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Properties

Label 18.0.33072058616...1159.1
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 $58579524$ (GRH)
Class group $[3, 18, 1084806]$ (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![18040528188928, -9636571236096, 1421970391296, -1761607827840, 455759740800, -73024540848, 59407364016, -888655572, 3893509620, -2063929, 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 - 2063929*x^9 + 3893509620*x^8 - 888655572*x^7 + 59407364016*x^6 - 73024540848*x^5 + 455759740800*x^4 - 1761607827840*x^3 + 1421970391296*x^2 - 9636571236096*x + 18040528188928)
 
gp: K = bnfinit(x^18 + 315*x^16 + 42903*x^14 + 3235470*x^12 + 145272897*x^10 - 2063929*x^9 + 3893509620*x^8 - 888655572*x^7 + 59407364016*x^6 - 73024540848*x^5 + 455759740800*x^4 - 1761607827840*x^3 + 1421970391296*x^2 - 9636571236096*x + 18040528188928, 1)
 

Normalized defining polynomial

\( x^{18} + 315 x^{16} + 42903 x^{14} + 3235470 x^{12} + 145272897 x^{10} - 2063929 x^{9} + 3893509620 x^{8} - 888655572 x^{7} + 59407364016 x^{6} - 73024540848 x^{5} + 455759740800 x^{4} - 1761607827840 x^{3} + 1421970391296 x^{2} - 9636571236096 x + 18040528188928 \)

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$$\chi_{2079}(1024,·)$, $\chi_{2079}(1,·)$, $\chi_{2079}(67,·)$, $\chi_{2079}(331,·)$, $\chi_{2079}(1748,·)$, $\chi_{2079}(2012,·)$, $\chi_{2079}(2078,·)$, $\chi_{2079}(1055,·)$, $\chi_{2079}(1319,·)$, $\chi_{2079}(1385,·)$, $\chi_{2079}(362,·)$, $\chi_{2079}(1387,·)$, $\chi_{2079}(1453,·)$, $\chi_{2079}(626,·)$, $\chi_{2079}(692,·)$, $\chi_{2079}(1717,·)$, $\chi_{2079}(694,·)$, $\chi_{2079}(760,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{7} a^{8}$, $\frac{1}{7} a^{9}$, $\frac{1}{14} a^{10} - \frac{1}{14} a^{8} - \frac{1}{14} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{28} a^{11} - \frac{1}{28} a^{9} - \frac{1}{28} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{392} a^{12} + \frac{1}{56} a^{10} - \frac{3}{56} a^{8} - \frac{1}{28} a^{6} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{784} a^{13} + \frac{1}{112} a^{11} - \frac{3}{112} a^{9} - \frac{1}{56} a^{7} - \frac{7}{16} a^{5} + \frac{7}{16} a^{4}$, $\frac{1}{167485761632} a^{14} + \frac{5233213}{83742880816} a^{13} + \frac{96244095}{167485761632} a^{12} + \frac{5330937}{11963268688} a^{11} + \frac{296781573}{23926537376} a^{10} + \frac{25489629}{11963268688} a^{9} - \frac{244864605}{11963268688} a^{8} + \frac{253359997}{5981634344} a^{7} - \frac{585893729}{23926537376} a^{6} - \frac{1532379823}{3418076768} a^{5} + \frac{499561975}{1709038384} a^{4} - \frac{40765705}{213629798} a^{3} - \frac{52569697}{427259596} a^{2} - \frac{32182597}{106814899} a + \frac{96719}{205019}$, $\frac{1}{12393946360768} a^{15} - \frac{13}{6196973180384} a^{14} + \frac{7811173499}{12393946360768} a^{13} + \frac{648347557}{6196973180384} a^{12} - \frac{1042542761}{252937680832} a^{11} - \frac{30832820121}{885281882912} a^{10} - \frac{48271137911}{885281882912} a^{9} + \frac{28630280813}{442640941456} a^{8} - \frac{107295355065}{1770563765824} a^{7} + \frac{90728040859}{1770563765824} a^{6} - \frac{60499883745}{126468840416} a^{5} - \frac{7533611845}{63234420208} a^{4} + \frac{185442564}{3952151263} a^{3} + \frac{3527391515}{7904302526} a^{2} - \frac{2058721067}{7904302526} a - \frac{51036}{205019}$, $\frac{1}{24787892721536} a^{16} - \frac{1}{669943046528} a^{14} + \frac{118975677}{442640941456} a^{13} + \frac{8063219511}{24787892721536} a^{12} - \frac{1999763625}{442640941456} a^{11} - \frac{61235099823}{1770563765824} a^{10} + \frac{8017472003}{442640941456} a^{9} + \frac{117440603191}{3541127531648} a^{8} + \frac{23410659121}{3541127531648} a^{7} - \frac{60526609405}{885281882912} a^{6} + \frac{5085257417}{126468840416} a^{5} + \frac{14927905419}{63234420208} a^{4} + \frac{1922504323}{31617210104} a^{3} + \frac{778163101}{15808605052} a^{2} - \frac{621042963}{7904302526} a + \frac{4839}{205019}$, $\frac{1}{106427404412297552076647604447863877183474734898497602816} a^{17} - \frac{3219676174027682495800442023337077406647}{182238706185441013829876035013465543122388244689208224} a^{16} + \frac{1462245756007507141166903293676168287673703}{106427404412297552076647604447863877183474734898497602816} a^{15} - \frac{197889147926207567237955797974079158094875}{91119353092720506914938017506732771561194122344604112} a^{14} - \frac{15484345747233324160260957726166212992282494454726245}{106427404412297552076647604447863877183474734898497602816} a^{13} + \frac{1091407539086345311435241247388265660357511746397}{11389919136590063364367252188341596445149265293075514} a^{12} + \frac{1000806560966565837047638413614001443279092301540097}{1085993922574464817108649024978202828402803417331608192} a^{11} - \frac{112145411813868901882931721065190959005896677331203}{26034100883634430547125147859066506160341177812744032} a^{10} + \frac{162833861289566386960697392455307982460605579376625903}{15203914916042507439521086349694839597639247842642514688} a^{9} - \frac{8777568727761827364085465800011905207213008060696895}{208272807069075444377001182872532049282729422501952256} a^{8} + \frac{69564326186156580966967172035086281376129367008934189}{1900489364505313429940135793711854949704905980330314336} a^{7} + \frac{1018132761773685592560958705105578451197633992854937}{26034100883634430547125147859066506160341177812744032} a^{6} - \frac{61196409076910583028996532906701706061962025257427707}{271498480643616204277162256244550707100700854332902048} a^{5} - \frac{84314997521097959760083649614371521689187793403151}{232447329318164558456474534455950947860189087613786} a^{4} + \frac{3051535554154500379609825976106810401712190699230625}{8484327520113006383661320507642209596896901697903189} a^{3} - \frac{87694128754953708194182093599888075441563208341345}{232447329318164558456474534455950947860189087613786} a^{2} + \frac{3207981488072847405633940753602486572177059142213841}{8484327520113006383661320507642209596896901697903189} a + \frac{2046114787281578811288409969286075329766913337}{6029136518082807450756718743994162677288714209}$

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

Class group and class number

$C_{3}\times C_{18}\times C_{1084806}$, which has order $58579524$ (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:  \( 4392158.291236831 \) (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, 9.9.3691950281939241.1

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 ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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