frac{1880683}{20314799} a + \frac{4609938}{20314799}$, $\frac{1}{237443439311208517199} a^{24} + \frac{4019853}{69452255135999} a^{13} - \frac{5090416}{69452255135999} a^{12} - \frac{5544828}{1615168724093} a^{11} - \frac{10046932}{1615168724093} a^{10} + \frac{119926}{37562063351} a^{9} - \frac{9283148}{37562063351} a^{8} - \frac{3245471}{37562063351} a^{7} + \frac{7897931}{37562063351} a^{6} + \frac{5153482}{873536357} a^{5} + \frac{9312651}{873536357} a^{4} - \frac{8641263}{873536357} a^{3} - \frac{693014}{20314799} a^{2} + \frac{7659145}{20314799} a + \frac{6943776}{20314799}$, $\frac{1}{237443439311208517199} a^{25} - \frac{2421116}{69452255135999} a^{13} + \frac{7840409}{69452255135999} a^{12} - \frac{4118234}{1615168724093} a^{11} + \frac{1616880}{1615168724093} a^{10} - \frac{8031916}{1615168724093} a^{9} - \frac{6518694}{37562063351} a^{8} - \frac{428550}{37562063351} a^{7} + \frac{4225851}{37562063351} a^{6} - \frac{4673720}{873536357} a^{5} + \frac{7614260}{873536357} a^{4} - \frac{3342412}{873536357} a^{3} + \frac{8673964}{20314799} a^{2} - \frac{5542165}{20314799} a + \frac{7754407}{20314799}$, $\frac{1}{237443439311208517199} a^{26} + \frac{6888707}{69452255135999} a^{13} - \frac{9411974}{69452255135999} a^{12} - \frac{6325793}{1615168724093} a^{11} + \frac{4252354}{1615168724093} a^{10} + \frac{5334923}{1615168724093} a^{9} - \frac{4382306}{37562063351} a^{8} + \frac{9395981}{37562063351} a^{7} + \frac{479513}{37562063351} a^{6} - \frac{5189158}{873536357} a^{5} - \frac{8981667}{873536357} a^{4} - \frac{1819629}{873536357} a^{3} - \frac{9211051}{20314799} a^{2} + \frac{6157107}{20314799} a - \frac{5583775}{20314799}$

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

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
43Data not computed