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Properties

Label 18.0.70283171793...0000.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{31}\cdot 5^{9}\cdot 7^{14}$
Root discriminant $240.09$
Ramified primes $2, 3, 5, 7$
Class number $48795648$ (GRH)
Class group $[2, 2, 2, 2, 304, 10032]$ (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![70597462733281, 81200252004, 15683707815702, -315811185954, 1569548611977, -60571863750, 93909017643, -4993969080, 3743115171, -234474428, 104127960, -6836730, 2042502, -124314, 27597, -1302, 237, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 237*x^16 - 1302*x^15 + 27597*x^14 - 124314*x^13 + 2042502*x^12 - 6836730*x^11 + 104127960*x^10 - 234474428*x^9 + 3743115171*x^8 - 4993969080*x^7 + 93909017643*x^6 - 60571863750*x^5 + 1569548611977*x^4 - 315811185954*x^3 + 15683707815702*x^2 + 81200252004*x + 70597462733281)
 
gp: K = bnfinit(x^18 - 6*x^17 + 237*x^16 - 1302*x^15 + 27597*x^14 - 124314*x^13 + 2042502*x^12 - 6836730*x^11 + 104127960*x^10 - 234474428*x^9 + 3743115171*x^8 - 4993969080*x^7 + 93909017643*x^6 - 60571863750*x^5 + 1569548611977*x^4 - 315811185954*x^3 + 15683707815702*x^2 + 81200252004*x + 70597462733281, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 237 x^{16} - 1302 x^{15} + 27597 x^{14} - 124314 x^{13} + 2042502 x^{12} - 6836730 x^{11} + 104127960 x^{10} - 234474428 x^{9} + 3743115171 x^{8} - 4993969080 x^{7} + 93909017643 x^{6} - 60571863750 x^{5} + 1569548611977 x^{4} - 315811185954 x^{3} + 15683707815702 x^{2} + 81200252004 x + 70597462733281 \)

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:  \(-7028317179364168027995195360411648000000000=-\,2^{33}\cdot 3^{31}\cdot 5^{9}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $240.09$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{54} a^{9} - \frac{1}{18} a^{8} - \frac{1}{18} a^{7} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{5}{18} a^{4} + \frac{1}{18} a^{3} - \frac{1}{18} a^{2} - \frac{2}{9} a - \frac{19}{54}$, $\frac{1}{54} a^{10} + \frac{1}{9} a^{8} - \frac{1}{18} a^{7} + \frac{1}{9} a^{6} - \frac{1}{18} a^{5} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{18} a^{2} - \frac{1}{54} a - \frac{7}{18}$, $\frac{1}{54} a^{11} - \frac{1}{18} a^{8} + \frac{1}{9} a^{7} - \frac{1}{18} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} + \frac{5}{18} a^{3} - \frac{1}{54} a^{2} - \frac{7}{18} a - \frac{2}{9}$, $\frac{1}{54} a^{12} - \frac{1}{18} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{5}{27} a^{3} + \frac{4}{9} a^{2} + \frac{4}{9} a - \frac{7}{18}$, $\frac{1}{54} a^{13} - \frac{1}{18} a^{8} + \frac{1}{18} a^{7} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{17}{54} a^{4} - \frac{7}{18} a^{3} - \frac{1}{18} a^{2} - \frac{1}{18} a + \frac{5}{18}$, $\frac{1}{162} a^{14} + \frac{1}{162} a^{13} + \frac{1}{162} a^{12} - \frac{1}{162} a^{11} - \frac{1}{162} a^{10} - \frac{1}{162} a^{9} + \frac{4}{27} a^{8} + \frac{4}{27} a^{7} - \frac{1}{54} a^{6} - \frac{2}{81} a^{5} + \frac{25}{81} a^{4} + \frac{77}{162} a^{3} - \frac{10}{81} a^{2} - \frac{37}{81} a + \frac{7}{162}$, $\frac{1}{162} a^{15} + \frac{1}{162} a^{12} + \frac{1}{162} a^{9} + \frac{1}{18} a^{8} + \frac{1}{18} a^{7} - \frac{1}{162} a^{6} - \frac{1}{9} a^{5} - \frac{5}{18} a^{4} + \frac{71}{162} a^{3} - \frac{4}{9} a^{2} - \frac{5}{18} a - \frac{23}{81}$, $\frac{1}{179982} a^{16} - \frac{137}{89991} a^{15} - \frac{359}{179982} a^{14} + \frac{310}{89991} a^{13} - \frac{163}{29997} a^{12} + \frac{1409}{179982} a^{11} + \frac{95}{29997} a^{10} - \frac{74}{8181} a^{9} - \frac{4613}{29997} a^{8} + \frac{24647}{179982} a^{7} + \frac{28693}{179982} a^{6} - \frac{6277}{179982} a^{5} + \frac{67381}{179982} a^{4} - \frac{22259}{59994} a^{3} - \frac{41183}{179982} a^{2} - \frac{10915}{29997} a - \frac{2485}{8181}$, $\frac{1}{458537700028344440638589516794150738872893650822980525133500546} a^{17} + \frac{84553960180751267211557619451479848790269195317179798193}{152845900009448146879529838931383579624297883607660175044500182} a^{16} + \frac{727127031726108964753452111358553439817342407959997091040741}{458537700028344440638589516794150738872893650822980525133500546} a^{15} + \frac{308311889653424904261200299916118216755398549305665619416245}{229268850014172220319294758397075369436446825411490262566750273} a^{14} - \frac{3505619481188400821258276805123124533107406011772354823465565}{458537700028344440638589516794150738872893650822980525133500546} a^{13} - \frac{49140106472687550582378157145086948810405492676067633405801}{5660959259609190625167771812273465912011032726209636112759266} a^{12} + \frac{25314809800202089578940079281724601305101202155131119018573}{4631693939680246875137267846405563018918117685080611364984854} a^{11} + \frac{312187505604507581156103418120464641175830344551676591389845}{458537700028344440638589516794150738872893650822980525133500546} a^{10} - \frac{3568418346464308068402377706428053711364088973388387801343761}{458537700028344440638589516794150738872893650822980525133500546} a^{9} - \frac{837173571314258517186449928452351931365123948313497403065107}{458537700028344440638589516794150738872893650822980525133500546} a^{8} - \frac{10240236802326889342346203787288154715460818952060533662458284}{76422950004724073439764919465691789812148941803830087522250091} a^{7} + \frac{34472877580372547027677258330331069281564685536691174658197753}{229268850014172220319294758397075369436446825411490262566750273} a^{6} - \frac{2797425441938874203690722266071020091304795515443980082126717}{229268850014172220319294758397075369436446825411490262566750273} a^{5} + \frac{40092007777172299212746403260675162408671436638363311474094615}{229268850014172220319294758397075369436446825411490262566750273} a^{4} - \frac{27175977121465776225920461886815356751963644410477838018666650}{76422950004724073439764919465691789812148941803830087522250091} a^{3} + \frac{8146436083876628803356359275879851377004663683223349969398502}{76422950004724073439764919465691789812148941803830087522250091} a^{2} + \frac{27627414115007821904814325954421849116171342858252607573426977}{458537700028344440638589516794150738872893650822980525133500546} a + \frac{6566694509760900947508304717391254258827570385554872781078614}{20842622728561110938117705308825033585131529582862751142431843}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{304}\times C_{10032}$, which has order $48795648$ (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

\(\Q(\sqrt{-30}) \), 3.3.756.1, 3.3.3969.2, 6.0.27433728000.9, 6.0.3024568512000.20, 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 R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$