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Properties

Label 18.0.53361084772...1648.7
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 $2899771392$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 28, 25284]$ (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![1331820844863488, 420721169799168, 193558495474176, 23218361290432, 6941687396868, 271774244448, 241142568884, 10024559160, 8415651069, 94202304, 177123093, -740880, 3256078, -4704, 38934, -168, 273, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 273*x^16 - 168*x^15 + 38934*x^14 - 4704*x^13 + 3256078*x^12 - 740880*x^11 + 177123093*x^10 + 94202304*x^9 + 8415651069*x^8 + 10024559160*x^7 + 241142568884*x^6 + 271774244448*x^5 + 6941687396868*x^4 + 23218361290432*x^3 + 193558495474176*x^2 + 420721169799168*x + 1331820844863488)
 
gp: K = bnfinit(x^18 + 273*x^16 - 168*x^15 + 38934*x^14 - 4704*x^13 + 3256078*x^12 - 740880*x^11 + 177123093*x^10 + 94202304*x^9 + 8415651069*x^8 + 10024559160*x^7 + 241142568884*x^6 + 271774244448*x^5 + 6941687396868*x^4 + 23218361290432*x^3 + 193558495474176*x^2 + 420721169799168*x + 1331820844863488, 1)
 

Normalized defining polynomial

\( x^{18} + 273 x^{16} - 168 x^{15} + 38934 x^{14} - 4704 x^{13} + 3256078 x^{12} - 740880 x^{11} + 177123093 x^{10} + 94202304 x^{9} + 8415651069 x^{8} + 10024559160 x^{7} + 241142568884 x^{6} + 271774244448 x^{5} + 6941687396868 x^{4} + 23218361290432 x^{3} + 193558495474176 x^{2} + 420721169799168 x + 1331820844863488 \)

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$$\chi_{4788}(1,·)$, $\chi_{4788}(2155,·)$, $\chi_{4788}(961,·)$, $\chi_{4788}(3979,·)$, $\chi_{4788}(4561,·)$, $\chi_{4788}(3649,·)$, $\chi_{4788}(31,·)$, $\chi_{4788}(1699,·)$, $\chi_{4788}(4453,·)$, $\chi_{4788}(1063,·)$, $\chi_{4788}(4225,·)$, $\chi_{4788}(2539,·)$, $\chi_{4788}(3313,·)$, $\chi_{4788}(2887,·)$, $\chi_{4788}(2995,·)$, $\chi_{4788}(2101,·)$, $\chi_{4788}(607,·)$, $\chi_{4788}(1873,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{28} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{56} a^{7} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{56} a^{8} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{112} a^{9} + \frac{1}{16} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{112} a^{10} - \frac{1}{112} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{1568} a^{11} - \frac{1}{224} a^{10} + \frac{1}{224} a^{7} + \frac{1}{224} a^{6} + \frac{5}{56} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{125440} a^{12} - \frac{67}{17920} a^{10} - \frac{1}{1120} a^{9} + \frac{11}{2560} a^{8} + \frac{3}{1120} a^{7} + \frac{319}{17920} a^{6} + \frac{11}{80} a^{5} - \frac{41}{640} a^{4} - \frac{31}{80} a^{3} + \frac{217}{640} a^{2} - \frac{17}{80} a + \frac{7}{20}$, $\frac{1}{250880} a^{13} + \frac{11}{250880} a^{11} - \frac{1}{2240} a^{10} + \frac{11}{5120} a^{9} - \frac{17}{2240} a^{8} + \frac{159}{35840} a^{7} - \frac{3}{1120} a^{6} - \frac{1247}{8960} a^{5} - \frac{21}{160} a^{4} + \frac{57}{1280} a^{3} + \frac{3}{160} a^{2} - \frac{3}{40} a$, $\frac{1}{501760} a^{14} - \frac{1}{501760} a^{13} - \frac{1}{501760} a^{12} - \frac{123}{501760} a^{11} - \frac{9}{10240} a^{10} + \frac{163}{71680} a^{9} - \frac{493}{71680} a^{8} + \frac{449}{71680} a^{7} - \frac{1}{896} a^{6} + \frac{2199}{17920} a^{5} - \frac{243}{2560} a^{4} + \frac{1023}{2560} a^{3} - \frac{281}{640} a^{2} - \frac{1}{5} a - \frac{1}{20}$, $\frac{1}{501760} a^{15} + \frac{1}{5120} a^{11} + \frac{1}{896} a^{10} - \frac{3}{896} a^{9} - \frac{401}{71680} a^{7} - \frac{15}{896} a^{6} - \frac{149}{640} a^{5} - \frac{3}{16} a^{4} - \frac{19}{2560} a^{3} - \frac{7}{64} a^{2} - \frac{13}{80} a - \frac{1}{5}$, $\frac{1}{12203373376954375745717951260549120} a^{16} + \frac{63232814406159102015625883}{435834763462656276632783973591040} a^{15} - \frac{552058663154874118064116907}{871669526925312553265567947182080} a^{14} + \frac{421042040259265191888345267}{435834763462656276632783973591040} a^{13} - \frac{272950349399827454235197033}{87166952692531255326556794718208} a^{12} + \frac{13726915776553831935715758629}{62262109066093753804683424798720} a^{11} - \frac{1085433562528581549538608443161}{871669526925312553265567947182080} a^{10} - \frac{55302758323606179269132623649}{62262109066093753804683424798720} a^{9} - \frac{37608553507532775033788556267}{7115669607553571863392391405568} a^{8} + \frac{2247357814028851765046688863}{31131054533046876902341712399360} a^{7} - \frac{71295621250531928650795725951}{6226210906609375380468342479872} a^{6} + \frac{306570944661305388088487368573}{2223646752360491207310122314240} a^{5} - \frac{11420932075755445811093987658573}{62262109066093753804683424798720} a^{4} - \frac{114096590719359734500796957653}{555911688090122801827530578560} a^{3} - \frac{48337783963964559646648783427}{555911688090122801827530578560} a^{2} - \frac{32701432462726518818367702617}{69488961011265350228441322320} a - \frac{6446802830088593276161940447}{17372240252816337557110330580}$, $\frac{1}{97277502561787279580272952576484687002785455063040} a^{17} + \frac{978184041854787}{97277502561787279580272952576484687002785455063040} a^{16} + \frac{2064384507682067717696205687593245371968639}{6948393040127662827162353755463191928770389647360} a^{15} + \frac{59144536243616101037746702559492512204149}{1389678608025532565432470751092638385754077929472} a^{14} + \frac{427204701998246141145906544252877423102751}{868549130015957853395294219432898991096298705920} a^{13} - \frac{1428174781938964033534621191864255894283239}{434274565007978926697647109716449495548149352960} a^{12} - \frac{863705121941651199974889220480819141245909047}{6948393040127662827162353755463191928770389647360} a^{11} - \frac{5679650400184026642642332295404542107416124289}{1389678608025532565432470751092638385754077929472} a^{10} + \frac{3647871866760178273505991109604715845527294867}{1985255154322189379189243930132340551077254184960} a^{9} - \frac{9307185588601503268293487560314325883521633687}{1985255154322189379189243930132340551077254184960} a^{8} + \frac{50672028803570251201840115700787812357734571}{15509805893142104524915968204158910555291048320} a^{7} + \frac{903356389841714547508740094616754863740306547}{248156894290273672398655491266542568884656773120} a^{6} + \frac{1450996846048269010822174900956890452594976331}{99262757716109468959462196506617027553862709248} a^{5} + \frac{32506030969077095155227312730611524010083776237}{496313788580547344797310982533085137769313546240} a^{4} - \frac{66032947677484336944780113746732665732172563}{4431373112326315578547419486902545872940299520} a^{3} - \frac{417875737799819904569598598149538070021337613}{2215686556163157789273709743451272936470149760} a^{2} - \frac{4078799173288663337925005945329708226529381}{276960819520394723659213717931409117058768720} a - \frac{1039752906977187089258320080364663013173910}{3462010244004934045740171474142613963234609}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{28}\times C_{25284}$, which has order $2899771392$ (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:  \( 10681224266.072006 \) (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

\(\Q(\sqrt{-133}) \), 3.3.29241.1, 3.3.1432809.2, 3.3.3969.1, 3.3.17689.2, 6.0.356625288952128.2, 6.0.17474639158654272.2, 6.0.48406202655552.5, 6.0.2663410937152.2, 9.9.2941473244627851129.8

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 ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
3Data not computed
7Data not computed
19Data not computed