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Properties

Label 18.0.21005238210...5344.7
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{15}\cdot 19^{12}$
Root discriminant $374.51$
Ramified primes $2, 3, 7, 19$
Class number $297851904$ (GRH)
Class group $[2, 2, 6, 6, 6, 6, 12, 4788]$ (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![63247295717733, 68911053919794, 37880347635009, 11074200636090, 1558542590511, -102611054952, -78907957919, -10162090644, 788783403, 362433664, 25244937, -5139900, -870492, 23100, 14436, 214, -123, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 123*x^16 + 214*x^15 + 14436*x^14 + 23100*x^13 - 870492*x^12 - 5139900*x^11 + 25244937*x^10 + 362433664*x^9 + 788783403*x^8 - 10162090644*x^7 - 78907957919*x^6 - 102611054952*x^5 + 1558542590511*x^4 + 11074200636090*x^3 + 37880347635009*x^2 + 68911053919794*x + 63247295717733)
 
gp: K = bnfinit(x^18 - 6*x^17 - 123*x^16 + 214*x^15 + 14436*x^14 + 23100*x^13 - 870492*x^12 - 5139900*x^11 + 25244937*x^10 + 362433664*x^9 + 788783403*x^8 - 10162090644*x^7 - 78907957919*x^6 - 102611054952*x^5 + 1558542590511*x^4 + 11074200636090*x^3 + 37880347635009*x^2 + 68911053919794*x + 63247295717733, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 123 x^{16} + 214 x^{15} + 14436 x^{14} + 23100 x^{13} - 870492 x^{12} - 5139900 x^{11} + 25244937 x^{10} + 362433664 x^{9} + 788783403 x^{8} - 10162090644 x^{7} - 78907957919 x^{6} - 102611054952 x^{5} + 1558542590511 x^{4} + 11074200636090 x^{3} + 37880347635009 x^{2} + 68911053919794 x + 63247295717733 \)

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:  \(-21005238210476463669563930923405113491534905344=-\,2^{18}\cdot 3^{27}\cdot 7^{15}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $374.51$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(2629,·)$, $\chi_{4788}(961,·)$, $\chi_{4788}(1033,·)$, $\chi_{4788}(3659,·)$, $\chi_{4788}(2063,·)$, $\chi_{4788}(467,·)$, $\chi_{4788}(4153,·)$, $\chi_{4788}(3611,·)$, $\chi_{4788}(1597,·)$, $\chi_{4788}(2015,·)$, $\chi_{4788}(419,·)$, $\chi_{4788}(4225,·)$, $\chi_{4788}(3503,·)$, $\chi_{4788}(1907,·)$, $\chi_{4788}(311,·)$, $\chi_{4788}(3193,·)$, $\chi_{4788}(2557,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{4} - \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{5} + \frac{3}{7} a^{2} + \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{49} a^{6} - \frac{2}{49} a^{5} - \frac{3}{49} a^{4} - \frac{1}{49} a^{3} + \frac{9}{49} a^{2} + \frac{10}{49} a - \frac{6}{49}$, $\frac{1}{49} a^{7} - \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{9}{49}$, $\frac{1}{686} a^{8} + \frac{1}{343} a^{7} - \frac{3}{98} a^{4} + \frac{1}{49} a^{3} + \frac{27}{98} a^{2} + \frac{22}{343} a + \frac{123}{686}$, $\frac{1}{686} a^{9} - \frac{2}{343} a^{7} - \frac{3}{98} a^{5} - \frac{3}{49} a^{4} - \frac{5}{98} a^{3} + \frac{78}{343} a^{2} - \frac{23}{98} a + \frac{73}{343}$, $\frac{1}{686} a^{10} - \frac{3}{343} a^{7} - \frac{1}{98} a^{6} + \frac{2}{49} a^{5} + \frac{5}{98} a^{4} + \frac{1}{343} a^{3} + \frac{19}{98} a^{2} - \frac{23}{49} a - \frac{55}{343}$, $\frac{1}{4802} a^{11} + \frac{1}{4802} a^{10} + \frac{3}{4802} a^{9} - \frac{3}{4802} a^{8} - \frac{47}{4802} a^{7} + \frac{5}{686} a^{6} + \frac{5}{343} a^{5} + \frac{99}{2401} a^{4} + \frac{78}{2401} a^{3} - \frac{837}{2401} a^{2} + \frac{1415}{4802} a - \frac{2285}{4802}$, $\frac{1}{43218} a^{12} - \frac{1}{14406} a^{11} + \frac{3}{4802} a^{10} - \frac{1}{43218} a^{9} - \frac{1}{2058} a^{8} + \frac{107}{14406} a^{7} - \frac{19}{3087} a^{6} + \frac{137}{2401} a^{5} + \frac{139}{7203} a^{4} - \frac{194}{7203} a^{3} + \frac{937}{4802} a^{2} + \frac{173}{686} a - \frac{1237}{7203}$, $\frac{1}{457721838} a^{13} - \frac{3971}{457721838} a^{12} - \frac{8905}{152573946} a^{11} + \frac{146537}{457721838} a^{10} - \frac{251095}{457721838} a^{9} - \frac{20317}{76286973} a^{8} + \frac{645278}{228860919} a^{7} - \frac{2274271}{228860919} a^{6} - \frac{1839764}{76286973} a^{5} - \frac{1090595}{152573946} a^{4} + \frac{10017977}{152573946} a^{3} + \frac{600815}{1495823} a^{2} + \frac{6081146}{76286973} a + \frac{71957077}{152573946}$, $\frac{1}{457721838} a^{14} - \frac{625}{65388834} a^{12} - \frac{5617}{65388834} a^{11} + \frac{14131}{21796278} a^{10} - \frac{8045}{65388834} a^{9} + \frac{239}{549486} a^{8} + \frac{952727}{152573946} a^{7} - \frac{15464}{4670631} a^{6} - \frac{244982}{10898139} a^{5} + \frac{677165}{10898139} a^{4} - \frac{758368}{10898139} a^{3} + \frac{4809295}{21796278} a^{2} - \frac{72007}{148274} a + \frac{48586397}{152573946}$, $\frac{1}{1373165514} a^{15} - \frac{20}{32694417} a^{12} - \frac{806}{10898139} a^{11} + \frac{2812}{32694417} a^{10} + \frac{43283}{65388834} a^{9} + \frac{17414}{25428991} a^{8} - \frac{284374}{32694417} a^{7} - \frac{103660}{14011893} a^{6} + \frac{352267}{21796278} a^{5} + \frac{1867321}{32694417} a^{4} - \frac{491635}{7265426} a^{3} - \frac{3200068}{10898139} a^{2} + \frac{885715}{4487469} a + \frac{3591905}{32694417}$, $\frac{1}{122720137922386921099932} a^{16} - \frac{3153626039057}{20453356320397820183322} a^{15} - \frac{7133285101322}{10226678160198910091661} a^{14} + \frac{366685801163}{417415435110159595578} a^{13} - \frac{497397494620802}{486984674295186194841} a^{12} - \frac{28695540528584450}{1460954022885558584523} a^{11} + \frac{30858360423138719}{162328224765062064947} a^{10} - \frac{497264852409487214}{3408892720066303363887} a^{9} + \frac{8704007453063270855}{40906712640795640366644} a^{8} - \frac{460188612103220245447}{61360068961193460549966} a^{7} + \frac{1510139061115055521}{417415435110159595578} a^{6} - \frac{75606249516539100619}{1460954022885558584523} a^{5} + \frac{100447405111873694405}{1947938697180744779364} a^{4} - \frac{47593898713420182421}{973969348590372389682} a^{3} - \frac{1245617392567827671521}{3408892720066303363887} a^{2} + \frac{4544997363294086858662}{10226678160198910091661} a - \frac{648380067794798393179}{13635570880265213455548}$, $\frac{1}{8550648665457343782084596297474154612} a^{17} + \frac{27882002387167}{8550648665457343782084596297474154612} a^{16} + \frac{163738277522931485851756589}{2137662166364335945521149074368538653} a^{15} + \frac{326320740652895478581496457}{1425108110909557297014099382912359102} a^{14} + \frac{11395272262077659661460213}{22620763665231068206573006077973954} a^{13} - \frac{426854399516067062062626489136}{101793436493539806929578527350882793} a^{12} - \frac{7100618867126311109241967449187}{203586872987079613859157054701765586} a^{11} - \frac{51695206528305581759367657039553}{83829888877032782177299963700727006} a^{10} - \frac{290577064640640430422813563452559}{950072073939704864676066255274906068} a^{9} - \frac{1151651633123528429572063638570725}{8550648665457343782084596297474154612} a^{8} - \frac{17333745238987436606321844430130653}{4275324332728671891042298148737077306} a^{7} - \frac{2887720307358594821188112053480279}{305380309480619420788735582052648379} a^{6} + \frac{21642998359193514945295749607951997}{407173745974159227718314109403531172} a^{5} - \frac{1787122317057972872031247498414937}{407173745974159227718314109403531172} a^{4} + \frac{260756313061674427986724744331112}{79172672828308738723005521272908839} a^{3} + \frac{185985430923939604391632691681526437}{1425108110909557297014099382912359102} a^{2} - \frac{710199642350859529301041520859761003}{2850216221819114594028198765824718204} a + \frac{601510961698538668637950111646964959}{2850216221819114594028198765824718204}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{6}\times C_{6}\times C_{12}\times C_{4788}$, which has order $297851904$ (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:  \( 204714035.62195203 \) (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

\(\Q(\sqrt{-21}) \), 3.3.1432809.4, 3.3.1432809.2, 3.3.17689.1, \(\Q(\zeta_{9})^+\), 6.0.2759153551366464.3, 6.0.2759153551366464.5, 6.0.3784847121216.1, 6.0.432081216.1, 9.9.2941473244627851129.12

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
7Data not computed
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$