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Properties

Label 18.0.21156460455...3776.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{12}\cdot 37^{15}\cdot 47^{9}$
Root discriminant $807.18$
Ramified primes $2, 7, 37, 47$
Class number $26407106460$ (GRH)
Class group $[3, 3, 3, 3, 3, 3, 777, 46620]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6328125825565187, 2634341655131550, 2270159537466107, 697432636061526, 128931293649937, 17062555046622, 550446017812, -265581279066, -18676098374, 4399169196, 412426863, -69040572, -7119103, 461226, 79167, -1770, -388, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 388*x^16 - 1770*x^15 + 79167*x^14 + 461226*x^13 - 7119103*x^12 - 69040572*x^11 + 412426863*x^10 + 4399169196*x^9 - 18676098374*x^8 - 265581279066*x^7 + 550446017812*x^6 + 17062555046622*x^5 + 128931293649937*x^4 + 697432636061526*x^3 + 2270159537466107*x^2 + 2634341655131550*x + 6328125825565187)
 
gp: K = bnfinit(x^18 - 388*x^16 - 1770*x^15 + 79167*x^14 + 461226*x^13 - 7119103*x^12 - 69040572*x^11 + 412426863*x^10 + 4399169196*x^9 - 18676098374*x^8 - 265581279066*x^7 + 550446017812*x^6 + 17062555046622*x^5 + 128931293649937*x^4 + 697432636061526*x^3 + 2270159537466107*x^2 + 2634341655131550*x + 6328125825565187, 1)
 

Normalized defining polynomial

\( x^{18} - 388 x^{16} - 1770 x^{15} + 79167 x^{14} + 461226 x^{13} - 7119103 x^{12} - 69040572 x^{11} + 412426863 x^{10} + 4399169196 x^{9} - 18676098374 x^{8} - 265581279066 x^{7} + 550446017812 x^{6} + 17062555046622 x^{5} + 128931293649937 x^{4} + 697432636061526 x^{3} + 2270159537466107 x^{2} + 2634341655131550 x + 6328125825565187 \)

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:  \(-21156460455554157443184358692743823704081665160523776=-\,2^{12}\cdot 7^{12}\cdot 37^{15}\cdot 47^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $807.18$
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:  $2, 7, 37, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{222} a^{9} - \frac{3}{74} a^{7} + \frac{1}{74} a^{6} + \frac{9}{74} a^{5} - \frac{3}{37} a^{4} - \frac{49}{111} a^{3} + \frac{9}{74} a^{2} - \frac{83}{222} a + \frac{25}{74}$, $\frac{1}{222} a^{10} - \frac{3}{74} a^{8} + \frac{1}{74} a^{7} + \frac{9}{74} a^{6} - \frac{3}{37} a^{5} - \frac{49}{111} a^{4} + \frac{9}{74} a^{3} - \frac{83}{222} a^{2} + \frac{25}{74} a$, $\frac{1}{222} a^{11} + \frac{1}{74} a^{8} - \frac{9}{37} a^{7} + \frac{3}{74} a^{6} - \frac{77}{222} a^{5} + \frac{29}{74} a^{4} - \frac{77}{222} a^{3} + \frac{16}{37} a^{2} - \frac{27}{74} a + \frac{3}{74}$, $\frac{1}{7770} a^{12} + \frac{11}{7770} a^{11} - \frac{17}{7770} a^{10} + \frac{17}{7770} a^{9} - \frac{304}{3885} a^{8} + \frac{317}{1295} a^{7} + \frac{229}{1110} a^{6} - \frac{1814}{3885} a^{5} - \frac{4}{21} a^{4} + \frac{1817}{3885} a^{3} + \frac{97}{555} a^{2} - \frac{157}{1110} a + \frac{439}{1110}$, $\frac{1}{54390} a^{13} + \frac{1}{54390} a^{12} - \frac{11}{27195} a^{11} + \frac{39}{18130} a^{10} + \frac{9}{18130} a^{9} + \frac{1249}{18130} a^{8} + \frac{8039}{27195} a^{7} + \frac{2561}{27195} a^{6} + \frac{362}{1813} a^{5} - \frac{11141}{54390} a^{4} + \frac{23783}{54390} a^{3} - \frac{317}{1295} a^{2} + \frac{1273}{2590} a - \frac{64}{259}$, $\frac{1}{54390} a^{14} - \frac{1}{27195} a^{12} - \frac{4}{1813} a^{11} + \frac{43}{54390} a^{10} - \frac{44}{27195} a^{9} - \frac{6317}{54390} a^{8} - \frac{4793}{18130} a^{7} - \frac{9332}{27195} a^{6} - \frac{442}{1295} a^{5} - \frac{5918}{27195} a^{4} + \frac{8837}{54390} a^{3} - \frac{326}{777} a^{2} - \frac{3121}{7770} a - \frac{506}{1295}$, $\frac{1}{3937509660} a^{15} - \frac{1105}{131250322} a^{14} + \frac{35681}{3937509660} a^{13} + \frac{5267}{106419180} a^{12} - \frac{101186}{328125805} a^{11} - \frac{265166}{984377415} a^{10} + \frac{202583}{1968754830} a^{9} - \frac{37612403}{1968754830} a^{8} - \frac{259339351}{1312503220} a^{7} + \frac{25169821}{196875483} a^{6} + \frac{981410471}{3937509660} a^{5} - \frac{140076467}{1312503220} a^{4} + \frac{225277281}{1312503220} a^{3} + \frac{19395366}{46875115} a^{2} - \frac{85133417}{187500460} a - \frac{193315231}{562501380}$, $\frac{1}{507938746140} a^{16} + \frac{1}{11287527692} a^{15} + \frac{198559}{33862583076} a^{14} + \frac{9649}{2288012370} a^{13} + \frac{2716649}{169312915380} a^{12} - \frac{88800022}{42328228845} a^{11} + \frac{40351357}{25396937307} a^{10} + \frac{1499672}{863841405} a^{9} + \frac{2378303429}{72562678020} a^{8} + \frac{53960872649}{169312915380} a^{7} + \frac{15676363729}{169312915380} a^{6} + \frac{18512096392}{42328228845} a^{5} - \frac{23410329167}{50793874614} a^{4} - \frac{63315660263}{169312915380} a^{3} + \frac{14228331353}{72562678020} a^{2} + \frac{479102177}{2418755934} a - \frac{58949489}{1687504140}$, $\frac{1}{917698754500282002022988956392969142088190696032769319953164974446930751901676555641323620} a^{17} - \frac{210738454636353590076229242218068176850475685403080539030437822893254936154259}{917698754500282002022988956392969142088190696032769319953164974446930751901676555641323620} a^{16} + \frac{9167792766567654110675375393916691433689678988730555493047328061442992935476784}{76474896208356833501915746366080761840682558002730776662763747870577562658473046303443635} a^{15} + \frac{267434978319372025884881466695332839332606766950924394672226796385570775549675846751}{30589958483342733400766298546432304736273023201092310665105499148231025063389218521377454} a^{14} + \frac{7777195722599427123189228236513474325583830687490932800711779073755059829873610306}{5098326413890455566794383091072050789378837200182051777517583191371837510564869753562909} a^{13} - \frac{4149074964792660075455729308486437262146444298082005726170895980975808660704988965061}{305899584833427334007662985464323047362730232010923106651054991482310250633892185213774540} a^{12} + \frac{2351799233743296504020018675321087127813571568028634546145810572018521950982865568297}{1872854601020983677597936645699937024669776930679121061128908111116185207962605215594538} a^{11} - \frac{5520079195423094861659158377547438943054812985769774021085863302171409213522695264573}{4209627314221477073499949341252152027927480257031051926390665020398764916980167686428090} a^{10} - \frac{1510694266578769152319397501783734794040036979549072406434994328910135307638861090331307}{917698754500282002022988956392969142088190696032769319953164974446930751901676555641323620} a^{9} - \frac{25751346974682269962698678681212177703989143083304516191848748913580385463426307532998491}{183539750900056400404597791278593828417638139206553863990632994889386150380335311128264724} a^{8} - \frac{76217780105551204139798504689825279557048892809956458829347675661119884028945201758203629}{152949792416713667003831492732161523681365116005461553325527495741155125316946092606887270} a^{7} + \frac{64128117965199695104483194040146904648975743867946877922698533745403754938665262501620819}{152949792416713667003831492732161523681365116005461553325527495741155125316946092606887270} a^{6} - \frac{182057477248026128558453175939223546874598533153302074344424930546351989659439646183483473}{917698754500282002022988956392969142088190696032769319953164974446930751901676555641323620} a^{5} + \frac{8934185514958777581245435996737700685242684531698777187592051430696800407160229011675421}{65549911035734428715927782599497795863442192573769237139511783889066482278691182545808830} a^{4} - \frac{102812845490408603673079934874585847661705766913203777787864199476484320713679986300469807}{229424688625070500505747239098242285522047674008192329988291243611732687975419138910330905} a^{3} + \frac{367986261859758956033125056917941002104936683610413204541896650628476026533226742836819}{9364273005104918387989683228499685123348884653395605305644540555580926039813026077972690} a^{2} - \frac{20266452736311753354378548444001212401868182840591975367106207324739517272065150436466251}{65549911035734428715927782599497795863442192573769237139511783889066482278691182545808830} a - \frac{2230224383076017218217042112414306212461240230251182664171957978332333463376194501413}{27970945609445030388703982333901342378255682770970444693625681198662889813821712202180}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{777}\times C_{46620}$, which has order $26407106460$ (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:  \( 508181773.5034319 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-1739}) \), 3.1.6956.1 x3, 3.3.67081.2, 6.0.84143142704.1, Deg 6, Deg 6 x2, 9.3.74211952229135601401024.1 x3

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

Sibling fields

Degree 6 sibling: data not computed
Degree 9 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
$37$37.6.5.3$x^{6} - 592$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.3$x^{6} - 592$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.3$x^{6} - 592$$6$$1$$5$$C_6$$[\ ]_{6}$
$47$47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$