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Properties

Label 18.0.12350939057...9792.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 17^{9}\cdot 37^{15}$
Root discriminant $689.33$
Ramified primes $2, 3, 17, 37$
Class number $10950652440$ (GRH)
Class group $[2, 78, 70196490]$ (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![23845975839012979, -8733433478069253, 2727446539429518, -423518882731740, 92885783574705, -15288000079185, 3789040141804, -512686378569, 77246982246, -5579169113, 865327146, -64843719, 12846028, -658197, 88605, -2272, 384, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 384*x^16 - 2272*x^15 + 88605*x^14 - 658197*x^13 + 12846028*x^12 - 64843719*x^11 + 865327146*x^10 - 5579169113*x^9 + 77246982246*x^8 - 512686378569*x^7 + 3789040141804*x^6 - 15288000079185*x^5 + 92885783574705*x^4 - 423518882731740*x^3 + 2727446539429518*x^2 - 8733433478069253*x + 23845975839012979)
 
gp: K = bnfinit(x^18 - 9*x^17 + 384*x^16 - 2272*x^15 + 88605*x^14 - 658197*x^13 + 12846028*x^12 - 64843719*x^11 + 865327146*x^10 - 5579169113*x^9 + 77246982246*x^8 - 512686378569*x^7 + 3789040141804*x^6 - 15288000079185*x^5 + 92885783574705*x^4 - 423518882731740*x^3 + 2727446539429518*x^2 - 8733433478069253*x + 23845975839012979, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 384 x^{16} - 2272 x^{15} + 88605 x^{14} - 658197 x^{13} + 12846028 x^{12} - 64843719 x^{11} + 865327146 x^{10} - 5579169113 x^{9} + 77246982246 x^{8} - 512686378569 x^{7} + 3789040141804 x^{6} - 15288000079185 x^{5} + 92885783574705 x^{4} - 423518882731740 x^{3} + 2727446539429518 x^{2} - 8733433478069253 x + 23845975839012979 \)

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:  \(-1235093905746889405718571099119702734980596317089792=-\,2^{12}\cdot 3^{27}\cdot 17^{9}\cdot 37^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $689.33$
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:  $2, 3, 17, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{333} a^{15} - \frac{19}{333} a^{14} - \frac{2}{333} a^{13} + \frac{23}{333} a^{12} + \frac{16}{111} a^{11} - \frac{20}{333} a^{10} + \frac{47}{333} a^{9} + \frac{16}{333} a^{8} + \frac{26}{111} a^{7} + \frac{6}{37} a^{6} + \frac{85}{333} a^{5} - \frac{103}{333} a^{4} + \frac{95}{333} a^{3} + \frac{44}{333} a^{2} + \frac{8}{37} a + \frac{11}{333}$, $\frac{1}{333} a^{16} - \frac{10}{111} a^{14} - \frac{5}{111} a^{13} + \frac{41}{333} a^{12} + \frac{4}{333} a^{11} + \frac{7}{111} a^{9} + \frac{160}{333} a^{8} - \frac{43}{111} a^{7} + \frac{1}{333} a^{6} - \frac{17}{37} a^{5} + \frac{136}{333} a^{4} - \frac{38}{333} a^{3} + \frac{131}{333} a^{2} + \frac{47}{333} a - \frac{13}{333}$, $\frac{1}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367} a^{17} + \frac{1294630458226592604312217332445618856327941573789689059922056529596538729853159940941662405694613173767690591029659616}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367} a^{16} - \frac{3102558082877793701187584559921730902551973144946503783921687596876867599741333529092435215606439963537142937765854288}{3940396750410779958837977390098367478153913376754756100443642743734721912446909884692226755771330432898650507785647513789} a^{15} - \frac{20599031298442464972890033047620016715366415305995685296507233352028400368493735125650340329319673976040664526849706646}{1313465583470259986279325796699455826051304458918252033481214247911573970815636628230742251923776810966216835928549171263} a^{14} - \frac{243851628451646334235437940370968807227493851341228247077931467593405762259923587404529710934332646836795865943227540283}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367} a^{13} - \frac{1945394868841784540143297627244444191620551213481379853258929543050423944377397326693492925246399558111306419669715681176}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367} a^{12} - \frac{692814160950185791203620069489687363612780616705785596756913843652456958717592599817504558351798086546259674074345853069}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367} a^{11} + \frac{56739131373111301447164012023156339174268457771965101977016148498849540427362391869009874329689691872502218444360723320}{1313465583470259986279325796699455826051304458918252033481214247911573970815636628230742251923776810966216835928549171263} a^{10} - \frac{1921156516505746437951472880534613014378886025113828017539864048527313600688301079375333854392435070371399283923800173089}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367} a^{9} + \frac{2888822628384727459616595858630320772368822745717139824561032790064794079165289772221407565947271776463659627789790247893}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367} a^{8} - \frac{1957507418057528059580486464310418883078716582071029600748014521354298527501304816628916865880598571165732177241079163004}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367} a^{7} - \frac{4866041677850568627793612097276920307499581432145931900936276132148201981335386394573994921240260008281479648762617001339}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367} a^{6} - \frac{5058230712552900889586477879192970471050281612173067936770069994063626988488413714513523406863653229765030602975296298141}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367} a^{5} - \frac{174281012461055701933983593328157347224302447695702066482274016599907860131562809454594720934353008030692752679322580462}{1313465583470259986279325796699455826051304458918252033481214247911573970815636628230742251923776810966216835928549171263} a^{4} - \frac{3370311211640140115453130609299731610724544281016552843402665886115258310527890512682907859782677395930983191053552125072}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367} a^{3} + \frac{447732286969739227544162801795746526374153677791352705247342373046200055421987604018650741414283939776712649715809453983}{3940396750410779958837977390098367478153913376754756100443642743734721912446909884692226755771330432898650507785647513789} a^{2} - \frac{1412063604058501613083308507006788458833285352716742936730293429630388866240049096012670145966130945012309349174100826219}{3940396750410779958837977390098367478153913376754756100443642743734721912446909884692226755771330432898650507785647513789} a - \frac{3236100097921755846052256834821784897101609581014673565803170853997574642082495062848366072347672599898660311186562347666}{11821190251232339876513932170295102434461740130264268301330928231204165737340729654076680267313991298695951523356942541367}$

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

Class group and class number

$C_{2}\times C_{78}\times C_{70196490}$, which has order $10950652440$ (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:  \( 118546543.87559307 \) (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

\(\Q(\sqrt{-1887}) \), 3.3.148.1, 3.3.110889.1, 6.0.6705739479965103.1, 6.0.107506737648.4, 9.9.3228844269788073792.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$17$17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.6.5.5$x^{6} + 296$$6$$1$$5$$C_6$$[\ ]_{6}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$