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Properties

Label 18.0.18253584485...0608.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 11^{9}\cdot 37^{14}$
Root discriminant $196.02$
Ramified primes $2, 11, 37$
Class number $30326688$ (GRH)
Class group $[2, 2, 2, 14, 270774]$ (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![3461536818169, 78764845432, 1048817590026, 2023454274, 145194850029, -2095429314, 12167724659, -317287568, 685380987, -22449952, 27080080, -929854, 755082, -23330, 14429, -330, 173, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 173*x^16 - 330*x^15 + 14429*x^14 - 23330*x^13 + 755082*x^12 - 929854*x^11 + 27080080*x^10 - 22449952*x^9 + 685380987*x^8 - 317287568*x^7 + 12167724659*x^6 - 2095429314*x^5 + 145194850029*x^4 + 2023454274*x^3 + 1048817590026*x^2 + 78764845432*x + 3461536818169)
 
gp: K = bnfinit(x^18 - 2*x^17 + 173*x^16 - 330*x^15 + 14429*x^14 - 23330*x^13 + 755082*x^12 - 929854*x^11 + 27080080*x^10 - 22449952*x^9 + 685380987*x^8 - 317287568*x^7 + 12167724659*x^6 - 2095429314*x^5 + 145194850029*x^4 + 2023454274*x^3 + 1048817590026*x^2 + 78764845432*x + 3461536818169, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 173 x^{16} - 330 x^{15} + 14429 x^{14} - 23330 x^{13} + 755082 x^{12} - 929854 x^{11} + 27080080 x^{10} - 22449952 x^{9} + 685380987 x^{8} - 317287568 x^{7} + 12167724659 x^{6} - 2095429314 x^{5} + 145194850029 x^{4} + 2023454274 x^{3} + 1048817590026 x^{2} + 78764845432 x + 3461536818169 \)

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:  \(-182535844857539433056785809958362947780608=-\,2^{33}\cdot 11^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $196.02$
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:  $2, 11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{3922995367712546300245728890965661454234164579164170136321771364163002} a^{17} + \frac{423917774221791766305567120335939426822633154960080683034334713331717}{1961497683856273150122864445482830727117082289582085068160885682081501} a^{16} + \frac{205825729258666669149653981371024569596931547235176975886785080243730}{1961497683856273150122864445482830727117082289582085068160885682081501} a^{15} - \frac{91582083532097588110117928059059629198541074006948281696482117172417}{1961497683856273150122864445482830727117082289582085068160885682081501} a^{14} - \frac{344935016780968538272003164700728703454179713580434785775709845838823}{1961497683856273150122864445482830727117082289582085068160885682081501} a^{13} - \frac{565269623746611218858694555289445517660038504649370929338787724431787}{3922995367712546300245728890965661454234164579164170136321771364163002} a^{12} + \frac{222172105163258767654773993588848055232233324857477059488691707608540}{1961497683856273150122864445482830727117082289582085068160885682081501} a^{11} + \frac{315002734378703514943905998627008022928274735212136586527972917868469}{1961497683856273150122864445482830727117082289582085068160885682081501} a^{10} + \frac{253781177431505027188255608653361423101959186798088900570768823022726}{1961497683856273150122864445482830727117082289582085068160885682081501} a^{9} - \frac{337012436085381796284689797560603079519893987154670928980912238217149}{1961497683856273150122864445482830727117082289582085068160885682081501} a^{8} + \frac{582846020734389541375021443969266352416034108331675709173971349462705}{3922995367712546300245728890965661454234164579164170136321771364163002} a^{7} + \frac{879892077348781320976841313427986168397460016301948214670776768906465}{1961497683856273150122864445482830727117082289582085068160885682081501} a^{6} + \frac{1384877401761272499502292354058413117651163108653074367413534638624133}{3922995367712546300245728890965661454234164579164170136321771364163002} a^{5} + \frac{112795846489746878988413078484563683987985036334412576919095397362181}{3922995367712546300245728890965661454234164579164170136321771364163002} a^{4} + \frac{1297814475626224129481755110148901550274451370380499272527513468089359}{3922995367712546300245728890965661454234164579164170136321771364163002} a^{3} + \frac{709325330246043610197361220518854321437126735497111232519490329280841}{3922995367712546300245728890965661454234164579164170136321771364163002} a^{2} - \frac{553571818354238525868597694624065087468853677262145600873429351404455}{3922995367712546300245728890965661454234164579164170136321771364163002} a - \frac{160501120836131335490972284544332134028182396992323177995258464358521}{1961497683856273150122864445482830727117082289582085068160885682081501}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{14}\times C_{270774}$, which has order $30326688$ (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:  \( 615797.1340659427 \) (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

\(\Q(\sqrt{-22}) \), 3.3.1369.1, 3.3.148.1, 6.0.3731740672.7, 6.0.1277188244992.3, 9.9.6075640136512.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$