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Properties

Label 18.0.15464650229...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 5^{9}\cdot 31^{14}$
Root discriminant $116.43$
Ramified primes $3, 5, 31$
Class number $428652$ (GRH)
Class group $[9, 126, 378]$ (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![232459264, -114413568, 173788800, -61439152, 51935352, -14204472, 9310034, -2223417, 1087623, -212016, 69558, -18729, 8007, -2379, -21, 142, 6, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 6*x^16 + 142*x^15 - 21*x^14 - 2379*x^13 + 8007*x^12 - 18729*x^11 + 69558*x^10 - 212016*x^9 + 1087623*x^8 - 2223417*x^7 + 9310034*x^6 - 14204472*x^5 + 51935352*x^4 - 61439152*x^3 + 173788800*x^2 - 114413568*x + 232459264)
 
gp: K = bnfinit(x^18 - 9*x^17 + 6*x^16 + 142*x^15 - 21*x^14 - 2379*x^13 + 8007*x^12 - 18729*x^11 + 69558*x^10 - 212016*x^9 + 1087623*x^8 - 2223417*x^7 + 9310034*x^6 - 14204472*x^5 + 51935352*x^4 - 61439152*x^3 + 173788800*x^2 - 114413568*x + 232459264, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 6 x^{16} + 142 x^{15} - 21 x^{14} - 2379 x^{13} + 8007 x^{12} - 18729 x^{11} + 69558 x^{10} - 212016 x^{9} + 1087623 x^{8} - 2223417 x^{7} + 9310034 x^{6} - 14204472 x^{5} + 51935352 x^{4} - 61439152 x^{3} + 173788800 x^{2} - 114413568 x + 232459264 \)

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:  \(-15464650229059143166953879566724609375=-\,3^{21}\cdot 5^{9}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.43$
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:  $3, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{9} + \frac{1}{6} a^{6} + \frac{1}{12} a^{3} + \frac{1}{3}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{5}{24} a^{4} + \frac{11}{24} a^{3} + \frac{1}{4} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{10} - \frac{1}{12} a^{8} - \frac{1}{24} a^{7} - \frac{1}{8} a^{6} + \frac{7}{48} a^{5} + \frac{11}{48} a^{4} + \frac{1}{8} a^{3} + \frac{1}{12} a^{2} + \frac{1}{6} a$, $\frac{1}{96} a^{12} + \frac{1}{96} a^{10} + \frac{1}{48} a^{9} - \frac{1}{16} a^{8} - \frac{1}{6} a^{7} + \frac{13}{96} a^{6} - \frac{3}{16} a^{5} + \frac{7}{96} a^{4} - \frac{1}{12} a^{3} - \frac{1}{12} a$, $\frac{1}{192} a^{13} - \frac{1}{192} a^{11} + \frac{1}{32} a^{9} - \frac{1}{24} a^{8} + \frac{11}{64} a^{7} + \frac{3}{32} a^{6} + \frac{17}{192} a^{5} - \frac{5}{96} a^{4} - \frac{1}{4} a^{3} + \frac{7}{24} a^{2} - \frac{5}{12} a$, $\frac{1}{384} a^{14} - \frac{1}{384} a^{13} + \frac{1}{384} a^{12} - \frac{1}{128} a^{11} - \frac{1}{96} a^{10} + \frac{1}{64} a^{9} - \frac{1}{128} a^{8} + \frac{25}{384} a^{7} - \frac{5}{128} a^{6} + \frac{5}{384} a^{5} - \frac{7}{32} a^{4} - \frac{1}{24} a^{3} + \frac{23}{48} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{2304} a^{15} + \frac{1}{2304} a^{14} + \frac{1}{768} a^{13} - \frac{1}{2304} a^{12} - \frac{7}{1152} a^{11} - \frac{17}{1152} a^{10} - \frac{31}{2304} a^{9} - \frac{13}{2304} a^{8} - \frac{281}{2304} a^{7} + \frac{37}{768} a^{6} + \frac{95}{384} a^{5} + \frac{2}{9} a^{4} + \frac{59}{288} a^{3} - \frac{67}{144} a^{2} + \frac{17}{36} a - \frac{2}{9}$, $\frac{1}{35495424} a^{16} - \frac{253}{35495424} a^{15} + \frac{10261}{35495424} a^{14} + \frac{59465}{35495424} a^{13} - \frac{5461}{1478976} a^{12} + \frac{27301}{5915904} a^{11} - \frac{689627}{35495424} a^{10} - \frac{1198787}{35495424} a^{9} + \frac{171055}{11831808} a^{8} - \frac{7960607}{35495424} a^{7} - \frac{150011}{1478976} a^{6} + \frac{253165}{4436928} a^{5} - \frac{17797}{1478976} a^{4} + \frac{44705}{369744} a^{3} + \frac{24115}{554616} a^{2} + \frac{1153}{30812} a + \frac{23396}{69327}$, $\frac{1}{4968763119276373900658359272421820185342451712} a^{17} - \frac{8281184676377046693912210935127869087}{2484381559638186950329179636210910092671225856} a^{16} - \frac{13249373743847588781077911974097205987971}{414063593273031158388196606035151682111870976} a^{15} + \frac{922924611863941490620723984820224221217121}{2484381559638186950329179636210910092671225856} a^{14} - \frac{109623406708538334076478118111976529876871}{552084791030708211184262141380202242815827968} a^{13} - \frac{232836142344474524408925596642857688744561}{77636923738693342197786863631590940395975808} a^{12} - \frac{14700194090984397814034287497508957494547667}{1656254373092124633552786424140606728447483904} a^{11} + \frac{12693651329726586554931868243899449929314569}{1242190779819093475164589818105455046335612928} a^{10} - \frac{82994726309561168448176560276305565459601695}{2484381559638186950329179636210910092671225856} a^{9} + \frac{159018010665553559237421032568364528166940403}{2484381559638186950329179636210910092671225856} a^{8} + \frac{397929852391967729788649491496180485443290345}{4968763119276373900658359272421820185342451712} a^{7} - \frac{254259370720514316520715234911113363911996219}{2484381559638186950329179636210910092671225856} a^{6} - \frac{631695072927669617420334861517274695218269}{8626324859854815799754095959065660043997312} a^{5} - \frac{84615211064162649508609741117004707396494871}{621095389909546737582294909052727523167806464} a^{4} + \frac{20829667029771674844655218139540584985121651}{103515898318257789597049151508787920527967744} a^{3} + \frac{1028797339185197921068058174430068933805869}{38818461869346671098893431815795470197987904} a^{2} - \frac{2037401887226982359438676128863599788327071}{4852307733668333887361678976974433774748488} a + \frac{2232863286158827937341215087015729550964303}{4852307733668333887361678976974433774748488}$

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

Class group and class number

$C_{9}\times C_{126}\times C_{378}$, which has order $428652$ (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:  \( 67295442.84873295 \) (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

\(\Q(\sqrt{-15}) \), 3.3.837.1, 3.3.961.1, 6.0.262713375.1, 6.0.3116883375.2, 9.9.541530783546813.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$31$31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$