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Properties

Label 18.0.14470224127...6223.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{30}\cdot 7^{15}\cdot 23^{6}$
Root discriminant $89.82$
Ramified primes $3, 7, 23$
Class number $12768$ (GRH)
Class group $[2, 2, 3192]$ (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![2097152, -4718592, 3244032, 1155072, -829440, -1445376, 1123200, 12960, -40512, -74840, 29988, 6528, -99, -1827, 822, -147, 18, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 18*x^16 - 147*x^15 + 822*x^14 - 1827*x^13 - 99*x^12 + 6528*x^11 + 29988*x^10 - 74840*x^9 - 40512*x^8 + 12960*x^7 + 1123200*x^6 - 1445376*x^5 - 829440*x^4 + 1155072*x^3 + 3244032*x^2 - 4718592*x + 2097152)
 
gp: K = bnfinit(x^18 - 3*x^17 + 18*x^16 - 147*x^15 + 822*x^14 - 1827*x^13 - 99*x^12 + 6528*x^11 + 29988*x^10 - 74840*x^9 - 40512*x^8 + 12960*x^7 + 1123200*x^6 - 1445376*x^5 - 829440*x^4 + 1155072*x^3 + 3244032*x^2 - 4718592*x + 2097152, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 18 x^{16} - 147 x^{15} + 822 x^{14} - 1827 x^{13} - 99 x^{12} + 6528 x^{11} + 29988 x^{10} - 74840 x^{9} - 40512 x^{8} + 12960 x^{7} + 1123200 x^{6} - 1445376 x^{5} - 829440 x^{4} + 1155072 x^{3} + 3244032 x^{2} - 4718592 x + 2097152 \)

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:  \(-144702241276662140714874256260206223=-\,3^{30}\cdot 7^{15}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.82$
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:  $3, 7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{10} + \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{32} a^{7} - \frac{1}{16} a^{6} + \frac{1}{32} a^{5} + \frac{5}{32} a^{4} - \frac{1}{4} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{3}{64} a^{8} - \frac{3}{64} a^{6} + \frac{7}{64} a^{5} + \frac{7}{32} a^{4} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{256} a^{12} - \frac{1}{256} a^{11} - \frac{3}{256} a^{9} - \frac{1}{16} a^{8} + \frac{13}{256} a^{7} - \frac{9}{256} a^{6} + \frac{31}{128} a^{5} - \frac{1}{16} a^{4} - \frac{7}{32} a^{3} - \frac{3}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{1024} a^{13} - \frac{1}{1024} a^{12} - \frac{3}{1024} a^{10} + \frac{3}{64} a^{9} - \frac{19}{1024} a^{8} - \frac{9}{1024} a^{7} + \frac{15}{512} a^{6} - \frac{9}{64} a^{5} - \frac{11}{128} a^{4} + \frac{9}{64} a^{3} - \frac{7}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{2048} a^{14} - \frac{1}{2048} a^{13} - \frac{3}{2048} a^{11} - \frac{1}{128} a^{10} - \frac{83}{2048} a^{9} + \frac{119}{2048} a^{8} - \frac{17}{1024} a^{7} - \frac{1}{128} a^{6} + \frac{45}{256} a^{5} + \frac{21}{128} a^{4} + \frac{1}{32} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16384} a^{15} + \frac{1}{16384} a^{14} + \frac{3}{8192} a^{13} - \frac{11}{16384} a^{12} + \frac{21}{8192} a^{11} + \frac{53}{16384} a^{10} + \frac{81}{16384} a^{9} - \frac{163}{4096} a^{8} + \frac{25}{4096} a^{7} - \frac{17}{2048} a^{6} - \frac{13}{512} a^{5} + \frac{37}{512} a^{4} + \frac{1}{16} a^{3} + \frac{3}{8} a$, $\frac{1}{3080192} a^{16} + \frac{25}{3080192} a^{15} + \frac{95}{1540096} a^{14} + \frac{997}{3080192} a^{13} + \frac{1809}{1540096} a^{12} + \frac{18245}{3080192} a^{11} - \frac{30903}{3080192} a^{10} + \frac{21003}{770048} a^{9} + \frac{47721}{770048} a^{8} + \frac{5331}{385024} a^{7} - \frac{1947}{96256} a^{6} - \frac{2043}{96256} a^{5} + \frac{437}{12032} a^{4} + \frac{109}{752} a^{3} + \frac{169}{1504} a^{2} + \frac{45}{188} a + \frac{20}{47}$, $\frac{1}{442973151049705185616544881521983488} a^{17} + \frac{66664522973283381941343621101}{442973151049705185616544881521983488} a^{16} - \frac{4778171757764488345368041520015}{221486575524852592808272440760991744} a^{15} - \frac{86226405114035046971813507536531}{442973151049705185616544881521983488} a^{14} + \frac{24088531908696304142907871046483}{221486575524852592808272440760991744} a^{13} - \frac{144413271802055882402442284377699}{442973151049705185616544881521983488} a^{12} + \frac{2602699309111220237565331710310541}{442973151049705185616544881521983488} a^{11} - \frac{153977757562047347900249180341411}{27685821940606574101034055095123968} a^{10} + \frac{5966638923028243623310940492509457}{110743287762426296404136220380495872} a^{9} + \frac{1109114785576266986274906154003981}{55371643881213148202068110190247936} a^{8} - \frac{216393262508589926989580033365649}{6921455485151643525258513773780992} a^{7} + \frac{1184662322626604621782122322347101}{13842910970303287050517027547561984} a^{6} + \frac{562177809996362135215431955480355}{3460727742575821762629256886890496} a^{5} - \frac{107938423577647699844433298021513}{865181935643955440657314221722624} a^{4} + \frac{33490696404747124774189052161817}{216295483910988860164328555430656} a^{3} - \frac{21883462930404172475021202363523}{54073870977747215041082138857664} a^{2} - \frac{77484516966970461696343459}{13518467744436803760270534714416} a + \frac{792134581176108067842065558749}{1689808468054600470033816839302}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3192}$, which has order $12768$ (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:  \( 237885673.5279126 \) (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

\(\Q(\sqrt{-7}) \), 3.3.3969.1, 3.3.621.1, 6.0.110270727.2, 6.0.132274863.5, 9.9.20539533187176381.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$7$7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.2$x^{12} + 35 x^{6} + 441$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$23$23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$