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Properties

Label 18.0.93291491561...2224.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{24}\cdot 67^{15}$
Root discriminant $406.85$
Ramified primes $2, 3, 67$
Class number $825118056$ (GRH)
Class group $[42, 42, 467754]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![151794199375503, 4181456863998, 38060351025159, 394357188862, 4343847853269, 16600209018, 293715212040, 413166918, 12479698206, 4558136, 340172631, -8580, 5927501, -714, 63651, -6, 384, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 384*x^16 - 6*x^15 + 63651*x^14 - 714*x^13 + 5927501*x^12 - 8580*x^11 + 340172631*x^10 + 4558136*x^9 + 12479698206*x^8 + 413166918*x^7 + 293715212040*x^6 + 16600209018*x^5 + 4343847853269*x^4 + 394357188862*x^3 + 38060351025159*x^2 + 4181456863998*x + 151794199375503)
 
gp: K = bnfinit(x^18 + 384*x^16 - 6*x^15 + 63651*x^14 - 714*x^13 + 5927501*x^12 - 8580*x^11 + 340172631*x^10 + 4558136*x^9 + 12479698206*x^8 + 413166918*x^7 + 293715212040*x^6 + 16600209018*x^5 + 4343847853269*x^4 + 394357188862*x^3 + 38060351025159*x^2 + 4181456863998*x + 151794199375503, 1)
 

Normalized defining polynomial

\( x^{18} + 384 x^{16} - 6 x^{15} + 63651 x^{14} - 714 x^{13} + 5927501 x^{12} - 8580 x^{11} + 340172631 x^{10} + 4558136 x^{9} + 12479698206 x^{8} + 413166918 x^{7} + 293715212040 x^{6} + 16600209018 x^{5} + 4343847853269 x^{4} + 394357188862 x^{3} + 38060351025159 x^{2} + 4181456863998 x + 151794199375503 \)

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:  \(-93291491561617523611597175490188150404729012224=-\,2^{27}\cdot 3^{24}\cdot 67^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $406.85$
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:  $2, 3, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4824=2^{3}\cdot 3^{2}\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{4824}(1,·)$, $\chi_{4824}(3589,·)$, $\chi_{4824}(1609,·)$, $\chi_{4824}(2509,·)$, $\chi_{4824}(1741,·)$, $\chi_{4824}(3217,·)$, $\chi_{4824}(3349,·)$, $\chi_{4824}(1369,·)$, $\chi_{4824}(4057,·)$, $\chi_{4824}(901,·)$, $\chi_{4824}(133,·)$, $\chi_{4824}(2977,·)$, $\chi_{4824}(2449,·)$, $\chi_{4824}(4585,·)$, $\chi_{4824}(373,·)$, $\chi_{4824}(841,·)$, $\chi_{4824}(1981,·)$, $\chi_{4824}(4117,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{18} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{18} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{4}{9} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{54} a^{11} + \frac{1}{54} a^{10} + \frac{1}{54} a^{9} + \frac{2}{9} a^{8} - \frac{2}{9} a^{7} - \frac{2}{9} a^{6} + \frac{13}{27} a^{5} + \frac{11}{54} a^{4} + \frac{1}{27} a^{3} - \frac{1}{6} a^{2} - \frac{5}{18} a - \frac{5}{18}$, $\frac{1}{4860} a^{12} - \frac{7}{810} a^{10} - \frac{1}{1215} a^{9} - \frac{77}{405} a^{8} - \frac{43}{270} a^{7} - \frac{139}{1215} a^{6} + \frac{37}{405} a^{5} + \frac{467}{1620} a^{4} - \frac{1177}{2430} a^{3} + \frac{64}{405} a^{2} + \frac{113}{270} a + \frac{323}{1620}$, $\frac{1}{160380} a^{13} - \frac{1}{16038} a^{12} - \frac{116}{13365} a^{11} + \frac{914}{40095} a^{10} - \frac{2197}{80190} a^{9} - \frac{1559}{26730} a^{8} + \frac{8047}{80190} a^{7} - \frac{4574}{40095} a^{6} + \frac{2407}{53460} a^{5} - \frac{3686}{40095} a^{4} - \frac{12011}{80190} a^{3} + \frac{949}{26730} a^{2} + \frac{25943}{53460} a - \frac{917}{5346}$, $\frac{1}{160380} a^{14} - \frac{7}{160380} a^{12} - \frac{677}{80190} a^{11} - \frac{193}{8910} a^{10} + \frac{784}{40095} a^{9} - \frac{8224}{40095} a^{8} + \frac{2977}{13365} a^{7} - \frac{12389}{160380} a^{6} - \frac{33637}{80190} a^{5} - \frac{5179}{53460} a^{4} - \frac{25193}{80190} a^{3} + \frac{24133}{53460} a^{2} - \frac{865}{1782} a - \frac{817}{10692}$, $\frac{1}{76501260} a^{15} + \frac{13}{8500140} a^{14} + \frac{1}{2125035} a^{13} + \frac{218}{3825063} a^{12} - \frac{221}{240570} a^{11} + \frac{1283}{77274} a^{10} - \frac{235883}{12750210} a^{9} + \frac{667661}{4250070} a^{8} + \frac{968731}{8500140} a^{7} + \frac{18309737}{76501260} a^{6} - \frac{112099}{850014} a^{5} + \frac{247339}{1416690} a^{4} - \frac{11155741}{76501260} a^{3} + \frac{8216843}{25500420} a^{2} - \frac{201823}{472230} a + \frac{23852}{120285}$, $\frac{1}{1737827276137095294736936344180} a^{16} + \frac{150475029037785112472}{39496074457661256698566735095} a^{15} + \frac{449441892134110452793}{3218198659513139434698030267} a^{14} + \frac{2049000246191355776843069}{868913638068547647368468172090} a^{13} + \frac{110466580421099426393620483}{1737827276137095294736936344180} a^{12} - \frac{236548783811341322083786972}{28963787935618254912282272403} a^{11} + \frac{5573730011771425576898149741}{289637879356182549122822724030} a^{10} - \frac{3623440583829492823344199108}{144818939678091274561411362015} a^{9} + \frac{19682717231994421185977268523}{193091919570788366081881816020} a^{8} + \frac{144514382512040684252383142701}{868913638068547647368468172090} a^{7} + \frac{171497967153753613218235851151}{868913638068547647368468172090} a^{6} + \frac{2810518746031588097896109375}{6436397319026278869396060534} a^{5} - \frac{113405902265035839900675376511}{868913638068547647368468172090} a^{4} + \frac{95586076340598057235052989891}{868913638068547647368468172090} a^{3} - \frac{31996283881201981857096545188}{144818939678091274561411362015} a^{2} - \frac{54440572427131137546979753418}{144818939678091274561411362015} a - \frac{4799258534695479415681772467}{10929731296459718834823499020}$, $\frac{1}{1890607529180382097021514280562175603458912620} a^{17} - \frac{82428522593069}{630202509726794032340504760187391867819637540} a^{16} + \frac{37854485861976867420510582577086761}{14322784311972591644102380913349815177719035} a^{15} - \frac{3230655083805522254972794511320720543793}{1890607529180382097021514280562175603458912620} a^{14} - \frac{17755233385368939155618585616922666073}{57291137247890366576409523653399260710876140} a^{13} + \frac{6323965251032896433452529810189317949017}{63020250972679403234050476018739186781963754} a^{12} - \frac{1146587511165412810631395856366547668020337}{315101254863397016170252380093695933909818770} a^{11} - \frac{113248971935572060223662383508698932341993}{7002250108075489248227830668748798531329306} a^{10} - \frac{1708204792182945153692992725269541767059207}{210067503242264677446834920062463955939879180} a^{9} + \frac{317780208431324363050344482017237258822152419}{1890607529180382097021514280562175603458912620} a^{8} - \frac{169560123416045675758886854460025506294138}{14322784311972591644102380913349815177719035} a^{7} + \frac{156178903713214430651301320228820296436068261}{630202509726794032340504760187391867819637540} a^{6} - \frac{79637549705495618087925439344475608616763854}{472651882295095524255378570140543900864728155} a^{5} + \frac{659432751079840684391187329995897097567383}{2378122678214317103171716076178837237055236} a^{4} + \frac{5764285535254687075712184164674019979208585}{31510125486339701617025238009369593390981877} a^{3} + \frac{18935593782457613375300331908843667794301577}{630202509726794032340504760187391867819637540} a^{2} - \frac{1394119373377457255918123321256298235653499}{210067503242264677446834920062463955939879180} a - \frac{473292536964667514080177444822209368634682}{990884449255965459654881698407848848773015}$

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

Class group and class number

$C_{42}\times C_{42}\times C_{467754}$, which has order $825118056$ (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:  \( 186599315.1231102 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-134}) \), \(\Q(\zeta_{9})^+\), 3.3.4489.1, 3.3.363609.2, 3.3.363609.1, 6.0.1010332694016.9, 6.0.691264054784.1, 6.0.4535383463437824.3, 6.0.4535383463437824.2, 9.9.48073293078275529.1

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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
67Data not computed