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Properties

Label 18.0.73390587140...1443.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 7^{12}\cdot 19^{12}\cdot 73^{12}$
Root discriminant $1639.68$
Ramified primes $3, 7, 19, 73$
Class number $13086203184$ (GRH)
Class group $[3, 3, 3, 3, 3, 3, 3, 9, 9, 36, 2052]$ (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![1397069806495609, 333613339887189, 412931389362222, -300600540176057, 52528838073972, -26874825967038, 9019520363287, -57748921875, 27813323421, -6203302885, -22759602, 29633925, -5041232, 57771, -15021, 1902, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 + 1902*x^15 - 15021*x^14 + 57771*x^13 - 5041232*x^12 + 29633925*x^11 - 22759602*x^10 - 6203302885*x^9 + 27813323421*x^8 - 57748921875*x^7 + 9019520363287*x^6 - 26874825967038*x^5 + 52528838073972*x^4 - 300600540176057*x^3 + 412931389362222*x^2 + 333613339887189*x + 1397069806495609)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 + 1902*x^15 - 15021*x^14 + 57771*x^13 - 5041232*x^12 + 29633925*x^11 - 22759602*x^10 - 6203302885*x^9 + 27813323421*x^8 - 57748921875*x^7 + 9019520363287*x^6 - 26874825967038*x^5 + 52528838073972*x^4 - 300600540176057*x^3 + 412931389362222*x^2 + 333613339887189*x + 1397069806495609, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} + 1902 x^{15} - 15021 x^{14} + 57771 x^{13} - 5041232 x^{12} + 29633925 x^{11} - 22759602 x^{10} - 6203302885 x^{9} + 27813323421 x^{8} - 57748921875 x^{7} + 9019520363287 x^{6} - 26874825967038 x^{5} + 52528838073972 x^{4} - 300600540176057 x^{3} + 412931389362222 x^{2} + 333613339887189 x + 1397069806495609 \)

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:  \(-7339058714058696548373894733037168312034761402182635301443=-\,3^{21}\cdot 7^{12}\cdot 19^{12}\cdot 73^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1639.68$
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:  $3, 7, 19, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{5} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{49} a^{6} - \frac{3}{49} a^{5} - \frac{1}{49} a^{4} + \frac{1}{7} a^{3} - \frac{15}{49} a^{2} + \frac{11}{49} a - \frac{6}{49}$, $\frac{1}{49} a^{7} - \frac{3}{49} a^{5} - \frac{3}{49} a^{4} + \frac{13}{49} a^{3} + \frac{22}{49} a^{2} + \frac{20}{49} a - \frac{11}{49}$, $\frac{1}{49} a^{8} + \frac{2}{49} a^{5} + \frac{3}{49} a^{4} - \frac{6}{49} a^{3} + \frac{10}{49} a^{2} - \frac{6}{49} a + \frac{3}{49}$, $\frac{1}{735} a^{9} - \frac{2}{245} a^{8} + \frac{2}{245} a^{7} - \frac{1}{245} a^{6} - \frac{11}{245} a^{5} - \frac{17}{245} a^{4} - \frac{352}{735} a^{3} + \frac{103}{245} a^{2} + \frac{13}{49} a + \frac{184}{735}$, $\frac{1}{5145} a^{10} + \frac{2}{5145} a^{9} - \frac{9}{1715} a^{8} - \frac{2}{343} a^{7} + \frac{1}{1715} a^{6} - \frac{16}{343} a^{5} + \frac{16}{1029} a^{4} - \frac{1472}{5145} a^{3} + \frac{229}{1715} a^{2} - \frac{446}{5145} a + \frac{1142}{5145}$, $\frac{1}{5145} a^{11} - \frac{1}{1715} a^{9} - \frac{13}{1715} a^{8} + \frac{1}{245} a^{7} - \frac{1}{343} a^{6} + \frac{38}{735} a^{5} - \frac{15}{343} a^{4} - \frac{149}{343} a^{3} - \frac{314}{735} a^{2} - \frac{162}{1715} a + \frac{711}{1715}$, $\frac{1}{468195} a^{12} - \frac{2}{156065} a^{11} + \frac{1}{22295} a^{10} - \frac{197}{468195} a^{9} - \frac{68}{156065} a^{8} - \frac{32}{4459} a^{7} - \frac{305}{93639} a^{6} - \frac{34}{22295} a^{5} - \frac{5997}{156065} a^{4} + \frac{60384}{156065} a^{3} - \frac{9561}{22295} a^{2} + \frac{398}{2401} a - \frac{89816}{468195}$, $\frac{1}{468195} a^{13} - \frac{1}{31213} a^{11} + \frac{4}{93639} a^{10} + \frac{2}{13377} a^{9} + \frac{295}{31213} a^{8} + \frac{2339}{468195} a^{7} - \frac{1286}{156065} a^{6} - \frac{54}{156065} a^{5} + \frac{851}{66885} a^{4} - \frac{97579}{468195} a^{3} - \frac{45271}{156065} a^{2} + \frac{1772}{9555} a - \frac{143228}{468195}$, $\frac{1}{16386825} a^{14} + \frac{1}{2340975} a^{13} + \frac{1}{2340975} a^{12} - \frac{9}{156065} a^{11} + \frac{2}{66885} a^{10} - \frac{211}{334425} a^{9} + \frac{1853}{334425} a^{8} - \frac{61678}{16386825} a^{7} - \frac{19477}{2340975} a^{6} + \frac{97141}{2340975} a^{5} + \frac{44748}{780325} a^{4} + \frac{666191}{2340975} a^{3} - \frac{11313}{156065} a^{2} - \frac{59972}{180075} a - \frac{4818841}{16386825}$, $\frac{1}{21808057706039793825} a^{15} - \frac{137547231356}{7269352568679931275} a^{14} + \frac{2757435192511}{3115436815148541975} a^{13} - \frac{55833108881}{89012480432815485} a^{12} - \frac{14612731160}{504524180590857} a^{11} - \frac{39956660192066}{445062402164077425} a^{10} - \frac{238226881259758}{1038478938382847325} a^{9} + \frac{11440585103141638}{2423117522893310425} a^{8} + \frac{167968645506866381}{21808057706039793825} a^{7} + \frac{7033212752621}{1179642868287975} a^{6} + \frac{538298196978968}{239648985780657075} a^{5} - \frac{2316004260223006}{346159646127615775} a^{4} + \frac{42892390355946203}{89012480432815485} a^{3} - \frac{1168088968879145966}{3115436815148541975} a^{2} + \frac{8551734061255084204}{21808057706039793825} a - \frac{265723577357316362}{872322308241591753}$, $\frac{1}{20303301724323048051075} a^{16} + \frac{67}{4060660344864609610215} a^{15} - \frac{380641665109481}{20303301724323048051075} a^{14} + \frac{2554328610965164}{2900471674903292578725} a^{13} + \frac{2653947128592581}{2900471674903292578725} a^{12} - \frac{49360338289550879}{966823891634430859575} a^{11} - \frac{3732026550676649}{82870619282951216535} a^{10} - \frac{597958358812501229}{2255922413813672005675} a^{9} - \frac{32745516103458823531}{20303301724323048051075} a^{8} - \frac{896065528421263856}{6767767241441016017025} a^{7} - \frac{280465417219199168}{27623539760983738845} a^{6} + \frac{57097440882765018349}{2900471674903292578725} a^{5} - \frac{52758404061087924728}{2900471674903292578725} a^{4} - \frac{5522252336841823172}{12890985221792411461} a^{3} + \frac{931659234242761086892}{2255922413813672005675} a^{2} - \frac{213885229266305362522}{1353553448288203203405} a - \frac{807318759374203876561}{20303301724323048051075}$, $\frac{1}{19019339223232442693437284470452526862628592503706896875} a^{17} - \frac{14279266815696797350687860864737}{2113259913692493632604142718939169651403176944856321875} a^{16} - \frac{167089944802059994511363721235218713}{19019339223232442693437284470452526862628592503706896875} a^{15} - \frac{243180821943018170709951600623942853629012228686}{19019339223232442693437284470452526862628592503706896875} a^{14} + \frac{3621141463409454395286196297520552051974964521}{143002550550619869875468304289116743328034530103059375} a^{13} + \frac{44712202187131383876922509765817095263087089486}{388149780065968218233413968784745446176093724565446875} a^{12} - \frac{43379353186904497843639023903024655105371113429008}{905682820153925842544632593831072707744218690652709375} a^{11} + \frac{147540955024487893020521799358782041566294071092044}{6339779741077480897812428156817508954209530834568965625} a^{10} + \frac{1422664371550112544133433569571527669993217761479556}{3803867844646488538687456894090505372525718500741379375} a^{9} - \frac{30051401818587890735711012475971377613202550075317296}{3803867844646488538687456894090505372525718500741379375} a^{8} - \frac{162956739545054638711222198346482649811970122030825934}{19019339223232442693437284470452526862628592503706896875} a^{7} + \frac{1650861392008008232537582881882389712152201548963632}{301894273384641947514877531277024235914739563550903125} a^{6} - \frac{47730785871809434908360754720463686660003520079629771}{2717048460461777527633897781493218123232656071958128125} a^{5} - \frac{29797378739659942851518270033426667035179949402734}{11089993716170520520954684822421298462174106416155625} a^{4} - \frac{5884311913952650634899418407918408431702647745010212338}{19019339223232442693437284470452526862628592503706896875} a^{3} - \frac{1516176470329563628197620670009939778990323341544354079}{3803867844646488538687456894090505372525718500741379375} a^{2} - \frac{31160725568901508889028854718433856106619813362300994}{162558454899422587123395593764551511646398226527409375} a + \frac{6116591461940402995450992347201665351275234192}{56538445245197080808065439092486019061283934375}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}\times C_{36}\times C_{2052}$, which has order $13086203184$ (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:  \( -\frac{2652662074527314360213183828124}{6367392019345468774644769269135113874281834375} a^{17} + \frac{3779201463795631620470720471825381}{592167457799128596041963542029565590308210596875} a^{16} - \frac{23569732124503527454587494317309684}{592167457799128596041963542029565590308210596875} a^{15} - \frac{411802938042223940239248365632932298}{592167457799128596041963542029565590308210596875} a^{14} + \frac{6765238861278837821427109696327516874}{592167457799128596041963542029565590308210596875} a^{13} - \frac{34874155445081581258632157915931385298}{592167457799128596041963542029565590308210596875} a^{12} + \frac{69132129865874051925936263728806766454}{31166708305217294528524396948924504753063715625} a^{11} - \frac{15020229614480371209840884887506239989724}{592167457799128596041963542029565590308210596875} a^{10} + \frac{8722291233886777869373611418710075833558}{118433491559825719208392708405913118061642119375} a^{9} + \frac{307033998475957259892233354953523280689972}{118433491559825719208392708405913118061642119375} a^{8} - \frac{5554170808861049224194995477573813075444954}{197389152599709532013987847343188530102736865625} a^{7} + \frac{16119742583469708802019493434064213909891046}{197389152599709532013987847343188530102736865625} a^{6} - \frac{39950402027982884049179228972789813248592378}{10388902768405764842841465649641501584354571875} a^{5} + \frac{4085880898782979951528535231731827109424860159}{118433491559825719208392708405913118061642119375} a^{4} - \frac{1304449340978037701493534307487746570042583014}{19102176058036406323934307807405341622845503125} a^{3} + \frac{25439780472619370706476913076988911830739511428}{118433491559825719208392708405913118061642119375} a^{2} - \frac{104335137246679271558287136368077994650328305113}{197389152599709532013987847343188530102736865625} a + \frac{3042587513598312496444508340591243529818}{5280976430747960646215889440968522503125} \) (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:  \( 2508774324153.621 \) (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

\(\Q(\sqrt{-3}) \), 3.1.2545146387.1 x3, 3.3.1923769.2, 6.0.19433310393777461307.1, Deg 6 x2, 6.0.99923953464747.4, 9.3.16486873245390750975847782603.1 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
$73$73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$