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Properties

Label 18.18.4747758522...2953.1
Degree $18$
Signature $[18, 0]$
Discriminant $73^{17}$
Root discriminant $57.52$
Ramified prime $73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -32, 224, 250, -2859, -1994, 9365, 4608, -13016, -3867, 8348, 1551, -2671, -311, 435, 29, -34, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 34*x^16 + 29*x^15 + 435*x^14 - 311*x^13 - 2671*x^12 + 1551*x^11 + 8348*x^10 - 3867*x^9 - 13016*x^8 + 4608*x^7 + 9365*x^6 - 1994*x^5 - 2859*x^4 + 250*x^3 + 224*x^2 - 32*x + 1)
 
gp: K = bnfinit(x^18 - x^17 - 34*x^16 + 29*x^15 + 435*x^14 - 311*x^13 - 2671*x^12 + 1551*x^11 + 8348*x^10 - 3867*x^9 - 13016*x^8 + 4608*x^7 + 9365*x^6 - 1994*x^5 - 2859*x^4 + 250*x^3 + 224*x^2 - 32*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 34 x^{16} + 29 x^{15} + 435 x^{14} - 311 x^{13} - 2671 x^{12} + 1551 x^{11} + 8348 x^{10} - 3867 x^{9} - 13016 x^{8} + 4608 x^{7} + 9365 x^{6} - 1994 x^{5} - 2859 x^{4} + 250 x^{3} + 224 x^{2} - 32 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(47477585226700098686074966922953=73^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.52$
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:  $73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(73\)
Dirichlet character group:    $\lbrace$$\chi_{73}(64,·)$, $\chi_{73}(1,·)$, $\chi_{73}(2,·)$, $\chi_{73}(4,·)$, $\chi_{73}(69,·)$, $\chi_{73}(65,·)$, $\chi_{73}(8,·)$, $\chi_{73}(9,·)$, $\chi_{73}(16,·)$, $\chi_{73}(18,·)$, $\chi_{73}(32,·)$, $\chi_{73}(36,·)$, $\chi_{73}(37,·)$, $\chi_{73}(41,·)$, $\chi_{73}(71,·)$, $\chi_{73}(72,·)$, $\chi_{73}(55,·)$, $\chi_{73}(57,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{1496110316163516441} a^{17} - \frac{40216416803060858}{498703438721172147} a^{16} + \frac{57643264178982656}{1496110316163516441} a^{15} + \frac{106137175703116900}{1496110316163516441} a^{14} + \frac{181624572251444800}{1496110316163516441} a^{13} - \frac{617611216924055161}{1496110316163516441} a^{12} + \frac{172520018343880942}{1496110316163516441} a^{11} + \frac{103406152484266816}{1496110316163516441} a^{10} + \frac{21938374372037048}{166234479573724049} a^{9} + \frac{60208375719954804}{166234479573724049} a^{8} - \frac{178810360112160035}{1496110316163516441} a^{7} - \frac{176693938482421166}{1496110316163516441} a^{6} + \frac{9338080058125579}{166234479573724049} a^{5} + \frac{398761835909844046}{1496110316163516441} a^{4} + \frac{611914114575342430}{1496110316163516441} a^{3} - \frac{165941408548999489}{1496110316163516441} a^{2} + \frac{434064321029204452}{1496110316163516441} a - \frac{454315324168616887}{1496110316163516441}$

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:  $17$
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:  \( 7147727796.24 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{73}) \), 3.3.5329.1, 6.6.2073071593.1, 9.9.806460091894081.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ $18$ $18$ $18$

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
73Data not computed