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Properties

Label 21.21.1191446152...9201.1
Degree $21$
Signature $[21, 0]$
Discriminant $127^{20}$
Root discriminant $100.84$
Ramified prime $127$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{21}$ (as 21T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14459, 50035, -205872, -380716, 1481414, -157753, -3188668, 3165855, 540263, -2307149, 832624, 489841, -380273, -2588, 62425, -10801, -4493, 1305, 133, -60, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 60*x^19 + 133*x^18 + 1305*x^17 - 4493*x^16 - 10801*x^15 + 62425*x^14 - 2588*x^13 - 380273*x^12 + 489841*x^11 + 832624*x^10 - 2307149*x^9 + 540263*x^8 + 3165855*x^7 - 3188668*x^6 - 157753*x^5 + 1481414*x^4 - 380716*x^3 - 205872*x^2 + 50035*x + 14459)
 
gp: K = bnfinit(x^21 - x^20 - 60*x^19 + 133*x^18 + 1305*x^17 - 4493*x^16 - 10801*x^15 + 62425*x^14 - 2588*x^13 - 380273*x^12 + 489841*x^11 + 832624*x^10 - 2307149*x^9 + 540263*x^8 + 3165855*x^7 - 3188668*x^6 - 157753*x^5 + 1481414*x^4 - 380716*x^3 - 205872*x^2 + 50035*x + 14459, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 60 x^{19} + 133 x^{18} + 1305 x^{17} - 4493 x^{16} - 10801 x^{15} + 62425 x^{14} - 2588 x^{13} - 380273 x^{12} + 489841 x^{11} + 832624 x^{10} - 2307149 x^{9} + 540263 x^{8} + 3165855 x^{7} - 3188668 x^{6} - 157753 x^{5} + 1481414 x^{4} - 380716 x^{3} - 205872 x^{2} + 50035 x + 14459 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1191446152405248657777607437681912764659201=127^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.84$
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:  $127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(127\)
Dirichlet character group:    $\lbrace$$\chi_{127}(64,·)$, $\chi_{127}(1,·)$, $\chi_{127}(2,·)$, $\chi_{127}(4,·)$, $\chi_{127}(8,·)$, $\chi_{127}(73,·)$, $\chi_{127}(76,·)$, $\chi_{127}(16,·)$, $\chi_{127}(19,·)$, $\chi_{127}(87,·)$, $\chi_{127}(25,·)$, $\chi_{127}(94,·)$, $\chi_{127}(32,·)$, $\chi_{127}(100,·)$, $\chi_{127}(38,·)$, $\chi_{127}(107,·)$, $\chi_{127}(47,·)$, $\chi_{127}(50,·)$, $\chi_{127}(117,·)$, $\chi_{127}(122,·)$, $\chi_{127}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} - \frac{4}{19} a^{14} + \frac{4}{19} a^{13} - \frac{9}{19} a^{12} + \frac{7}{19} a^{11} + \frac{2}{19} a^{10} + \frac{1}{19} a^{9} + \frac{2}{19} a^{8} + \frac{4}{19} a^{7} + \frac{9}{19} a^{6} - \frac{7}{19} a^{5} - \frac{2}{19} a^{4} - \frac{2}{19} a^{3} + \frac{2}{19} a^{2} - \frac{8}{19} a$, $\frac{1}{19} a^{16} + \frac{7}{19} a^{14} + \frac{7}{19} a^{13} + \frac{9}{19} a^{12} - \frac{8}{19} a^{11} + \frac{9}{19} a^{10} + \frac{6}{19} a^{9} - \frac{7}{19} a^{8} + \frac{6}{19} a^{7} - \frac{9}{19} a^{6} + \frac{8}{19} a^{5} + \frac{9}{19} a^{4} - \frac{6}{19} a^{3} + \frac{6}{19} a$, $\frac{1}{19} a^{17} - \frac{3}{19} a^{14} - \frac{2}{19} a^{12} - \frac{2}{19} a^{11} - \frac{8}{19} a^{10} + \frac{5}{19} a^{9} - \frac{8}{19} a^{8} + \frac{1}{19} a^{7} + \frac{2}{19} a^{6} + \frac{1}{19} a^{5} + \frac{8}{19} a^{4} - \frac{5}{19} a^{3} - \frac{8}{19} a^{2} - \frac{1}{19} a$, $\frac{1}{19} a^{18} + \frac{7}{19} a^{14} - \frac{9}{19} a^{13} + \frac{9}{19} a^{12} - \frac{6}{19} a^{11} - \frac{8}{19} a^{10} - \frac{5}{19} a^{9} + \frac{7}{19} a^{8} - \frac{5}{19} a^{7} + \frac{9}{19} a^{6} + \frac{6}{19} a^{5} + \frac{8}{19} a^{4} + \frac{5}{19} a^{3} + \frac{5}{19} a^{2} - \frac{5}{19} a$, $\frac{1}{361} a^{19} - \frac{5}{361} a^{18} + \frac{9}{361} a^{16} - \frac{4}{361} a^{15} - \frac{127}{361} a^{14} - \frac{155}{361} a^{13} - \frac{42}{361} a^{12} - \frac{108}{361} a^{11} - \frac{58}{361} a^{10} - \frac{134}{361} a^{9} - \frac{163}{361} a^{8} + \frac{44}{361} a^{7} + \frac{142}{361} a^{6} - \frac{63}{361} a^{5} - \frac{122}{361} a^{4} - \frac{52}{361} a^{3} - \frac{147}{361} a^{2} - \frac{156}{361} a + \frac{9}{19}$, $\frac{1}{91645163994998292140053} a^{20} + \frac{81390783967321781390}{91645163994998292140053} a^{19} + \frac{53091926664796047464}{91645163994998292140053} a^{18} - \frac{50378893164675068582}{91645163994998292140053} a^{17} - \frac{2397456927174675725}{91645163994998292140053} a^{16} - \frac{196482339406237010687}{91645163994998292140053} a^{15} + \frac{38717242443039573702791}{91645163994998292140053} a^{14} - \frac{38777672873334661276611}{91645163994998292140053} a^{13} - \frac{3122878917901743698913}{91645163994998292140053} a^{12} + \frac{2783615504292133431868}{91645163994998292140053} a^{11} + \frac{208242830477412135940}{4823429683947278533687} a^{10} + \frac{972731103264732496901}{91645163994998292140053} a^{9} + \frac{25312409631927821423823}{91645163994998292140053} a^{8} + \frac{10671189488722492710378}{91645163994998292140053} a^{7} - \frac{38385407135289506130505}{91645163994998292140053} a^{6} + \frac{5995836102839197366698}{91645163994998292140053} a^{5} + \frac{1961701664912517456919}{91645163994998292140053} a^{4} + \frac{379321316136623540071}{4823429683947278533687} a^{3} - \frac{16114473854175013026833}{91645163994998292140053} a^{2} - \frac{7324239103892453380369}{91645163994998292140053} a + \frac{1230061077287477004782}{4823429683947278533687}$

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:  $20$
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:  \( 399125736989601.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{21}$ (as 21T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

3.3.16129.1, 7.7.4195872914689.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{21}$ $21$ $21$ $21$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ $21$ $21$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
127Data not computed