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Properties

Label 18.0.53361084772...1648.9
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 7^{15}\cdot 19^{15}$
Root discriminant $509.41$
Ramified primes $2, 3, 7, 19$
Class number $1926043392$ (GRH)
Class group $[2, 2, 2, 2, 28, 4299204]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43904083722291521, 3316967769886236, 3925005531309753, 125502023684818, 58784088299895, -127425006738, 341322297569, 4324050780, 17029175391, 223754298, 167764281, -1975392, 1534972, 17328, 25992, -114, 57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 57*x^16 - 114*x^15 + 25992*x^14 + 17328*x^13 + 1534972*x^12 - 1975392*x^11 + 167764281*x^10 + 223754298*x^9 + 17029175391*x^8 + 4324050780*x^7 + 341322297569*x^6 - 127425006738*x^5 + 58784088299895*x^4 + 125502023684818*x^3 + 3925005531309753*x^2 + 3316967769886236*x + 43904083722291521)
 
gp: K = bnfinit(x^18 + 57*x^16 - 114*x^15 + 25992*x^14 + 17328*x^13 + 1534972*x^12 - 1975392*x^11 + 167764281*x^10 + 223754298*x^9 + 17029175391*x^8 + 4324050780*x^7 + 341322297569*x^6 - 127425006738*x^5 + 58784088299895*x^4 + 125502023684818*x^3 + 3925005531309753*x^2 + 3316967769886236*x + 43904083722291521, 1)
 

Normalized defining polynomial

\( x^{18} + 57 x^{16} - 114 x^{15} + 25992 x^{14} + 17328 x^{13} + 1534972 x^{12} - 1975392 x^{11} + 167764281 x^{10} + 223754298 x^{9} + 17029175391 x^{8} + 4324050780 x^{7} + 341322297569 x^{6} - 127425006738 x^{5} + 58784088299895 x^{4} + 125502023684818 x^{3} + 3925005531309753 x^{2} + 3316967769886236 x + 43904083722291521 \)

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:  \(-5336108477246594974427370451986506423645848731648=-\,2^{18}\cdot 3^{24}\cdot 7^{15}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $509.41$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(4483,·)$, $\chi_{4788}(2053,·)$, $\chi_{4788}(961,·)$, $\chi_{4788}(3751,·)$, $\chi_{4788}(2383,·)$, $\chi_{4788}(277,·)$, $\chi_{4788}(3799,·)$, $\chi_{4788}(1369,·)$, $\chi_{4788}(31,·)$, $\chi_{4788}(1699,·)$, $\chi_{4788}(4135,·)$, $\chi_{4788}(4225,·)$, $\chi_{4788}(2857,·)$, $\chi_{4788}(1063,·)$, $\chi_{4788}(3697,·)$, $\chi_{4788}(1399,·)$, $\chi_{4788}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{19} a^{3}$, $\frac{1}{19} a^{4}$, $\frac{1}{19} a^{5}$, $\frac{1}{361} a^{6}$, $\frac{1}{361} a^{7}$, $\frac{1}{13718} a^{8} - \frac{1}{361} a^{5} - \frac{1}{38} a^{4} + \frac{1}{38} a^{2} - \frac{1}{2}$, $\frac{1}{13718} a^{9} - \frac{1}{38} a^{5} - \frac{1}{38} a^{3} - \frac{1}{2} a$, $\frac{1}{13718} a^{10} - \frac{1}{722} a^{6} - \frac{1}{38} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{260642} a^{11} - \frac{1}{722} a^{7} - \frac{3}{722} a^{5} - \frac{1}{38} a^{3} + \frac{1}{19} a^{2}$, $\frac{1}{1303210} a^{12} + \frac{1}{651605} a^{11} + \frac{1}{34295} a^{10} + \frac{1}{1805} a^{7} - \frac{1}{722} a^{6} + \frac{16}{1805} a^{5} + \frac{23}{190} a^{2} + \frac{1}{10}$, $\frac{1}{24760990} a^{13} + \frac{1}{1303210} a^{11} - \frac{2}{651605} a^{10} + \frac{1}{34295} a^{8} + \frac{3}{34295} a^{7} - \frac{1}{1805} a^{6} + \frac{69}{3610} a^{5} - \frac{8}{361} a^{4} + \frac{1}{95} a^{3} - \frac{17}{95} a^{2} + \frac{1}{190} a + \frac{1}{5}$, $\frac{1}{24760990} a^{14} - \frac{1}{1303210} a^{11} - \frac{1}{34295} a^{10} + \frac{1}{34295} a^{9} + \frac{1}{68590} a^{8} + \frac{1}{3610} a^{7} + \frac{2}{1805} a^{6} + \frac{73}{3610} a^{5} - \frac{3}{190} a^{4} + \frac{1}{190} a^{3} - \frac{17}{190} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{24760990} a^{15} - \frac{1}{1303210} a^{11} - \frac{1}{68590} a^{10} + \frac{1}{68590} a^{9} - \frac{1}{68590} a^{8} + \frac{1}{3610} a^{7} + \frac{3}{3610} a^{6} - \frac{9}{361} a^{5} - \frac{2}{95} a^{4} - \frac{1}{95} a^{3} + \frac{8}{95} a^{2} + \frac{2}{5} a + \frac{1}{10}$, $\frac{1}{54677502831703041743439113820020} a^{16} - \frac{4991741753455884341668}{719440826732934759782093602895} a^{15} + \frac{1458818082252080522061}{143888165346586951956418720579} a^{14} - \frac{18535047172183602013267}{1438881653465869519564187205790} a^{13} - \frac{3166166756202428647879}{15146122668061784416465128482} a^{12} + \frac{9669312747798433269908}{7573061334030892208232564241} a^{11} + \frac{405984225676889369308121}{75730613340308922082325642410} a^{10} + \frac{59260563280268209520437}{3985821754753101162227665390} a^{9} - \frac{109993114689156411519}{1594328701901240464891066156} a^{8} + \frac{4933244914738447725079479}{3985821754753101162227665390} a^{7} + \frac{271100370346713952974431}{209780092355426376959350810} a^{6} + \frac{1714555200626046204415849}{209780092355426376959350810} a^{5} + \frac{26151409634601033122771}{2046635047370013433749764} a^{4} - \frac{19752251432445822034253}{1104105749239086194522899} a^{3} - \frac{306925922424708575170347}{2208211498478172389045798} a^{2} + \frac{335536896990680096635872}{1104105749239086194522899} a - \frac{469648154441904604189131}{1162216578146406520550420}$, $\frac{1}{6509068847593757819559489351810133164907123940} a^{17} + \frac{50801768932319}{6509068847593757819559489351810133164907123940} a^{16} + \frac{2752355106117754751869027294597675779}{171291285462993626830512877679214030655450630} a^{15} + \frac{1012849901353884070842189178042559633}{171291285462993626830512877679214030655450630} a^{14} + \frac{1402551969655864813450106878596936584}{85645642731496813415256438839607015327725315} a^{13} - \frac{2959654119752442781270192264655846627}{9015330813841769833184888298906001613444770} a^{12} - \frac{8549567758902596433895762020278659241}{4507665406920884916592444149453000806722385} a^{11} + \frac{85213083943504912150397301553666367191}{4507665406920884916592444149453000806722385} a^{10} + \frac{2481215418647123817077863609949133175}{189796438186142522803892385240126349756732} a^{9} - \frac{6270137081603822476443081225370498671}{948982190930712614019461926200631748783660} a^{8} - \frac{506277799601721088414551533816214271483}{474491095465356307009730963100315874391830} a^{7} + \frac{2855331360338099432094063548641950493}{2497321555080822668472268226843767759957} a^{6} + \frac{525735257533360186118825415148445057359}{49946431101616453369445364536875355199140} a^{5} - \frac{1075569947186600483075683576356079120527}{49946431101616453369445364536875355199140} a^{4} - \frac{5827856484773424451505253928430252814}{657189882916005965387439007064149410515} a^{3} + \frac{224483499921455968357157404700452326717}{1314379765832011930774878014128298821030} a^{2} + \frac{487130117957186856128610255670384525717}{2628759531664023861549756028256597642060} a - \frac{50223073138183842459010411809903109181}{138355764824422308502618738329294612740}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{28}\times C_{4299204}$, which has order $1926043392$ (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:  \( 42198260.232521206 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-133}) \), 3.3.29241.2, 3.3.1432809.2, \(\Q(\zeta_{7})^+\), 3.3.1432809.1, 6.0.356625288952128.1, 6.0.17474639158654272.2, 6.0.7377869632.2, 6.0.17474639158654272.4, 9.9.2941473244627851129.4

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ 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 ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
7Data not computed
19Data not computed