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Properties

Label 18.0.10537754066...5744.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{30}\cdot 3^{31}\cdot 7^{9}\cdot 13^{14}$
Root discriminant $409.61$
Ramified primes $2, 3, 7, 13$
Class number $179822592$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 24, 58536]$ (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![4180912696, 6016806408, 5415393588, 2219455920, 683861808, 50331552, 55674768, -5868096, 6632250, -2916182, 971853, 131052, 180798, 5892, 5322, -300, 60, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 60*x^16 - 300*x^15 + 5322*x^14 + 5892*x^13 + 180798*x^12 + 131052*x^11 + 971853*x^10 - 2916182*x^9 + 6632250*x^8 - 5868096*x^7 + 55674768*x^6 + 50331552*x^5 + 683861808*x^4 + 2219455920*x^3 + 5415393588*x^2 + 6016806408*x + 4180912696)
 
gp: K = bnfinit(x^18 - 6*x^17 + 60*x^16 - 300*x^15 + 5322*x^14 + 5892*x^13 + 180798*x^12 + 131052*x^11 + 971853*x^10 - 2916182*x^9 + 6632250*x^8 - 5868096*x^7 + 55674768*x^6 + 50331552*x^5 + 683861808*x^4 + 2219455920*x^3 + 5415393588*x^2 + 6016806408*x + 4180912696, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 60 x^{16} - 300 x^{15} + 5322 x^{14} + 5892 x^{13} + 180798 x^{12} + 131052 x^{11} + 971853 x^{10} - 2916182 x^{9} + 6632250 x^{8} - 5868096 x^{7} + 55674768 x^{6} + 50331552 x^{5} + 683861808 x^{4} + 2219455920 x^{3} + 5415393588 x^{2} + 6016806408 x + 4180912696 \)

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:  \(-105377540665610698021122402053144975812532895744=-\,2^{30}\cdot 3^{31}\cdot 7^{9}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $409.61$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{18} a^{10} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{18} a^{11} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{72} a^{12} - \frac{1}{36} a^{11} + \frac{1}{72} a^{10} - \frac{1}{36} a^{9} - \frac{1}{8} a^{8} - \frac{5}{12} a^{7} - \frac{7}{24} a^{6} + \frac{1}{4} a^{5} + \frac{1}{12} a^{4} + \frac{5}{18} a^{3} - \frac{5}{36} a^{2} + \frac{5}{18} a - \frac{1}{18}$, $\frac{1}{144} a^{13} - \frac{1}{144} a^{12} + \frac{1}{48} a^{11} - \frac{1}{144} a^{10} - \frac{1}{48} a^{9} + \frac{7}{48} a^{8} + \frac{11}{48} a^{7} - \frac{1}{48} a^{6} + \frac{1}{3} a^{5} - \frac{17}{72} a^{4} + \frac{5}{72} a^{3} + \frac{1}{8} a^{2} - \frac{1}{18} a + \frac{1}{4}$, $\frac{1}{144} a^{14} - \frac{1}{72} a^{11} + \frac{1}{72} a^{10} - \frac{1}{72} a^{9} + \frac{11}{24} a^{7} - \frac{11}{48} a^{6} + \frac{25}{72} a^{5} - \frac{5}{12} a^{4} - \frac{5}{12} a^{3} + \frac{31}{72} a^{2} + \frac{1}{36} a - \frac{13}{36}$, $\frac{1}{144} a^{15} - \frac{1}{72} a^{11} - \frac{1}{36} a^{9} - \frac{1}{6} a^{8} + \frac{17}{48} a^{7} + \frac{1}{18} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{7}{24} a^{3} - \frac{1}{9} a^{2} - \frac{1}{12} a - \frac{1}{18}$, $\frac{1}{15696} a^{16} - \frac{53}{15696} a^{15} + \frac{1}{15696} a^{14} - \frac{5}{2616} a^{13} + \frac{5}{1962} a^{12} + \frac{35}{7848} a^{11} + \frac{5}{981} a^{10} - \frac{49}{3924} a^{9} + \frac{619}{5232} a^{8} + \frac{3557}{15696} a^{7} + \frac{41}{144} a^{6} - \frac{2723}{7848} a^{5} + \frac{351}{872} a^{4} - \frac{899}{7848} a^{3} - \frac{305}{7848} a^{2} + \frac{17}{1962} a - \frac{1145}{3924}$, $\frac{1}{496466317610377932230279706278855779453486376444998953826304810392224} a^{17} - \frac{356575549397222686371878271443737439185057701954780077830728809}{62058289701297241528784963284856972431685797055624869228288101299028} a^{16} - \frac{84899972856743018519372466184813145846951209014342181052987901989}{27581462089465440679459983682158654414082576469166608545905822799568} a^{15} - \frac{71063629691160963868692476627323758945340499699159616915125112827}{27581462089465440679459983682158654414082576469166608545905822799568} a^{14} + \frac{292425129524794116937303692394554441527586555002199730552291241329}{124116579402594483057569926569713944863371594111249738456576202598056} a^{13} - \frac{567738000642236874474536296042169680720241503002530069566852927585}{124116579402594483057569926569713944863371594111249738456576202598056} a^{12} - \frac{582401647054167813304361058943465597646346199677365200249302298187}{27581462089465440679459983682158654414082576469166608545905822799568} a^{11} + \frac{11553842601279586548359859105782632817334614561749414812997804783}{569342107351350839713623516374834609465007312436925405764111021092} a^{10} - \frac{7383401996016261447507437867121151691045730659387662104577391201975}{496466317610377932230279706278855779453486376444998953826304810392224} a^{9} + \frac{14539845913946638561610204969465496053954021567151410389575266380229}{124116579402594483057569926569713944863371594111249738456576202598056} a^{8} - \frac{6918904527119732898977682963864260963404325890234151913677449169767}{62058289701297241528784963284856972431685797055624869228288101299028} a^{7} - \frac{32513833898639776712185689484750625023640254504236893105753523180221}{82744386268396322038379951046475963242247729407499825637717468398704} a^{6} - \frac{9430810702163877632804444871697689676304486379155146787157224081381}{27581462089465440679459983682158654414082576469166608545905822799568} a^{5} + \frac{7746020464681612700684244841742002188939803089597279879780509191551}{15514572425324310382196240821214243107921449263906217307072025324757} a^{4} - \frac{60770446060129555033181315106342393598651372452271754511235716400751}{124116579402594483057569926569713944863371594111249738456576202598056} a^{3} - \frac{4355195341585978107146671513589194677828375368867277321161165471789}{13790731044732720339729991841079327207041288234583304272952911399784} a^{2} + \frac{27493475921368232947802939485992489340503457403897005348937445688067}{62058289701297241528784963284856972431685797055624869228288101299028} a + \frac{9674684484186129645771610141550922305331198471365842404977574979789}{62058289701297241528784963284856972431685797055624869228288101299028}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{24}\times C_{58536}$, which has order $179822592$ (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:  \( 3533133948.6916637 \) (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

\(\Q(\sqrt{-21}) \), 3.3.2808.1, 3.3.13689.1, 6.0.129816400896.2, 6.0.12340671610176.3, 9.9.460990789028310528.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
3Data not computed
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.12.10.2$x^{12} + 39 x^{6} + 676$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$