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Properties

Label 18.0.94186295366...0000.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 5^{9}\cdot 163^{14}$
Root discriminant $407.07$
Ramified primes $2, 3, 5, 163$
Class number $165923264$ (GRH)
Class group $[2, 2, 4, 10370204]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4397661349, 874624481, 1085934684, -289152616, 155971711, 3616657, 37954104, -6535669, -905428, 39697, 492348, 38123, -44338, -605, 2995, 38, -82, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 82*x^16 + 38*x^15 + 2995*x^14 - 605*x^13 - 44338*x^12 + 38123*x^11 + 492348*x^10 + 39697*x^9 - 905428*x^8 - 6535669*x^7 + 37954104*x^6 + 3616657*x^5 + 155971711*x^4 - 289152616*x^3 + 1085934684*x^2 + 874624481*x + 4397661349)
 
gp: K = bnfinit(x^18 - x^17 - 82*x^16 + 38*x^15 + 2995*x^14 - 605*x^13 - 44338*x^12 + 38123*x^11 + 492348*x^10 + 39697*x^9 - 905428*x^8 - 6535669*x^7 + 37954104*x^6 + 3616657*x^5 + 155971711*x^4 - 289152616*x^3 + 1085934684*x^2 + 874624481*x + 4397661349, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 82 x^{16} + 38 x^{15} + 2995 x^{14} - 605 x^{13} - 44338 x^{12} + 38123 x^{11} + 492348 x^{10} + 39697 x^{9} - 905428 x^{8} - 6535669 x^{7} + 37954104 x^{6} + 3616657 x^{5} + 155971711 x^{4} - 289152616 x^{3} + 1085934684 x^{2} + 874624481 x + 4397661349 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-94186295366043352700011964780992402944000000000=-\,2^{18}\cdot 3^{9}\cdot 5^{9}\cdot 163^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $407.07$
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:  $2, 3, 5, 163$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{74} a^{14} - \frac{7}{74} a^{13} - \frac{21}{74} a^{11} + \frac{13}{37} a^{10} - \frac{15}{74} a^{9} - \frac{4}{37} a^{8} - \frac{25}{74} a^{7} - \frac{5}{37} a^{6} - \frac{27}{74} a^{5} + \frac{17}{37} a^{4} - \frac{7}{74} a^{3} + \frac{11}{37} a^{2} - \frac{35}{74} a - \frac{7}{74}$, $\frac{1}{74} a^{15} - \frac{6}{37} a^{13} + \frac{8}{37} a^{12} - \frac{5}{37} a^{11} - \frac{9}{37} a^{10} - \frac{1}{37} a^{9} + \frac{15}{37} a^{8} + \frac{7}{37} a^{6} + \frac{15}{37} a^{5} - \frac{14}{37} a^{4} + \frac{5}{37} a^{3} + \frac{4}{37} a^{2} + \frac{7}{74} a - \frac{6}{37}$, $\frac{1}{68929594} a^{16} + \frac{87191}{68929594} a^{15} - \frac{81701}{68929594} a^{14} - \frac{17114093}{68929594} a^{13} - \frac{2891995}{34464797} a^{12} + \frac{17913861}{68929594} a^{11} - \frac{16647121}{34464797} a^{10} - \frac{19835443}{68929594} a^{9} + \frac{8287591}{34464797} a^{8} + \frac{20536467}{68929594} a^{7} + \frac{16988034}{34464797} a^{6} - \frac{10521559}{68929594} a^{5} + \frac{13370548}{34464797} a^{4} - \frac{2495501}{68929594} a^{3} + \frac{12198029}{68929594} a^{2} - \frac{13474870}{34464797} a + \frac{11195183}{68929594}$, $\frac{1}{3829014615064525103130213119894325116443201990381031868987849254} a^{17} - \frac{35937710576729918880487306079000892081236166051581005}{10971388581846776799800037592820415806427512866421294753546846} a^{16} - \frac{607597705850178125590126008839414336702621691210341441715113}{103486881488230408192708462699846624768735188929217077540212142} a^{15} + \frac{83967267452583873986789405286701577299352381728424836186089}{112618076913662503033241562349833091660094176187677407911407331} a^{14} - \frac{553554646008993813587081866283162750442690272213125838067151029}{3829014615064525103130213119894325116443201990381031868987849254} a^{13} - \frac{14557833589031827205974402106985230241758274625359967151286479}{72245558774802360436419115469704247480060414912849657905431118} a^{12} - \frac{1099553715229014143256690418245491052082761189608242977781689057}{3829014615064525103130213119894325116443201990381031868987849254} a^{11} + \frac{810292432383384860686013626290096634688960962897817992095576183}{3829014615064525103130213119894325116443201990381031868987849254} a^{10} - \frac{269769046465076511997964771305090010104705911898797326079849021}{3829014615064525103130213119894325116443201990381031868987849254} a^{9} - \frac{1090889507368125379101544139322294610225795502981961427861741223}{3829014615064525103130213119894325116443201990381031868987849254} a^{8} - \frac{154938024785259552454168454981949362511816104800294915913803723}{3829014615064525103130213119894325116443201990381031868987849254} a^{7} - \frac{1051950798853698988963299578250050541187398922014328067475727797}{3829014615064525103130213119894325116443201990381031868987849254} a^{6} + \frac{1051176624101480729460647484294584350609622808519845095817640967}{3829014615064525103130213119894325116443201990381031868987849254} a^{5} + \frac{1809905716544780301987045184407405176487193783802486190578187339}{3829014615064525103130213119894325116443201990381031868987849254} a^{4} + \frac{555936065793617551714472141634527090542239280126210681745528389}{1914507307532262551565106559947162558221600995190515934493924627} a^{3} - \frac{37195538578407143845672193252187648736835120064913166277048543}{1914507307532262551565106559947162558221600995190515934493924627} a^{2} - \frac{902115681329724275975432698855606935718806980378556451034140267}{1914507307532262551565106559947162558221600995190515934493924627} a - \frac{1703605377496096877389602654756715037306505091347833252809611191}{3829014615064525103130213119894325116443201990381031868987849254}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{10370204}$, which has order $165923264$ (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:  $8$
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:  \( 87986390.20265311 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.3.26569.1, 3.3.1304.1, 6.0.2382452193375.3, 6.0.5738904000.3, 9.9.1565248123502319104.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$163$163.6.4.1$x^{6} + 5216 x^{3} + 35363339$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
163.12.10.1$x^{12} + 266994 x^{6} + 47068604209$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$