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Properties

Label 18.0.11044596577...0528.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 13^{9}\cdot 37^{15}$
Root discriminant $602.80$
Ramified primes $2, 3, 13, 37$
Class number $1857334752$ (GRH)
Class group $[2, 12, 77388948]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5065118682021109, -2315082332910126, 710612754783045, -110436829949061, 19540700837241, -3682933439697, 1114840641631, -192359931897, 27515504667, -1868512448, 231643500, -23816085, 5761489, -388629, 46269, -1210, 249, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 249*x^16 - 1210*x^15 + 46269*x^14 - 388629*x^13 + 5761489*x^12 - 23816085*x^11 + 231643500*x^10 - 1868512448*x^9 + 27515504667*x^8 - 192359931897*x^7 + 1114840641631*x^6 - 3682933439697*x^5 + 19540700837241*x^4 - 110436829949061*x^3 + 710612754783045*x^2 - 2315082332910126*x + 5065118682021109)
 
gp: K = bnfinit(x^18 - 9*x^17 + 249*x^16 - 1210*x^15 + 46269*x^14 - 388629*x^13 + 5761489*x^12 - 23816085*x^11 + 231643500*x^10 - 1868512448*x^9 + 27515504667*x^8 - 192359931897*x^7 + 1114840641631*x^6 - 3682933439697*x^5 + 19540700837241*x^4 - 110436829949061*x^3 + 710612754783045*x^2 - 2315082332910126*x + 5065118682021109, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 249 x^{16} - 1210 x^{15} + 46269 x^{14} - 388629 x^{13} + 5761489 x^{12} - 23816085 x^{11} + 231643500 x^{10} - 1868512448 x^{9} + 27515504667 x^{8} - 192359931897 x^{7} + 1114840641631 x^{6} - 3682933439697 x^{5} + 19540700837241 x^{4} - 110436829949061 x^{3} + 710612754783045 x^{2} - 2315082332910126 x + 5065118682021109 \)

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:  \(-110445965776445517995022588915789180791979915030528=-\,2^{12}\cdot 3^{27}\cdot 13^{9}\cdot 37^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $602.80$
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:  $2, 3, 13, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{333} a^{15} - \frac{19}{333} a^{14} - \frac{26}{333} a^{13} - \frac{7}{333} a^{12} + \frac{10}{111} a^{11} + \frac{4}{333} a^{10} - \frac{4}{333} a^{9} + \frac{34}{333} a^{8} - \frac{4}{37} a^{7} - \frac{10}{111} a^{6} - \frac{158}{333} a^{5} - \frac{106}{333} a^{4} + \frac{53}{333} a^{3} + \frac{125}{333} a^{2} + \frac{47}{111} a - \frac{121}{333}$, $\frac{1}{46287} a^{16} + \frac{53}{46287} a^{15} - \frac{4502}{46287} a^{14} - \frac{6430}{46287} a^{13} + \frac{712}{5143} a^{12} - \frac{6272}{46287} a^{11} - \frac{3823}{46287} a^{10} - \frac{698}{46287} a^{9} - \frac{2470}{5143} a^{8} + \frac{1534}{5143} a^{7} + \frac{18328}{46287} a^{6} + \frac{16268}{46287} a^{5} + \frac{1301}{46287} a^{4} + \frac{15374}{46287} a^{3} + \frac{3491}{15429} a^{2} + \frac{2594}{46287} a + \frac{882}{5143}$, $\frac{1}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{17} + \frac{509139446494576659947068315122417191464529984587097078060465767599339300725714254338596538627380231303744141}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{16} + \frac{224532614070315585928372734127402943938707600586914582954737335572535151296748386905982937451637261392145687660}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{15} + \frac{19243862777935607818999127315610434261658512678235185997153663230182266039354322395188670523724716494564762088874}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{14} - \frac{26789659606001841138463408689638581353169105459652812700882328458192337053477570379377975767169107208045788339096}{308843475141091634822230946068849949538806676343646044743440717567007302249630156899584493680803347696528670185653} a^{13} - \frac{16980504618063394432150411348638177980700412250047431353658002965894885370018797997785296882631164072925426583438}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{12} + \frac{15584724550713643074820133284561631442597872230006357883256102548431324556127435799064003908925896172144426275532}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{11} + \frac{137080320594058947073767347570316434658847548087512647531616102567854225785028813682655623784071078097235717203373}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{10} + \frac{1092695363645381479601730343833383232104951291861463843877665153996327318115792725631551754139392066758819251764}{8347120949759233373573809353212160798346126387666109317390289663973170331071085321610391721102793180987261356369} a^{9} + \frac{111081178985530655197325257734202460839593595666359508166246711863941070445461679147747151490852639385517280276307}{308843475141091634822230946068849949538806676343646044743440717567007302249630156899584493680803347696528670185653} a^{8} - \frac{103203568221208528398191450208207506281702938983646498135665980285535721640635098052029838806696768208139354037135}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{7} - \frac{16826757316893137550588242794500722369876797340042906022647483357747334053630791714793970468576057320268361749583}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{6} - \frac{232411046403833914583952365376619876334451976416670408161631826412355589042146916987927199768966937139219997734686}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{5} - \frac{21755161199494947187120548240065166626597574225810948118667428262970016310617328882900868783309714469429730544874}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{4} - \frac{42711083821685697557956100936992314827956759914596871625706202304799826341063529297252443755306810697386845779887}{102947825047030544940743648689616649846268892114548681581146905855669100749876718966528164560267782565509556728551} a^{3} + \frac{447288498097897975222245727619381385782935640116124423743582441544599186042070402218050794206031002370015356320965}{926530425423274904466692838206549848616420029030938134230322152701021906748890470698753481042410043089586010556959} a^{2} + \frac{15224188422213093474827398205330418540619841483168530249603102481822452855621869565873120167338904651319796232131}{102947825047030544940743648689616649846268892114548681581146905855669100749876718966528164560267782565509556728551} a + \frac{115226579069998971031038529685644696219886168088024912387622864943067175899094045706699576857716840060403015790080}{308843475141091634822230946068849949538806676343646044743440717567007302249630156899584493680803347696528670185653}$

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

Class group and class number

$C_{2}\times C_{12}\times C_{77388948}$, which has order $1857334752$ (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:  \( 118546543.87559307 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-1443}) \), 3.3.110889.1, 3.3.148.1, 6.0.2998678941071307.1, 6.0.48074964912.4, 9.9.3228844269788073792.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$13$13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.6.5.5$x^{6} + 296$$6$$1$$5$$C_6$$[\ ]_{6}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$