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Properties

Label 18.0.15122562715...8416.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{30}\cdot 7^{14}\cdot 31^{9}$
Root discriminant $250.53$
Ramified primes $2, 3, 7, 31$
Class number $59439744$ (GRH)
Class group $[3, 12, 12, 137592]$ (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![4059023093, -3727057221, 3760200192, -2071336188, 1169945709, -449775009, 180078636, -51892869, 16474170, -3697991, 983802, -157041, 32856, -1725, 393, 42, 24, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 24*x^16 + 42*x^15 + 393*x^14 - 1725*x^13 + 32856*x^12 - 157041*x^11 + 983802*x^10 - 3697991*x^9 + 16474170*x^8 - 51892869*x^7 + 180078636*x^6 - 449775009*x^5 + 1169945709*x^4 - 2071336188*x^3 + 3760200192*x^2 - 3727057221*x + 4059023093)
 
gp: K = bnfinit(x^18 - 3*x^17 + 24*x^16 + 42*x^15 + 393*x^14 - 1725*x^13 + 32856*x^12 - 157041*x^11 + 983802*x^10 - 3697991*x^9 + 16474170*x^8 - 51892869*x^7 + 180078636*x^6 - 449775009*x^5 + 1169945709*x^4 - 2071336188*x^3 + 3760200192*x^2 - 3727057221*x + 4059023093, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 24 x^{16} + 42 x^{15} + 393 x^{14} - 1725 x^{13} + 32856 x^{12} - 157041 x^{11} + 983802 x^{10} - 3697991 x^{9} + 16474170 x^{8} - 51892869 x^{7} + 180078636 x^{6} - 449775009 x^{5} + 1169945709 x^{4} - 2071336188 x^{3} + 3760200192 x^{2} - 3727057221 x + 4059023093 \)

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:  \(-15122562715473759157739145863562318625468416=-\,2^{12}\cdot 3^{30}\cdot 7^{14}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $250.53$
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:  $2, 3, 7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{21} a^{15} + \frac{1}{21} a^{12} - \frac{1}{7} a^{11} - \frac{1}{21} a^{10} - \frac{2}{21} a^{9} - \frac{2}{21} a^{8} + \frac{1}{3} a^{7} + \frac{8}{21} a^{6} + \frac{1}{3} a^{5} - \frac{1}{7} a^{4} - \frac{5}{21} a^{3} - \frac{4}{21} a^{2} - \frac{5}{21} a - \frac{2}{7}$, $\frac{1}{231} a^{16} + \frac{2}{231} a^{15} + \frac{4}{33} a^{14} - \frac{2}{77} a^{13} - \frac{12}{77} a^{12} - \frac{1}{11} a^{11} - \frac{1}{21} a^{10} + \frac{3}{7} a^{9} - \frac{20}{77} a^{8} + \frac{8}{231} a^{7} + \frac{2}{231} a^{6} - \frac{2}{7} a^{5} - \frac{1}{21} a^{4} - \frac{5}{33} a^{3} + \frac{43}{231} a^{2} - \frac{79}{231} a - \frac{47}{231}$, $\frac{1}{56190473879470335121680880678098994655729884665276887199} a^{17} - \frac{106720561154091973623461045325178692630460546107842882}{56190473879470335121680880678098994655729884665276887199} a^{16} + \frac{257755862931950885167471942632145255388526481942341376}{56190473879470335121680880678098994655729884665276887199} a^{15} - \frac{734512808991332704452022050130660210267388895047384834}{5108224898133666829243716425281726786884534969570626109} a^{14} - \frac{105579815158233477997288510854827036723749963009737236}{8027210554210047874525840096871284950818554952182412457} a^{13} - \frac{2229315559002531523801572404988256731149727759700412593}{18730157959823445040560293559366331551909961555092295733} a^{12} - \frac{990700873779567868047837548405001608037964302460805132}{56190473879470335121680880678098994655729884665276887199} a^{11} - \frac{810551578089833609380257773157566999804391853792001035}{5108224898133666829243716425281726786884534969570626109} a^{10} + \frac{11362170367276492398482083894541317873436139666924693068}{56190473879470335121680880678098994655729884665276887199} a^{9} - \frac{23260560632529840683154016752433761581002957718482218235}{56190473879470335121680880678098994655729884665276887199} a^{8} - \frac{23622585340592637024271038820764724075178180358875503363}{56190473879470335121680880678098994655729884665276887199} a^{7} + \frac{21931121588578630057651051460907739826946115448389297768}{56190473879470335121680880678098994655729884665276887199} a^{6} - \frac{195199167029833403983356655993115015026413321465441892}{729746414019095261320530917897389540983504995652946587} a^{5} + \frac{2412544043295329083795043411700899478019946822038740514}{56190473879470335121680880678098994655729884665276887199} a^{4} - \frac{3372459796541659413258565422010106294722253992090915571}{56190473879470335121680880678098994655729884665276887199} a^{3} - \frac{12362522447914025061943579083132685127705194234190419427}{56190473879470335121680880678098994655729884665276887199} a^{2} + \frac{67133576770006522947872433040323196497108805894676659}{263805041687654155500849205061497627491689599367497123} a - \frac{49170174034847367868118859519248209113162539168185135}{247535127222336278069078769507044029320395967688444437}$

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

Class group and class number

$C_{3}\times C_{12}\times C_{12}\times C_{137592}$, which has order $59439744$ (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

\(\Q(\sqrt{-31}) \), 3.3.756.1, 3.3.3969.2, 6.0.17026628976.7, 6.0.469296461151.7, 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
$31$31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.12.6.1$x^{12} + 178746 x^{6} - 114516604 x^{2} + 7987533129$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$