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Properties

Label 18.0.12520069450...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 5^{12}\cdot 107^{12}$
Root discriminant $783.99$
Ramified primes $2, 3, 5, 107$
Class number $972$ (GRH)
Class group $[3, 3, 3, 6, 6]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51961428, -202915692, 468467820, -805226586, 1079608554, -810558162, 272888029, -4698873, 3378711, -812226, -233499, 131103, -30962, 2259, -489, -18, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 18*x^15 - 489*x^14 + 2259*x^13 - 30962*x^12 + 131103*x^11 - 233499*x^10 - 812226*x^9 + 3378711*x^8 - 4698873*x^7 + 272888029*x^6 - 810558162*x^5 + 1079608554*x^4 - 805226586*x^3 + 468467820*x^2 - 202915692*x + 51961428)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 18*x^15 - 489*x^14 + 2259*x^13 - 30962*x^12 + 131103*x^11 - 233499*x^10 - 812226*x^9 + 3378711*x^8 - 4698873*x^7 + 272888029*x^6 - 810558162*x^5 + 1079608554*x^4 - 805226586*x^3 + 468467820*x^2 - 202915692*x + 51961428, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 18 x^{15} - 489 x^{14} + 2259 x^{13} - 30962 x^{12} + 131103 x^{11} - 233499 x^{10} - 812226 x^{9} + 3378711 x^{8} - 4698873 x^{7} + 272888029 x^{6} - 810558162 x^{5} + 1079608554 x^{4} - 805226586 x^{3} + 468467820 x^{2} - 202915692 x + 51961428 \)

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:  \(-12520069450520138163679370498940927487923000000000000=-\,2^{12}\cdot 3^{33}\cdot 5^{12}\cdot 107^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $783.99$
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:  $2, 3, 5, 107$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{6} + \frac{1}{6} a^{3}$, $\frac{1}{114} a^{13} - \frac{7}{114} a^{12} - \frac{5}{57} a^{11} + \frac{3}{38} a^{10} + \frac{11}{114} a^{9} + \frac{4}{19} a^{8} + \frac{35}{114} a^{7} - \frac{53}{114} a^{6} - \frac{13}{57} a^{5} + \frac{5}{38} a^{4} + \frac{49}{114} a^{3} + \frac{2}{19} a^{2} + \frac{9}{19} a$, $\frac{1}{114} a^{14} - \frac{1}{57} a^{12} + \frac{5}{38} a^{11} - \frac{1}{57} a^{10} + \frac{1}{19} a^{9} - \frac{25}{114} a^{8} - \frac{6}{19} a^{7} + \frac{1}{57} a^{6} - \frac{5}{38} a^{5} + \frac{1}{57} a^{4} - \frac{1}{19} a^{3} + \frac{4}{19} a^{2} + \frac{6}{19} a$, $\frac{1}{11826997830} a^{15} - \frac{1035493}{394233261} a^{14} + \frac{2011717}{11826997830} a^{13} - \frac{6043808}{1971166305} a^{12} - \frac{177982739}{1182699783} a^{11} + \frac{59192003}{3942332610} a^{10} + \frac{661997149}{5913498915} a^{9} - \frac{203322298}{1971166305} a^{8} + \frac{3895168841}{11826997830} a^{7} - \frac{144268411}{394233261} a^{6} + \frac{1814825524}{5913498915} a^{5} + \frac{51221933}{207491190} a^{4} + \frac{696098569}{3942332610} a^{3} - \frac{144442034}{394233261} a^{2} - \frac{157694586}{657055435} a + \frac{15804568}{34581865}$, $\frac{1}{5130295217490940443810} a^{16} + \frac{146762103661}{5130295217490940443810} a^{15} - \frac{1966041892805403184}{2565147608745470221905} a^{14} - \frac{12792005943130201361}{5130295217490940443810} a^{13} - \frac{155310983279622288709}{2565147608745470221905} a^{12} - \frac{367434159888494815123}{2565147608745470221905} a^{11} - \frac{306431755230764762413}{5130295217490940443810} a^{10} - \frac{46478687248922084890}{513029521749094044381} a^{9} + \frac{476743986028627135459}{2565147608745470221905} a^{8} + \frac{309358647200204418671}{5130295217490940443810} a^{7} + \frac{57527939242976694331}{135007768881340537995} a^{6} - \frac{658100316546320383133}{2565147608745470221905} a^{5} + \frac{132327219428044111706}{285016400971718913545} a^{4} - \frac{454774571990211197141}{1710098405830313481270} a^{3} + \frac{118042732004923904657}{855049202915156740635} a^{2} + \frac{8689503329241706406}{285016400971718913545} a + \frac{4588216377541588903}{15000863209037837555}$, $\frac{1}{29128268297324103213529139861119762890} a^{17} - \frac{392772132479377}{29128268297324103213529139861119762890} a^{16} + \frac{116032501612618211018959993}{4854711382887350535588189976853293815} a^{15} - \frac{55408838044693646311485783596833106}{14564134148662051606764569930559881445} a^{14} - \frac{7009196536236421128986834428544}{6905706092300640875658876211740105} a^{13} + \frac{53941040366754833294394888708710744}{856713773450708918044974701797640085} a^{12} - \frac{225126484965539504046233889560063684}{2912826829732410321352913986111976289} a^{11} - \frac{256930661124152710108889220449111075}{2912826829732410321352913986111976289} a^{10} + \frac{15358462465501992616927039993410103}{323647425525823369039212665123552921} a^{9} + \frac{104652930308118245699080587023107015}{2912826829732410321352913986111976289} a^{8} + \frac{705151741160378158105586936074823643}{1618237127629116845196063325617764605} a^{7} + \frac{817570612087469503803910068191741047}{2912826829732410321352913986111976289} a^{6} - \frac{11021221957940512525425187930272016127}{29128268297324103213529139861119762890} a^{5} - \frac{2696502962248572086951044919504316271}{9709422765774701071176379953706587630} a^{4} - \frac{250695738132718688984657020609746677}{1618237127629116845196063325617764605} a^{3} - \frac{370447550561991552966143463619012898}{4854711382887350535588189976853293815} a^{2} + \frac{91803679295924229359911201315756772}{323647425525823369039212665123552921} a + \frac{25081191828306489782353368008206093}{85170375138374570799792806611461295}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{6}\times C_{6}$, which has order $972$ (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:  \( -\frac{8058671157661604}{283476462326445795582345} a^{17} + \frac{7316099243876069728}{30331981468929700127310915} a^{16} - \frac{1559818702115762654}{1596420077312089480384785} a^{15} - \frac{305464420980937072}{6066396293785940025462183} a^{14} + \frac{427748703421045465811}{30331981468929700127310915} a^{13} - \frac{202909187785966083277}{3568468408109376485565990} a^{12} + \frac{25703734635250753575313}{30331981468929700127310915} a^{11} - \frac{99404895550218975718432}{30331981468929700127310915} a^{10} + \frac{95752481691775189742097}{20221320979286466751540610} a^{9} + \frac{797027536410549499460228}{30331981468929700127310915} a^{8} - \frac{133259822047767466071251}{1596420077312089480384785} a^{7} + \frac{5077005508348693024371497}{60663962937859400254621830} a^{6} - \frac{4095532978869815971543649}{532140025770696493461595} a^{5} + \frac{192701669310732543050043082}{10110660489643233375770305} a^{4} - \frac{1145954402734249491407027687}{60663962937859400254621830} a^{3} + \frac{93100331298587281159012866}{10110660489643233375770305} a^{2} - \frac{46769674374079514408915124}{10110660489643233375770305} a + \frac{205143826598086758694278}{106428005154139298692319} \) (order $6$)
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:  \( 181961253025547170 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.34347.1 x3, 6.0.1771470000.6, Deg 6, Deg 6, 6.0.3539149227.1

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

Sibling fields

Degree 18 siblings: 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 R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$107$107.6.4.1$x^{6} + 1498 x^{3} + 1431125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
107.6.4.1$x^{6} + 1498 x^{3} + 1431125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
107.6.4.1$x^{6} + 1498 x^{3} + 1431125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$