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Properties

Label 18.0.14752461749...9283.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 13^{12}\cdot 41^{12}\cdot 103^{12}$
Root discriminant $2501.90$
Ramified primes $3, 13, 41, 103$
Class number $26737275663$ (GRH)
Class group $[3, 63, 6867, 20601]$ (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![762516782753267677, -1294085044187005725, 1031399763020665000, -510173162432207521, 174370588263468377, -43369484085540759, 8043217290890512, -1118467036430126, 115084114183121, -8399255282261, 387409454753, -6602474036, -368833646, 23671983, -337117, 437, -440, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 440*x^16 + 437*x^15 - 337117*x^14 + 23671983*x^13 - 368833646*x^12 - 6602474036*x^11 + 387409454753*x^10 - 8399255282261*x^9 + 115084114183121*x^8 - 1118467036430126*x^7 + 8043217290890512*x^6 - 43369484085540759*x^5 + 174370588263468377*x^4 - 510173162432207521*x^3 + 1031399763020665000*x^2 - 1294085044187005725*x + 762516782753267677)
 
gp: K = bnfinit(x^18 - 3*x^17 - 440*x^16 + 437*x^15 - 337117*x^14 + 23671983*x^13 - 368833646*x^12 - 6602474036*x^11 + 387409454753*x^10 - 8399255282261*x^9 + 115084114183121*x^8 - 1118467036430126*x^7 + 8043217290890512*x^6 - 43369484085540759*x^5 + 174370588263468377*x^4 - 510173162432207521*x^3 + 1031399763020665000*x^2 - 1294085044187005725*x + 762516782753267677, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 440 x^{16} + 437 x^{15} - 337117 x^{14} + 23671983 x^{13} - 368833646 x^{12} - 6602474036 x^{11} + 387409454753 x^{10} - 8399255282261 x^{9} + 115084114183121 x^{8} - 1118467036430126 x^{7} + 8043217290890512 x^{6} - 43369484085540759 x^{5} + 174370588263468377 x^{4} - 510173162432207521 x^{3} + 1031399763020665000 x^{2} - 1294085044187005725 x + 762516782753267677 \)

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:  \(-14752461749278147190952026748722919230164585269517740875639283=-\,3^{9}\cdot 13^{12}\cdot 41^{12}\cdot 103^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2501.90$
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:  $3, 13, 41, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} + \frac{5}{19} a^{12} + \frac{1}{19} a^{9} - \frac{8}{19} a^{8} - \frac{2}{19} a^{7} + \frac{3}{19} a^{5} - \frac{2}{19} a^{4} - \frac{6}{19} a^{3} + \frac{5}{19} a^{2} + \frac{3}{19} a$, $\frac{1}{57} a^{14} + \frac{1}{57} a^{13} - \frac{1}{57} a^{12} - \frac{1}{3} a^{11} + \frac{1}{57} a^{10} + \frac{26}{57} a^{9} - \frac{9}{19} a^{8} + \frac{9}{19} a^{7} + \frac{1}{19} a^{6} + \frac{5}{57} a^{5} - \frac{17}{57} a^{4} - \frac{28}{57} a^{3} + \frac{2}{57} a^{2} + \frac{7}{57} a + \frac{1}{3}$, $\frac{1}{65379} a^{15} + \frac{349}{65379} a^{14} + \frac{623}{65379} a^{13} - \frac{1153}{65379} a^{12} - \frac{3077}{65379} a^{11} - \frac{22540}{65379} a^{10} - \frac{986}{21793} a^{9} - \frac{7488}{21793} a^{8} + \frac{5172}{21793} a^{7} - \frac{20782}{65379} a^{6} + \frac{382}{2109} a^{5} - \frac{2962}{65379} a^{4} - \frac{18808}{65379} a^{3} - \frac{4700}{65379} a^{2} + \frac{4480}{65379} a + \frac{509}{1147}$, $\frac{1}{1058309396626894047} a^{16} + \frac{8058525052001}{1058309396626894047} a^{15} - \frac{1229542453685258}{1058309396626894047} a^{14} - \frac{512421139847887}{28602956665591731} a^{13} + \frac{13717705291145225}{28602956665591731} a^{12} - \frac{171699302111994499}{1058309396626894047} a^{11} + \frac{245540735637365}{18566831519770071} a^{10} - \frac{4148389988222779}{55700494559310213} a^{9} + \frac{39329143563282015}{117589932958543783} a^{8} - \frac{143521169268828181}{1058309396626894047} a^{7} - \frac{27051680167615862}{117589932958543783} a^{6} + \frac{77028853855123502}{1058309396626894047} a^{5} - \frac{68022671602186126}{1058309396626894047} a^{4} + \frac{506913334336399853}{1058309396626894047} a^{3} - \frac{99192566245596074}{1058309396626894047} a^{2} - \frac{361030856085367963}{1058309396626894047} a - \frac{20692516822848701}{55700494559310213}$, $\frac{1}{146081551774316752713565563312177516777897806609841541909698849402356169} a^{17} + \frac{339543539661526404639062237410993356779934052411843}{854278080551559957389272300071213548408759102981529484852040054984539} a^{16} + \frac{199660031959889803483451427819277461121926986735923577128121681965}{48693850591438917571188521104059172259299268869947180636566283134118723} a^{15} - \frac{18839818958729234157010539697535029145906131029357612084458338952830}{2562834241654679872167816900213640645226277308944588454556120164953617} a^{14} + \frac{92672325622811428776862893831012783554561248318652531284125551087614}{4712308121752153313340824622973468283157993761607791674506414496850199} a^{13} - \frac{25846611354504704605192353037660764588211517194140435476311247612602235}{146081551774316752713565563312177516777897806609841541909698849402356169} a^{12} + \frac{66265011333191137280378146896154553507175175828753010430578550252482984}{146081551774316752713565563312177516777897806609841541909698849402356169} a^{11} + \frac{49892828080275627860848659160574914931598042390583774935412571646865811}{146081551774316752713565563312177516777897806609841541909698849402356169} a^{10} - \frac{14616324188992892124506001490937587768437217330055763913878832013958949}{146081551774316752713565563312177516777897806609841541909698849402356169} a^{9} - \frac{46552524084958205787645177186411128588478932040074565521194265286878650}{146081551774316752713565563312177516777897806609841541909698849402356169} a^{8} + \frac{17678150827863516341268254584058348228415914766817054417160321805102773}{146081551774316752713565563312177516777897806609841541909698849402356169} a^{7} - \frac{4283291874098723059319207964341055974217827652121041603397518701242318}{146081551774316752713565563312177516777897806609841541909698849402356169} a^{6} - \frac{55858837260935869243120381653018563761773485507139552463893486154973267}{146081551774316752713565563312177516777897806609841541909698849402356169} a^{5} + \frac{65777661820241601179257389895791631054293573449771079873430197345064534}{146081551774316752713565563312177516777897806609841541909698849402356169} a^{4} + \frac{425256949201095146123208530849139328583045354291167794502447972774145}{16231283530479639190396173701353057419766422956649060212188761044706241} a^{3} + \frac{775470654817533709754362623012946044551832499628212106350920408563309}{2562834241654679872167816900213640645226277308944588454556120164953617} a^{2} - \frac{84161717589581562911392595430760235395929847260548571764637454644928}{523589791305794812593424958108163142573110417956421297167379388538911} a + \frac{33442752833175035627928109501873944115252129037255065891470512511253}{105321955136493693376759598638916738844915505847037881694087130066587}$

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

Class group and class number

$C_{3}\times C_{63}\times C_{6867}\times C_{20601}$, which has order $26737275663$ (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{1899341481477423821130892358293274309946857621945146472}{41254321314407442167061723612588962659671789497272392518977364982309} a^{17} - \frac{6188553004047535049897450031164844607243556228748397205}{41254321314407442167061723612588962659671789497272392518977364982309} a^{16} + \frac{795292857985600553435760432370470102934733672283003489122}{41254321314407442167061723612588962659671789497272392518977364982309} a^{15} + \frac{4131140755688766847959019127345250218906797565311656835477}{41254321314407442167061723612588962659671789497272392518977364982309} a^{14} + \frac{11697496067301279274816980130301258045820866747136969835123}{723760023059779687141433747589280046660908587671445482789076578637} a^{13} - \frac{4531230681094691775319316719289177363032439582041684005090156}{4583813479378604685229080401398773628852421055252488057664151664701} a^{12} + \frac{14385860603350873861391118827345291486751654969712798916426209}{1330784558529272327969733019760934279344251274105561048999269838139} a^{11} + \frac{15299059013578540923858387981668527365485719344430932482431046222}{41254321314407442167061723612588962659671789497272392518977364982309} a^{10} - \frac{639879918037330076084645169774400712277309399095105418028205356439}{41254321314407442167061723612588962659671789497272392518977364982309} a^{9} + \frac{11963274721946024650190276424621632298959037593408311736302783517537}{41254321314407442167061723612588962659671789497272392518977364982309} a^{8} - \frac{144191463178787570582575791776721157097587311236928741594138855811292}{41254321314407442167061723612588962659671789497272392518977364982309} a^{7} + \frac{1229663105154510427301639740338088146970543407320371372511697216541385}{41254321314407442167061723612588962659671789497272392518977364982309} a^{6} - \frac{851378550895654341881851337470588632879000524831553183350933071899128}{4583813479378604685229080401398773628852421055252488057664151664701} a^{5} + \frac{125522644148221376613505998969427390735608625541072287785030475551634}{147864950947696925329970335528992697704916808233951227666585537571} a^{4} - \frac{115209260790568831718238242900225480976099246337881553341246970664037292}{41254321314407442167061723612588962659671789497272392518977364982309} a^{3} + \frac{260010327050821170290540230648937411815222661817136122756170855676995980}{41254321314407442167061723612588962659671789497272392518977364982309} a^{2} - \frac{362660498055500359868997262527253410813579625502642884896563955277982287}{41254321314407442167061723612588962659671789497272392518977364982309} a + \frac{171082321320559921845572500459705539296734708706863716225122402238485}{29743562591497795361976729352984111506612681685127896552975749807} \) (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:  \( 54845796471897.65 \) (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

\(\Q(\sqrt{-3}) \), Deg 3 x3, 3.3.1792921.1, 6.0.245257049382871690827.1, 6.0.136791888422787.1 x2, 6.0.86793274230507.3, Deg 9 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
41Data not computed
$103$103.9.6.1$x^{9} + 8961 x^{6} + 26755898 x^{3} + 26650518803$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
103.9.6.1$x^{9} + 8961 x^{6} + 26755898 x^{3} + 26650518803$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$