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Properties

Label 27.27.3287590482...1209.1
Degree $27$
Signature $[27, 0]$
Discriminant $163^{26}$
Root discriminant $134.98$
Ramified prime $163$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
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![-255583, 3203525, -12922671, 8685910, 70412054, -184127684, 92044084, 239709074, -355135692, 20494295, 284942103, -176991137, -63556949, 103371060, -17166283, -23282154, 10489594, 1782518, -1968453, 127774, 180016, -33013, -8354, 2388, 175, -78, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - x^26 - 78*x^25 + 175*x^24 + 2388*x^23 - 8354*x^22 - 33013*x^21 + 180016*x^20 + 127774*x^19 - 1968453*x^18 + 1782518*x^17 + 10489594*x^16 - 23282154*x^15 - 17166283*x^14 + 103371060*x^13 - 63556949*x^12 - 176991137*x^11 + 284942103*x^10 + 20494295*x^9 - 355135692*x^8 + 239709074*x^7 + 92044084*x^6 - 184127684*x^5 + 70412054*x^4 + 8685910*x^3 - 12922671*x^2 + 3203525*x - 255583)
 
gp: K = bnfinit(x^27 - x^26 - 78*x^25 + 175*x^24 + 2388*x^23 - 8354*x^22 - 33013*x^21 + 180016*x^20 + 127774*x^19 - 1968453*x^18 + 1782518*x^17 + 10489594*x^16 - 23282154*x^15 - 17166283*x^14 + 103371060*x^13 - 63556949*x^12 - 176991137*x^11 + 284942103*x^10 + 20494295*x^9 - 355135692*x^8 + 239709074*x^7 + 92044084*x^6 - 184127684*x^5 + 70412054*x^4 + 8685910*x^3 - 12922671*x^2 + 3203525*x - 255583, 1)
 

Normalized defining polynomial

\( x^{27} - x^{26} - 78 x^{25} + 175 x^{24} + 2388 x^{23} - 8354 x^{22} - 33013 x^{21} + 180016 x^{20} + 127774 x^{19} - 1968453 x^{18} + 1782518 x^{17} + 10489594 x^{16} - 23282154 x^{15} - 17166283 x^{14} + 103371060 x^{13} - 63556949 x^{12} - 176991137 x^{11} + 284942103 x^{10} + 20494295 x^{9} - 355135692 x^{8} + 239709074 x^{7} + 92044084 x^{6} - 184127684 x^{5} + 70412054 x^{4} + 8685910 x^{3} - 12922671 x^{2} + 3203525 x - 255583 \)

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:  \(3287590482487420232312619225433719761322768037503974231209=163^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.98$
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:  $163$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(163\)
Dirichlet character group:    $\lbrace$$\chi_{163}(64,·)$, $\chi_{163}(1,·)$, $\chi_{163}(126,·)$, $\chi_{163}(132,·)$, $\chi_{163}(133,·)$, $\chi_{163}(6,·)$, $\chi_{163}(135,·)$, $\chi_{163}(136,·)$, $\chi_{163}(140,·)$, $\chi_{163}(77,·)$, $\chi_{163}(146,·)$, $\chi_{163}(21,·)$, $\chi_{163}(22,·)$, $\chi_{163}(25,·)$, $\chi_{163}(155,·)$, $\chi_{163}(150,·)$, $\chi_{163}(36,·)$, $\chi_{163}(38,·)$, $\chi_{163}(40,·)$, $\chi_{163}(104,·)$, $\chi_{163}(115,·)$, $\chi_{163}(158,·)$, $\chi_{163}(58,·)$, $\chi_{163}(65,·)$, $\chi_{163}(61,·)$, $\chi_{163}(85,·)$, $\chi_{163}(53,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{59} a^{20} - \frac{28}{59} a^{19} - \frac{29}{59} a^{18} - \frac{10}{59} a^{17} - \frac{8}{59} a^{16} - \frac{9}{59} a^{15} - \frac{19}{59} a^{14} - \frac{5}{59} a^{13} + \frac{15}{59} a^{12} - \frac{12}{59} a^{11} + \frac{15}{59} a^{10} - \frac{26}{59} a^{9} + \frac{23}{59} a^{8} + \frac{26}{59} a^{7} + \frac{24}{59} a^{6} - \frac{9}{59} a^{5} - \frac{20}{59} a^{4} - \frac{4}{59} a^{3} - \frac{24}{59} a^{2} + \frac{18}{59} a + \frac{22}{59}$, $\frac{1}{59} a^{21} + \frac{13}{59} a^{19} + \frac{4}{59} a^{18} + \frac{7}{59} a^{17} + \frac{3}{59} a^{16} + \frac{24}{59} a^{15} - \frac{6}{59} a^{14} - \frac{7}{59} a^{13} - \frac{5}{59} a^{12} - \frac{26}{59} a^{11} - \frac{19}{59} a^{10} + \frac{3}{59} a^{9} + \frac{21}{59} a^{8} - \frac{15}{59} a^{7} + \frac{14}{59} a^{6} + \frac{23}{59} a^{5} + \frac{26}{59} a^{4} - \frac{18}{59} a^{3} - \frac{5}{59} a^{2} - \frac{5}{59} a + \frac{26}{59}$, $\frac{1}{59} a^{22} + \frac{14}{59} a^{19} - \frac{29}{59} a^{18} + \frac{15}{59} a^{17} + \frac{10}{59} a^{16} - \frac{7}{59} a^{15} + \frac{4}{59} a^{14} + \frac{1}{59} a^{13} + \frac{15}{59} a^{12} + \frac{19}{59} a^{11} - \frac{15}{59} a^{10} + \frac{5}{59} a^{9} - \frac{19}{59} a^{8} - \frac{29}{59} a^{7} + \frac{6}{59} a^{6} + \frac{25}{59} a^{5} + \frac{6}{59} a^{4} - \frac{12}{59} a^{3} + \frac{12}{59} a^{2} + \frac{28}{59} a + \frac{9}{59}$, $\frac{1}{59} a^{23} + \frac{9}{59} a^{19} + \frac{8}{59} a^{18} - \frac{27}{59} a^{17} - \frac{13}{59} a^{16} + \frac{12}{59} a^{15} - \frac{28}{59} a^{14} + \frac{26}{59} a^{13} - \frac{14}{59} a^{12} - \frac{24}{59} a^{11} - \frac{28}{59} a^{10} - \frac{9}{59} a^{9} + \frac{3}{59} a^{8} - \frac{4}{59} a^{7} - \frac{16}{59} a^{6} + \frac{14}{59} a^{5} - \frac{27}{59} a^{4} + \frac{9}{59} a^{3} + \frac{10}{59} a^{2} - \frac{7}{59} a - \frac{13}{59}$, $\frac{1}{59} a^{24} + \frac{24}{59} a^{19} - \frac{2}{59} a^{18} + \frac{18}{59} a^{17} + \frac{25}{59} a^{16} - \frac{6}{59} a^{15} + \frac{20}{59} a^{14} - \frac{28}{59} a^{13} + \frac{18}{59} a^{12} + \frac{21}{59} a^{11} - \frac{26}{59} a^{10} + \frac{1}{59} a^{9} + \frac{25}{59} a^{8} - \frac{14}{59} a^{7} - \frac{25}{59} a^{6} - \frac{5}{59} a^{5} + \frac{12}{59} a^{4} - \frac{13}{59} a^{3} - \frac{27}{59} a^{2} + \frac{2}{59} a - \frac{21}{59}$, $\frac{1}{227327} a^{25} - \frac{1104}{227327} a^{24} + \frac{1582}{227327} a^{23} - \frac{1862}{227327} a^{22} - \frac{398}{227327} a^{21} - \frac{729}{227327} a^{20} - \frac{94044}{227327} a^{19} + \frac{21924}{227327} a^{18} + \frac{75559}{227327} a^{17} + \frac{39341}{227327} a^{16} + \frac{18010}{227327} a^{15} + \frac{75534}{227327} a^{14} + \frac{88492}{227327} a^{13} - \frac{63363}{227327} a^{12} + \frac{285}{3853} a^{11} - \frac{19124}{227327} a^{10} + \frac{51813}{227327} a^{9} - \frac{67329}{227327} a^{8} + \frac{83713}{227327} a^{7} - \frac{102212}{227327} a^{6} + \frac{61825}{227327} a^{5} - \frac{59603}{227327} a^{4} + \frac{69992}{227327} a^{3} - \frac{96423}{227327} a^{2} - \frac{53580}{227327} a + \frac{43906}{227327}$, $\frac{1}{34265140496558977978967442901569483958672369} a^{26} - \frac{48634750496464743306265612644136739809}{34265140496558977978967442901569483958672369} a^{25} - \frac{18630366978569424307585258185664337190934}{34265140496558977978967442901569483958672369} a^{24} + \frac{208776264025912577388849426239004810174454}{34265140496558977978967442901569483958672369} a^{23} + \frac{251213271145924443133769985072745611463068}{34265140496558977978967442901569483958672369} a^{22} - \frac{114086934558265944751039035575932182712457}{34265140496558977978967442901569483958672369} a^{21} - \frac{164794141636143917303426701784305720891216}{34265140496558977978967442901569483958672369} a^{20} + \frac{15673373829892194490944114145684009055447532}{34265140496558977978967442901569483958672369} a^{19} + \frac{7857094007526767389616448444362546469219141}{34265140496558977978967442901569483958672369} a^{18} - \frac{13825883238755905262003953190130254603677827}{34265140496558977978967442901569483958672369} a^{17} - \frac{11985028235402338722816826269294269662411608}{34265140496558977978967442901569483958672369} a^{16} + \frac{12419597719884502540394099396199041575733415}{34265140496558977978967442901569483958672369} a^{15} - \frac{2286147866538140453745429635693253980660385}{34265140496558977978967442901569483958672369} a^{14} - \frac{2701418896698492872180823241905269426857639}{34265140496558977978967442901569483958672369} a^{13} + \frac{10272452527282782038680158964489229149301241}{34265140496558977978967442901569483958672369} a^{12} - \frac{13109645274714049918870375848845748907016560}{34265140496558977978967442901569483958672369} a^{11} + \frac{754446553536329102330000616177530165214279}{34265140496558977978967442901569483958672369} a^{10} - \frac{13662372455784657483289454263977895794808691}{34265140496558977978967442901569483958672369} a^{9} - \frac{8762716107734221019685072110557033716672448}{34265140496558977978967442901569483958672369} a^{8} - \frac{371671395969676686786310006242595044710416}{34265140496558977978967442901569483958672369} a^{7} - \frac{7060007312119239883509840827006537104659359}{34265140496558977978967442901569483958672369} a^{6} + \frac{10356969344509707200830356344198423675995796}{34265140496558977978967442901569483958672369} a^{5} + \frac{7498214327597505462743251878050233752944096}{34265140496558977978967442901569483958672369} a^{4} + \frac{15732024470331421174106885585166957330331324}{34265140496558977978967442901569483958672369} a^{3} - \frac{13931904987758959825195254234408136750604739}{34265140496558977978967442901569483958672369} a^{2} - \frac{11564990017577628014837030874168220392783569}{34265140496558977978967442901569483958672369} a + \frac{1203508288274670812003510567057049098214}{79501486070902501111293371001321308488799}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 36699744145500324000 \) (assuming GRH)
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

3.3.26569.1, 9.9.498311414318121121.1

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 $27$ $27$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ $27$ $27$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{9}$ $27$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{27}$

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
163Data not computed