↑

Properties

Label 18.0.15537225845...1664.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 13^{9}\cdot 127^{14}$
Root discriminant $540.57$
Ramified primes $2, 3, 13, 127$
Class number $60002572896$ (GRH)
Class group $[2, 84, 357158172]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![194888648704, -137349245952, 87450314880, -31071279168, 10903299792, -2810738664, 838542252, -191602578, 45186124, -7028429, 1442715, -272982, 72318, -9060, 648, 72, 22, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 22*x^16 + 72*x^15 + 648*x^14 - 9060*x^13 + 72318*x^12 - 272982*x^11 + 1442715*x^10 - 7028429*x^9 + 45186124*x^8 - 191602578*x^7 + 838542252*x^6 - 2810738664*x^5 + 10903299792*x^4 - 31071279168*x^3 + 87450314880*x^2 - 137349245952*x + 194888648704)
 
gp: K = bnfinit(x^18 - 7*x^17 + 22*x^16 + 72*x^15 + 648*x^14 - 9060*x^13 + 72318*x^12 - 272982*x^11 + 1442715*x^10 - 7028429*x^9 + 45186124*x^8 - 191602578*x^7 + 838542252*x^6 - 2810738664*x^5 + 10903299792*x^4 - 31071279168*x^3 + 87450314880*x^2 - 137349245952*x + 194888648704, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 22 x^{16} + 72 x^{15} + 648 x^{14} - 9060 x^{13} + 72318 x^{12} - 272982 x^{11} + 1442715 x^{10} - 7028429 x^{9} + 45186124 x^{8} - 191602578 x^{7} + 838542252 x^{6} - 2810738664 x^{5} + 10903299792 x^{4} - 31071279168 x^{3} + 87450314880 x^{2} - 137349245952 x + 194888648704 \)

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:  \(-15537225845019417529672325781592151434950300401664=-\,2^{18}\cdot 3^{9}\cdot 13^{9}\cdot 127^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $540.57$
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:  $2, 3, 13, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} + \frac{1}{16} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{1}{64} a^{10} - \frac{3}{64} a^{8} + \frac{3}{32} a^{7} + \frac{5}{64} a^{6} - \frac{3}{16} a^{5} + \frac{3}{32} a^{4} - \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{1}{32} a^{10} + \frac{1}{64} a^{9} + \frac{5}{64} a^{7} - \frac{1}{32} a^{6} + \frac{5}{32} a^{5} - \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{13} - \frac{1}{128} a^{12} + \frac{3}{128} a^{11} + \frac{3}{128} a^{10} + \frac{3}{128} a^{9} + \frac{5}{128} a^{8} - \frac{3}{128} a^{7} - \frac{1}{32} a^{6} + \frac{11}{64} a^{5} + \frac{3}{32} a^{4} + \frac{5}{16} a^{3} + \frac{3}{8} a^{2}$, $\frac{1}{128} a^{15} - \frac{1}{32} a^{10} - \frac{3}{64} a^{9} - \frac{1}{16} a^{8} - \frac{1}{128} a^{7} + \frac{3}{32} a^{6} - \frac{1}{64} a^{5} - \frac{1}{4} a^{4} - \frac{7}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1024} a^{16} - \frac{1}{256} a^{15} - \frac{1}{512} a^{14} + \frac{1}{512} a^{13} + \frac{3}{512} a^{12} - \frac{5}{512} a^{11} - \frac{1}{128} a^{10} - \frac{11}{512} a^{9} - \frac{39}{1024} a^{8} + \frac{35}{512} a^{7} - \frac{55}{512} a^{6} + \frac{27}{256} a^{5} - \frac{29}{128} a^{4} + \frac{15}{64} a^{3} + \frac{3}{8} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{1709112062982812366249372969582272640277606836873682359026688} a^{17} + \frac{770552403282925936559547049636538176170427328931388409689}{1709112062982812366249372969582272640277606836873682359026688} a^{16} - \frac{1460199111001407113038787453166316253081189500491337716211}{854556031491406183124686484791136320138803418436841179513344} a^{15} + \frac{16852141553545030680312338127385344998398882675124487681}{213639007872851545781171621197784080034700854609210294878336} a^{14} + \frac{221836066540652413472375145517133944143442575834005704453}{106819503936425772890585810598892040017350427304605147439168} a^{13} + \frac{1836523671695187478437999833734937804558623794476826594297}{427278015745703091562343242395568160069401709218420589756672} a^{12} - \frac{4410884193095006577930705842978230320361612569845173071325}{854556031491406183124686484791136320138803418436841179513344} a^{11} - \frac{8345098922855261713440202998758227026292231982903721420559}{854556031491406183124686484791136320138803418436841179513344} a^{10} - \frac{6450108785453428477584565329294360584941397660122755116533}{1709112062982812366249372969582272640277606836873682359026688} a^{9} + \frac{103603330448626354594377537864747229270191141106250115503067}{1709112062982812366249372969582272640277606836873682359026688} a^{8} - \frac{270878297054376449702849984484878893239090535205610648109}{26704875984106443222646452649723010004337606826151286859792} a^{7} + \frac{78304582053988690271156189567126154185462716165858918395275}{854556031491406183124686484791136320138803418436841179513344} a^{6} - \frac{21540392109182817748480075318355359964238415532839148336331}{427278015745703091562343242395568160069401709218420589756672} a^{5} + \frac{10748248074723246773896519778575926028103255454609272627357}{213639007872851545781171621197784080034700854609210294878336} a^{4} - \frac{41610996991845458042995572161426501260622801792327170637389}{106819503936425772890585810598892040017350427304605147439168} a^{3} - \frac{406515034582960216281630581250755318989172065549171833443}{6676218996026610805661613162430752501084401706537821714948} a^{2} - \frac{5512699900849386119718079257931556264137401431832136095535}{13352437992053221611323226324861505002168803413075643429896} a - \frac{624578561661004319056272676915587040410413677628194332693}{3338109498013305402830806581215376250542200853268910857474}$

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

Class group and class number

$C_{2}\times C_{84}\times C_{357158172}$, which has order $60002572896$ (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:  \( 5546046730.2947445 \) (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

\(\Q(\sqrt{-39}) \), 3.3.16129.1, 3.3.1016.1, 6.0.61232393664.2, 6.0.15431519959479.2, 9.9.272832440404737536.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$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$$[\ ]_{2}^{6}$
$127$127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.6.5.1$x^{6} - 127$$6$$1$$5$$C_6$$[\ ]_{6}$
127.6.5.1$x^{6} - 127$$6$$1$$5$$C_6$$[\ ]_{6}$