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Properties

Label 18.0.33072058616...1159.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 7^{15}\cdot 11^{9}$
Root discriminant $261.67$
Ramified primes $3, 7, 11$
Class number $38949012$ (GRH)
Class group $[3, 18, 721278]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18089129916928, 7312629429504, 1421970391296, 682703468160, 455759740800, -1188816048, 59407364016, -600770772, 3893509620, -6297529, 145272897, 0, 3235470, 0, 42903, 0, 315, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 315*x^16 + 42903*x^14 + 3235470*x^12 + 145272897*x^10 - 6297529*x^9 + 3893509620*x^8 - 600770772*x^7 + 59407364016*x^6 - 1188816048*x^5 + 455759740800*x^4 + 682703468160*x^3 + 1421970391296*x^2 + 7312629429504*x + 18089129916928)
 
gp: K = bnfinit(x^18 + 315*x^16 + 42903*x^14 + 3235470*x^12 + 145272897*x^10 - 6297529*x^9 + 3893509620*x^8 - 600770772*x^7 + 59407364016*x^6 - 1188816048*x^5 + 455759740800*x^4 + 682703468160*x^3 + 1421970391296*x^2 + 7312629429504*x + 18089129916928, 1)
 

Normalized defining polynomial

\( x^{18} + 315 x^{16} + 42903 x^{14} + 3235470 x^{12} + 145272897 x^{10} - 6297529 x^{9} + 3893509620 x^{8} - 600770772 x^{7} + 59407364016 x^{6} - 1188816048 x^{5} + 455759740800 x^{4} + 682703468160 x^{3} + 1421970391296 x^{2} + 7312629429504 x + 18089129916928 \)

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:  \(-33072058616555172267704055543460613736681159=-\,3^{45}\cdot 7^{15}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $261.67$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2079=3^{3}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2079}(1,·)$, $\chi_{2079}(131,·)$, $\chi_{2079}(1222,·)$, $\chi_{2079}(1550,·)$, $\chi_{2079}(529,·)$, $\chi_{2079}(857,·)$, $\chi_{2079}(1948,·)$, $\chi_{2079}(2078,·)$, $\chi_{2079}(164,·)$, $\chi_{2079}(1255,·)$, $\chi_{2079}(1385,·)$, $\chi_{2079}(1387,·)$, $\chi_{2079}(1517,·)$, $\chi_{2079}(562,·)$, $\chi_{2079}(692,·)$, $\chi_{2079}(694,·)$, $\chi_{2079}(824,·)$, $\chi_{2079}(1915,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{7} a^{8}$, $\frac{1}{7} a^{9}$, $\frac{1}{14} a^{10} - \frac{1}{14} a^{8} - \frac{1}{14} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{28} a^{11} - \frac{1}{28} a^{9} - \frac{1}{28} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{392} a^{12} + \frac{1}{56} a^{10} - \frac{3}{56} a^{8} - \frac{1}{28} a^{6} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{14896} a^{13} + \frac{9}{7448} a^{12} + \frac{1}{2128} a^{11} - \frac{11}{1064} a^{10} + \frac{11}{304} a^{9} + \frac{9}{1064} a^{8} - \frac{1}{1064} a^{7} + \frac{13}{532} a^{6} + \frac{105}{304} a^{5} - \frac{135}{304} a^{4} - \frac{65}{152} a^{3} - \frac{15}{38} a^{2} + \frac{7}{38} a$, $\frac{1}{167051817632} a^{14} - \frac{120629}{5220369301} a^{13} + \frac{22614939}{167051817632} a^{12} - \frac{1073337}{1491534086} a^{11} - \frac{678979311}{23864545376} a^{10} - \frac{9314535}{213076298} a^{9} + \frac{626041217}{11932272688} a^{8} - \frac{71797269}{1491534086} a^{7} - \frac{1536013417}{23864545376} a^{6} - \frac{595071657}{3409220768} a^{5} + \frac{41635053}{852305192} a^{4} + \frac{184286087}{852305192} a^{3} - \frac{17275764}{106538149} a^{2} - \frac{81577935}{213076298} a - \frac{2683985}{5607271}$, $\frac{1}{334103635264} a^{15} - \frac{287779}{17584401856} a^{13} - \frac{25315183}{41762954408} a^{12} + \frac{688267373}{47729090752} a^{11} + \frac{96038965}{5966136344} a^{10} - \frac{584844201}{23864545376} a^{9} - \frac{69600071}{5966136344} a^{8} + \frac{2261288079}{47729090752} a^{7} + \frac{393065873}{47729090752} a^{6} + \frac{36179091}{213076298} a^{5} + \frac{218670101}{852305192} a^{4} + \frac{347009107}{852305192} a^{3} + \frac{41120014}{106538149} a^{2} - \frac{64460351}{213076298} a - \frac{2333503}{5607271}$, $\frac{1}{668207270528} a^{16} - \frac{1}{668207270528} a^{14} + \frac{621833}{41762954408} a^{13} + \frac{243369603}{668207270528} a^{12} + \frac{24857603}{5966136344} a^{11} - \frac{1704388893}{47729090752} a^{10} + \frac{26415565}{852305192} a^{9} + \frac{1266124735}{95458181504} a^{8} - \frac{796806527}{95458181504} a^{7} + \frac{759352961}{11932272688} a^{6} + \frac{734136235}{1704610384} a^{5} + \frac{353378067}{852305192} a^{4} + \frac{8533553}{426152596} a^{3} - \frac{7084581}{426152596} a^{2} + \frac{63035439}{213076298} a - \frac{2236824}{5607271}$, $\frac{1}{199815596251321475435675574072500599530540343296982141634816} a^{17} + \frac{56568972992649884773664060259171389242012168327}{99907798125660737717837787036250299765270171648491070817408} a^{16} + \frac{138032431708608798461773124445470662639799326787}{199815596251321475435675574072500599530540343296982141634816} a^{15} - \frac{140116277541223359959727587056753737349329813239}{99907798125660737717837787036250299765270171648491070817408} a^{14} - \frac{5343358860079079266670996697714897741407612430986093889}{199815596251321475435675574072500599530540343296982141634816} a^{13} - \frac{42691379802208778975372630228771800911400180088230177755}{99907798125660737717837787036250299765270171648491070817408} a^{12} - \frac{94284294606711179526899613277690260229631921201198965619}{14272542589380105388262541005178614252181453092641581545344} a^{11} - \frac{24326860024964640744650992916529138395474000970426276213}{1019467327812864670590181500369901018012960935188684396096} a^{10} - \frac{1873185221550339777925673274542732760448324471010282430249}{28545085178760210776525082010357228504362906185283163090688} a^{9} + \frac{18381896178302447451073295315242933730733927523471928243}{28545085178760210776525082010357228504362906185283163090688} a^{8} - \frac{252878485028898595816399630381009809888930268703440816471}{14272542589380105388262541005178614252181453092641581545344} a^{7} - \frac{51137966024096109609560685778202209880740370001616881997}{7136271294690052694131270502589307126090726546320790772672} a^{6} + \frac{80606518513314723117842181845746917067067226693406743359}{254866831953216167647545375092475254503240233797171099024} a^{5} - \frac{29665560035777073182494059274715369758017514228818956331}{127433415976608083823772687546237627251620116898585549512} a^{4} + \frac{5392466053120480651756091072517452989338279415603321377}{31858353994152020955943171886559406812905029224646387378} a^{3} + \frac{25712175460263571379724710331908560269309408369350595603}{63716707988304041911886343773118813625810058449292774756} a^{2} + \frac{6189642423059692525312041217525009024669871304044108785}{15929176997076010477971585943279703406452514612323193689} a + \frac{11621089571873629626868754642112943963667377690966709}{838377736688211077787978207541037021392237611174904931}$

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

Class group and class number

$C_{3}\times C_{18}\times C_{721278}$, which has order $38949012$ (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:  \( 10392888.21418944 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-231}) \), \(\Q(\zeta_{9})^+\), 6.0.8985939039.8, 9.9.3691950281939241.2

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
7Data not computed
11Data not computed