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Properties

Label 18.0.56175392424...2768.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 7^{14}\cdot 41^{9}$
Root discriminant $306.26$
Ramified primes $2, 3, 7, 41$
Class number $333777024$ (GRH)
Class group $[6, 72, 772632]$ (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![542046171673, -254465350584, 224587222647, -9490822527, 24611916219, -2813326287, 5911054881, -274927533, 738641697, -16485100, 45514758, -812343, 1500495, -32223, 25869, -426, 237, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 237*x^16 - 426*x^15 + 25869*x^14 - 32223*x^13 + 1500495*x^12 - 812343*x^11 + 45514758*x^10 - 16485100*x^9 + 738641697*x^8 - 274927533*x^7 + 5911054881*x^6 - 2813326287*x^5 + 24611916219*x^4 - 9490822527*x^3 + 224587222647*x^2 - 254465350584*x + 542046171673)
 
gp: K = bnfinit(x^18 - 3*x^17 + 237*x^16 - 426*x^15 + 25869*x^14 - 32223*x^13 + 1500495*x^12 - 812343*x^11 + 45514758*x^10 - 16485100*x^9 + 738641697*x^8 - 274927533*x^7 + 5911054881*x^6 - 2813326287*x^5 + 24611916219*x^4 - 9490822527*x^3 + 224587222647*x^2 - 254465350584*x + 542046171673, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 237 x^{16} - 426 x^{15} + 25869 x^{14} - 32223 x^{13} + 1500495 x^{12} - 812343 x^{11} + 45514758 x^{10} - 16485100 x^{9} + 738641697 x^{8} - 274927533 x^{7} + 5911054881 x^{6} - 2813326287 x^{5} + 24611916219 x^{4} - 9490822527 x^{3} + 224587222647 x^{2} - 254465350584 x + 542046171673 \)

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:  \(-561753924246716084864628955145607304138272768=-\,2^{12}\cdot 3^{31}\cdot 7^{14}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $306.26$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{1}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{2}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{99} a^{15} - \frac{2}{99} a^{14} - \frac{1}{33} a^{13} + \frac{5}{99} a^{12} - \frac{1}{33} a^{11} + \frac{7}{99} a^{10} + \frac{4}{33} a^{9} - \frac{1}{33} a^{8} - \frac{5}{33} a^{7} - \frac{8}{99} a^{6} + \frac{16}{99} a^{5} - \frac{1}{33} a^{4} + \frac{29}{99} a^{3} + \frac{4}{11} a^{2} - \frac{41}{99} a - \frac{4}{11}$, $\frac{1}{297} a^{16} + \frac{1}{297} a^{15} - \frac{1}{33} a^{14} - \frac{4}{297} a^{13} - \frac{10}{297} a^{12} + \frac{1}{33} a^{11} + \frac{2}{27} a^{10} - \frac{1}{27} a^{9} + \frac{1}{33} a^{8} + \frac{46}{297} a^{7} - \frac{8}{297} a^{6} + \frac{16}{33} a^{5} - \frac{13}{297} a^{4} + \frac{35}{297} a^{3} + \frac{5}{33} a^{2} - \frac{5}{297} a - \frac{119}{297}$, $\frac{1}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a^{17} + \frac{6070911245232871202608051227433488762057682729846392204340885529746346067093220811980390}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a^{16} - \frac{6288932720733459208302632338179672955963824598118780503082766941226122676524479173815675}{1246631011242997119523473465986530078646744630670069420639962678045527006215348550905313893} a^{15} + \frac{179632118798339351337071164199329208665313698871742567560927517287207390806583660387819912}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a^{14} + \frac{1222648462694554538854769702781324092356151852434128678496596966642957099192935656274336}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a^{13} + \frac{34765378725191970400957662161425580756668408292779798832446037603853593171083873030528606}{1246631011242997119523473465986530078646744630670069420639962678045527006215348550905313893} a^{12} - \frac{70591087182264755606532078098501293597873736615373741867836961415023240169576080555760199}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a^{11} - \frac{546625877159001306480046163399885948105995156985640740922038637774433643425313441422211159}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a^{10} - \frac{196084596254610045537555173004688375463735173567858932551935374602314818014934402929993278}{1246631011242997119523473465986530078646744630670069420639962678045527006215348550905313893} a^{9} + \frac{237635642339560732710480136852084023205376291351175649126373935562327676558896476526356222}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a^{8} + \frac{165342576194394425855738927063656599784927885300423417440991510155461160205557453309252295}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a^{7} + \frac{27024121310109929651807903848468248294378892236066299301738784587058774382995601873624659}{1246631011242997119523473465986530078646744630670069420639962678045527006215348550905313893} a^{6} + \frac{57616804323463747930298416246699161884132964334911657457599645550268667299221708306901228}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a^{5} - \frac{1197610152996013513258183955770569199711841044573274656007070059911206332922340732305376690}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a^{4} + \frac{9544381345497468624035259802139252760129502278577929256613381767906125300876503508158425}{1246631011242997119523473465986530078646744630670069420639962678045527006215348550905313893} a^{3} - \frac{611232876210699691167746372924120424993056225623617462811149807543122531096114170308110990}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a^{2} - \frac{1036602801647332910771674966091023138778811844992702940549052704887389546240166382369408289}{3739893033728991358570420397959590235940233892010208261919888034136581018646045652715941679} a - \frac{346848204241481037984118703059071372765208631405843003615816200523545389806263286184993601}{1246631011242997119523473465986530078646744630670069420639962678045527006215348550905313893}$

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

Class group and class number

$C_{6}\times C_{72}\times C_{772632}$, which has order $333777024$ (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:  \( 4695974.091249611 \) (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

\(\Q(\sqrt{-123}) \), 3.3.3969.2, 3.3.756.1, 6.0.118172497968.3, 6.0.3257129475243.6, 9.9.756284282720064.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
$7$7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$41$41.6.3.2$x^{6} - 1681 x^{2} + 895973$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
41.6.3.2$x^{6} - 1681 x^{2} + 895973$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
41.6.3.2$x^{6} - 1681 x^{2} + 895973$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$