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Properties

Label 18.0.29701346406...0003.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 37^{10}$
Root discriminant $49.31$
Ramified primes $3, 37$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 43008, 112896, -132864, 49968, -119904, 104028, -57408, 54399, -31018, 16053, -10746, 4191, -1584, 669, -150, 39, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 39*x^16 - 150*x^15 + 669*x^14 - 1584*x^13 + 4191*x^12 - 10746*x^11 + 16053*x^10 - 31018*x^9 + 54399*x^8 - 57408*x^7 + 104028*x^6 - 119904*x^5 + 49968*x^4 - 132864*x^3 + 112896*x^2 + 43008*x + 4096)
 
gp: K = bnfinit(x^18 - 6*x^17 + 39*x^16 - 150*x^15 + 669*x^14 - 1584*x^13 + 4191*x^12 - 10746*x^11 + 16053*x^10 - 31018*x^9 + 54399*x^8 - 57408*x^7 + 104028*x^6 - 119904*x^5 + 49968*x^4 - 132864*x^3 + 112896*x^2 + 43008*x + 4096, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 39 x^{16} - 150 x^{15} + 669 x^{14} - 1584 x^{13} + 4191 x^{12} - 10746 x^{11} + 16053 x^{10} - 31018 x^{9} + 54399 x^{8} - 57408 x^{7} + 104028 x^{6} - 119904 x^{5} + 49968 x^{4} - 132864 x^{3} + 112896 x^{2} + 43008 x + 4096 \)

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:  \(-2970134640629244629640134970003=-\,3^{31}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.31$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{16} a^{5} - \frac{3}{16} a^{4} + \frac{1}{16} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{11} + \frac{1}{32} a^{10} + \frac{1}{8} a^{7} - \frac{1}{32} a^{6} + \frac{5}{32} a^{5} - \frac{1}{32} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{12} + \frac{1}{64} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{15}{64} a^{7} - \frac{11}{64} a^{6} - \frac{1}{64} a^{5} - \frac{1}{4} a^{4} + \frac{3}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{13} + \frac{1}{128} a^{12} - \frac{1}{32} a^{11} + \frac{1}{32} a^{10} - \frac{1}{32} a^{9} + \frac{15}{128} a^{8} - \frac{11}{128} a^{7} + \frac{31}{128} a^{6} + \frac{1}{8} a^{5} - \frac{7}{32} a^{4} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{15} + \frac{1}{128} a^{12} - \frac{1}{32} a^{11} + \frac{1}{32} a^{10} - \frac{5}{128} a^{9} + \frac{1}{32} a^{8} - \frac{7}{32} a^{7} - \frac{21}{128} a^{6} + \frac{1}{16} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{221696} a^{16} + \frac{291}{110848} a^{15} - \frac{517}{221696} a^{14} + \frac{437}{110848} a^{13} - \frac{3207}{221696} a^{12} - \frac{1259}{55424} a^{11} - \frac{13729}{221696} a^{10} - \frac{6875}{110848} a^{9} + \frac{12609}{221696} a^{8} - \frac{24437}{110848} a^{7} - \frac{1725}{221696} a^{6} + \frac{11015}{55424} a^{5} + \frac{3351}{55424} a^{4} - \frac{6789}{13856} a^{3} - \frac{4213}{13856} a^{2} - \frac{1261}{3464} a - \frac{243}{866}$, $\frac{1}{2907626681873310790051616768} a^{17} + \frac{2624247072738851774123}{1453813340936655395025808384} a^{16} - \frac{204206549561964718670545}{2907626681873310790051616768} a^{15} - \frac{5258345282239643864671481}{1453813340936655395025808384} a^{14} - \frac{7722823006290085092519243}{2907626681873310790051616768} a^{13} + \frac{7103992643925421158467259}{726906670468327697512904192} a^{12} + \frac{17430087368884691700455551}{2907626681873310790051616768} a^{11} - \frac{22995752890408190921717899}{1453813340936655395025808384} a^{10} + \frac{99847305829466169208689773}{2907626681873310790051616768} a^{9} + \frac{164709816326494696183291585}{1453813340936655395025808384} a^{8} + \frac{647376365987361560218387111}{2907626681873310790051616768} a^{7} - \frac{144262204045703904882940963}{726906670468327697512904192} a^{6} + \frac{161151723758438245967507215}{726906670468327697512904192} a^{5} + \frac{31175235177708695087707699}{181726667617081924378226048} a^{4} - \frac{5347827782462351603926885}{181726667617081924378226048} a^{3} + \frac{5500207603989303655415861}{45431666904270481094556512} a^{2} - \frac{293023014568047646402943}{11357916726067620273639128} a - \frac{675845476137651493267695}{1419739590758452534204891}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{11978143602513}{74609822057633792} a^{17} - \frac{29976971569341}{37304911028816896} a^{16} + \frac{415793453734303}{74609822057633792} a^{15} - \frac{710713571392785}{37304911028816896} a^{14} + \frac{6879586266563397}{74609822057633792} a^{13} - \frac{3263500535138257}{18652455514408448} a^{12} + \frac{41924747111947407}{74609822057633792} a^{11} - \frac{47734083028939779}{37304911028816896} a^{10} + \frac{128324784685056701}{74609822057633792} a^{9} - \frac{155646134439642183}{37304911028816896} a^{8} + \frac{436686128753512791}{74609822057633792} a^{7} - \frac{127624973750072423}{18652455514408448} a^{6} + \frac{262718668465992135}{18652455514408448} a^{5} - \frac{50069247771771669}{4663113878602112} a^{4} + \frac{39161496666829331}{4663113878602112} a^{3} - \frac{21554835277888431}{1165778469650528} a^{2} + \frac{2370215554964301}{291444617412632} a + \frac{75676383861303}{36430577176579} \) (order $6$)
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:  \( 810773555.0000257 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.999.1, 6.0.26946027.1, 6.0.2994003.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$