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Properties

Label 18.0.14186194402...4631.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{15}\cdot 19^{15}$
Root discriminant $101.96$
Ramified primes $3, 7, 19$
Class number $109312$ (GRH)
Class group $[2, 2, 2, 2, 2, 3416]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31109176, -17042844, 19805226, -10269721, 10119676, -1063041, 2032311, 341949, 143808, 186817, 32703, 32172, 20098, -2100, 2190, -83, 84, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 84*x^16 - 83*x^15 + 2190*x^14 - 2100*x^13 + 20098*x^12 + 32172*x^11 + 32703*x^10 + 186817*x^9 + 143808*x^8 + 341949*x^7 + 2032311*x^6 - 1063041*x^5 + 10119676*x^4 - 10269721*x^3 + 19805226*x^2 - 17042844*x + 31109176)
 
gp: K = bnfinit(x^18 - x^17 + 84*x^16 - 83*x^15 + 2190*x^14 - 2100*x^13 + 20098*x^12 + 32172*x^11 + 32703*x^10 + 186817*x^9 + 143808*x^8 + 341949*x^7 + 2032311*x^6 - 1063041*x^5 + 10119676*x^4 - 10269721*x^3 + 19805226*x^2 - 17042844*x + 31109176, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 84 x^{16} - 83 x^{15} + 2190 x^{14} - 2100 x^{13} + 20098 x^{12} + 32172 x^{11} + 32703 x^{10} + 186817 x^{9} + 143808 x^{8} + 341949 x^{7} + 2032311 x^{6} - 1063041 x^{5} + 10119676 x^{4} - 10269721 x^{3} + 19805226 x^{2} - 17042844 x + 31109176 \)

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:  \(-1418619440298203088067262326207274631=-\,3^{9}\cdot 7^{15}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.96$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(399=3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{399}(64,·)$, $\chi_{399}(1,·)$, $\chi_{399}(172,·)$, $\chi_{399}(398,·)$, $\chi_{399}(335,·)$, $\chi_{399}(227,·)$, $\chi_{399}(277,·)$, $\chi_{399}(278,·)$, $\chi_{399}(122,·)$, $\chi_{399}(163,·)$, $\chi_{399}(164,·)$, $\chi_{399}(293,·)$, $\chi_{399}(106,·)$, $\chi_{399}(235,·)$, $\chi_{399}(236,·)$, $\chi_{399}(121,·)$, $\chi_{399}(58,·)$, $\chi_{399}(341,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{5} - \frac{1}{16} a^{3} + \frac{5}{16} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{16} a^{15} - \frac{1}{8} a^{12} + \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{3}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} - \frac{1}{8} a^{5} + \frac{3}{16} a^{4} - \frac{1}{16} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{97966480} a^{16} - \frac{2078163}{97966480} a^{15} + \frac{738771}{24491620} a^{14} + \frac{763071}{48983240} a^{13} - \frac{423193}{97966480} a^{12} + \frac{9780919}{97966480} a^{11} - \frac{2082707}{97966480} a^{10} - \frac{53103}{433480} a^{9} + \frac{6291671}{97966480} a^{8} - \frac{6340891}{48983240} a^{7} - \frac{393419}{6122905} a^{6} + \frac{23422729}{97966480} a^{5} + \frac{26932227}{97966480} a^{4} + \frac{47231771}{97966480} a^{3} - \frac{45350023}{97966480} a^{2} - \frac{13552563}{48983240} a + \frac{419787}{12245810}$, $\frac{1}{32101996230792575907945960754994250590094555822252433600} a^{17} - \frac{66876825595674657969701780111085476597238471187}{32101996230792575907945960754994250590094555822252433600} a^{16} + \frac{301615837830384146359144336822328583617254320790112733}{16050998115396287953972980377497125295047277911126216800} a^{15} + \frac{418240481358884070795017429770090376359636699433055941}{32101996230792575907945960754994250590094555822252433600} a^{14} - \frac{206524891727837242290661593822596564190655795445813717}{4012749528849071988493245094374281323761819477781554200} a^{13} - \frac{82718050525132849240158046431108868710784906454043919}{2006374764424535994246622547187140661880909738890777100} a^{12} + \frac{1575725067210648382922050573822273572271070615608301871}{16050998115396287953972980377497125295047277911126216800} a^{11} - \frac{38366269903469590928393271140035208319346252718207341}{802549905769814397698649018874856264752363895556310840} a^{10} - \frac{7586730276586194801238889400545514949863638164681484057}{32101996230792575907945960754994250590094555822252433600} a^{9} + \frac{4434729191176458295605558142892750288313951049273033519}{32101996230792575907945960754994250590094555822252433600} a^{8} + \frac{2653409607875723215132247473377849873332008401809112637}{16050998115396287953972980377497125295047277911126216800} a^{7} - \frac{2098131897508642319122987312922278757022167379828525803}{6420399246158515181589192150998850118018911164450486720} a^{6} - \frac{950661476725234880229034588151647931668155008899378499}{32101996230792575907945960754994250590094555822252433600} a^{5} + \frac{11307223517343856999989373596993185070584836120390802773}{32101996230792575907945960754994250590094555822252433600} a^{4} - \frac{3449628103462744890898195707090667896188898689751104701}{16050998115396287953972980377497125295047277911126216800} a^{3} - \frac{2881094540543025398177288970653360560744880928199759849}{32101996230792575907945960754994250590094555822252433600} a^{2} - \frac{229964148039677621509803406067240854596722597082385753}{1605099811539628795397298037749712529504727791112621680} a - \frac{2277188540204015674876104444624079988458250575850065521}{8025499057698143976986490188748562647523638955563108400}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{3416}$, which has order $109312$ (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:  \( 833965.2438558349 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-399}) \), 3.3.361.1, 3.3.17689.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 6.0.22931152839.1, 6.0.1123626489111.1, 6.0.1123626489111.2, 6.0.3112538751.2, 9.9.5534900853769.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
19Data not computed