↑

Properties

Label 18.0.67943190949...2336.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 23^{9}\cdot 853^{15}$
Root discriminant $2108.67$
Ramified primes $2, 23, 853$
Class number $16205712768$ (GRH)
Class group $[2, 2, 42, 294, 328104]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22113695846279691, 26421806277378576, 9961068503950641, 512206086889224, -391238980093323, -53767855256436, 16704551289024, 3937918314038, 77131235840, -51385464532, -1783988569, 468459422, 1101203, -3015554, 91669, 10918, -404, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 404*x^16 + 10918*x^15 + 91669*x^14 - 3015554*x^13 + 1101203*x^12 + 468459422*x^11 - 1783988569*x^10 - 51385464532*x^9 + 77131235840*x^8 + 3937918314038*x^7 + 16704551289024*x^6 - 53767855256436*x^5 - 391238980093323*x^4 + 512206086889224*x^3 + 9961068503950641*x^2 + 26421806277378576*x + 22113695846279691)
 
gp: K = bnfinit(x^18 - 8*x^17 - 404*x^16 + 10918*x^15 + 91669*x^14 - 3015554*x^13 + 1101203*x^12 + 468459422*x^11 - 1783988569*x^10 - 51385464532*x^9 + 77131235840*x^8 + 3937918314038*x^7 + 16704551289024*x^6 - 53767855256436*x^5 - 391238980093323*x^4 + 512206086889224*x^3 + 9961068503950641*x^2 + 26421806277378576*x + 22113695846279691, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 404 x^{16} + 10918 x^{15} + 91669 x^{14} - 3015554 x^{13} + 1101203 x^{12} + 468459422 x^{11} - 1783988569 x^{10} - 51385464532 x^{9} + 77131235840 x^{8} + 3937918314038 x^{7} + 16704551289024 x^{6} - 53767855256436 x^{5} - 391238980093323 x^{4} + 512206086889224 x^{3} + 9961068503950641 x^{2} + 26421806277378576 x + 22113695846279691 \)

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:  \(-679431909497141409089207993258291620226476010831347538702336=-\,2^{12}\cdot 23^{9}\cdot 853^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2108.67$
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:  $2, 23, 853$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{7} - \frac{2}{27} a^{5} - \frac{1}{9} a^{4} + \frac{1}{27} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{27} a^{8} + \frac{1}{27} a^{6} - \frac{1}{9} a^{5} + \frac{4}{27} a^{4} + \frac{1}{9} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{162} a^{9} - \frac{1}{54} a^{7} - \frac{1}{18} a^{6} + \frac{1}{54} a^{5} + \frac{1}{9} a^{4} - \frac{5}{81} a^{3} - \frac{1}{18} a^{2} + \frac{1}{18} a - \frac{1}{2}$, $\frac{1}{1458} a^{10} - \frac{1}{486} a^{9} + \frac{7}{486} a^{8} - \frac{4}{81} a^{6} + \frac{23}{162} a^{5} - \frac{101}{729} a^{4} + \frac{49}{486} a^{3} + \frac{14}{81} a^{2} - \frac{11}{27} a + \frac{1}{6}$, $\frac{1}{1458} a^{11} + \frac{1}{486} a^{9} + \frac{1}{162} a^{8} + \frac{1}{162} a^{7} + \frac{1}{81} a^{6} - \frac{20}{729} a^{5} - \frac{1}{54} a^{4} + \frac{7}{54} a^{3} - \frac{1}{6} a^{2} - \frac{4}{9} a$, $\frac{1}{100602} a^{12} + \frac{5}{33534} a^{11} - \frac{5}{50301} a^{10} + \frac{31}{33534} a^{9} + \frac{83}{33534} a^{8} - \frac{31}{11178} a^{7} - \frac{767}{50301} a^{6} - \frac{1489}{16767} a^{5} + \frac{8360}{50301} a^{4} + \frac{137}{33534} a^{3} + \frac{581}{11178} a^{2} - \frac{28}{1863} a + \frac{29}{414}$, $\frac{1}{301806} a^{13} - \frac{14}{150903} a^{11} - \frac{11}{100602} a^{10} + \frac{101}{100602} a^{9} + \frac{359}{33534} a^{8} + \frac{2257}{150903} a^{7} - \frac{17}{729} a^{6} - \frac{24409}{150903} a^{5} - \frac{1700}{50301} a^{4} - \frac{3991}{33534} a^{3} + \frac{2717}{5589} a^{2} + \frac{107}{1242} a - \frac{11}{46}$, $\frac{1}{301806} a^{14} - \frac{1}{301806} a^{12} - \frac{7}{50301} a^{11} + \frac{11}{100602} a^{10} + \frac{43}{16767} a^{9} - \frac{3667}{301806} a^{8} + \frac{451}{33534} a^{7} - \frac{4132}{150903} a^{6} - \frac{5609}{50301} a^{5} + \frac{5183}{33534} a^{4} - \frac{227}{11178} a^{3} - \frac{70}{621} a^{2} + \frac{29}{414} a - \frac{17}{46}$, $\frac{1}{10555061238} a^{15} + \frac{2224}{1759176873} a^{14} + \frac{16303}{10555061238} a^{13} + \frac{13859}{3518353746} a^{12} - \frac{1468651}{5277530619} a^{11} + \frac{922229}{3518353746} a^{10} + \frac{8817587}{10555061238} a^{9} + \frac{47056417}{3518353746} a^{8} + \frac{43996073}{10555061238} a^{7} + \frac{91383962}{1759176873} a^{6} + \frac{397791220}{5277530619} a^{5} + \frac{51326461}{3518353746} a^{4} - \frac{36367264}{586392291} a^{3} + \frac{75541873}{195464097} a^{2} - \frac{264721}{1888542} a + \frac{70547}{804379}$, $\frac{1}{474977755710} a^{16} - \frac{104617}{94995551142} a^{14} + \frac{68518}{79162959285} a^{13} + \frac{2211913}{474977755710} a^{12} - \frac{31766651}{158325918570} a^{11} - \frac{69813122}{237488877855} a^{10} + \frac{1083667}{1147289265} a^{9} + \frac{3684160537}{237488877855} a^{8} - \frac{220417123}{79162959285} a^{7} + \frac{323060327}{10325603385} a^{6} + \frac{15764911139}{158325918570} a^{5} - \frac{2182833931}{17591768730} a^{4} - \frac{2463964127}{17591768730} a^{3} + \frac{2760300877}{5863922910} a^{2} - \frac{272698489}{651546990} a + \frac{1708577}{4021895}$, $\frac{1}{654789314538742333045539649419114355998715468533387639786439743387664796893277213584350} a^{17} - \frac{23515462530399829338299342816395605372809423637834466500104282402561986289}{109131552423123722174256608236519059333119244755564606631073290564610799482212868930725} a^{16} + \frac{1791757619488102349858005619262756000760138254181926049711769102779809604303}{65478931453874233304553964941911435599871546853338763978643974338766479689327721358435} a^{15} - \frac{158308461873430373332383588834748482709426463188549971754516587829793410933875427}{109131552423123722174256608236519059333119244755564606631073290564610799482212868930725} a^{14} - \frac{620795072728074215118224117323659672380031993365098142103017248681370773110388619}{654789314538742333045539649419114355998715468533387639786439743387664796893277213584350} a^{13} - \frac{22830788061070688912826860911247770124728374975141556237193616265381216342174861}{4850291218805498763300293699400847081471966433580649183603257358427146643653905285810} a^{12} + \frac{123140888150398422856702294301733200931862798721096014291467022210132259390638799213}{654789314538742333045539649419114355998715468533387639786439743387664796893277213584350} a^{11} - \frac{28073045373142860371490005324186546280822588322295977329083916061743919870813518511}{109131552423123722174256608236519059333119244755564606631073290564610799482212868930725} a^{10} + \frac{124501080106066441805211049868310617379350094807135486322855845462988669080201646397}{654789314538742333045539649419114355998715468533387639786439743387664796893277213584350} a^{9} + \frac{8402284507207544985543877570078107725015155663805282305113930318564365895485441567}{805398910871761787263886407649587153749957525871325510192422808594913649315224124950} a^{8} + \frac{444211149011668983252799241756200308938463875791481284215392130720254699884729413997}{327394657269371166522769824709557177999357734266693819893219871693832398446638606792175} a^{7} - \frac{2866774457422966085397100009509302873789313213603060976919551970377861449487851457109}{72754368282082481449504405491012706222079496503709737754048860376407199654808579287150} a^{6} - \frac{1956223669919173242707241867907672125327019275480718389520623570495251548374340924681}{14550873656416496289900881098202541244415899300741947550809772075281439930961715857430} a^{5} - \frac{659821783392946491091752510727921875196942774727699043580475236818505513263554741629}{12125728047013746908250734248502117703679916083951622959008143396067866609134763214525} a^{4} - \frac{133077319411889598844303366791384821636636067581245576246180990295779487119832555549}{2694606232669721535166829833000470600817759129767027324224031865792859246474391825450} a^{3} + \frac{306612732906126185280692723034808250855475978567210092265782892461671578064521245071}{1347303116334860767583414916500235300408879564883513662112015932896429623237195912725} a^{2} - \frac{12368919720745362520882093507210095936410050406188138480811494465608775635415047562}{29940069251885794835186998144449673342419545886300303602489242953253991627493242505} a - \frac{3390830322441874881523715907502964623863966052460754788439334302970844488051779299}{16633371806603219352881665635805374079121969936833502001382912751807773126385134725}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{42}\times C_{294}\times C_{328104}$, which has order $16205712768$ (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:  \( 2283911350493462.0 \) (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

\(\Q(\sqrt{-19619}) \), 3.1.78476.2 x3, 3.3.727609.1, 6.0.120823269658544.1, Deg 6, Deg 6 x2, 9.3.255862536056690165995365056.1 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 R ${\href{/LocalNumberField/3.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
853Data not computed