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Properties

Label 18.0.17215978016...5792.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 7^{9}\cdot 79^{14}$
Root discriminant $543.66$
Ramified primes $2, 3, 7, 79$
Class number $16288183296$ (GRH)
Class group $[2, 2, 4, 12, 84, 1009932]$ (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![4021085863936, 8595850543104, 9183109401600, 5797665555712, 2034413940288, 224932172208, -102727801808, -40253540568, -2094160692, 1419179555, 237244221, -16784976, -6149376, -37218, 73770, 1984, -420, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 420*x^16 + 1984*x^15 + 73770*x^14 - 37218*x^13 - 6149376*x^12 - 16784976*x^11 + 237244221*x^10 + 1419179555*x^9 - 2094160692*x^8 - 40253540568*x^7 - 102727801808*x^6 + 224932172208*x^5 + 2034413940288*x^4 + 5797665555712*x^3 + 9183109401600*x^2 + 8595850543104*x + 4021085863936)
 
gp: K = bnfinit(x^18 - 9*x^17 - 420*x^16 + 1984*x^15 + 73770*x^14 - 37218*x^13 - 6149376*x^12 - 16784976*x^11 + 237244221*x^10 + 1419179555*x^9 - 2094160692*x^8 - 40253540568*x^7 - 102727801808*x^6 + 224932172208*x^5 + 2034413940288*x^4 + 5797665555712*x^3 + 9183109401600*x^2 + 8595850543104*x + 4021085863936, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 420 x^{16} + 1984 x^{15} + 73770 x^{14} - 37218 x^{13} - 6149376 x^{12} - 16784976 x^{11} + 237244221 x^{10} + 1419179555 x^{9} - 2094160692 x^{8} - 40253540568 x^{7} - 102727801808 x^{6} + 224932172208 x^{5} + 2034413940288 x^{4} + 5797665555712 x^{3} + 9183109401600 x^{2} + 8595850543104 x + 4021085863936 \)

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:  \(-17215978016843899580835684099700667905564175265792=-\,2^{12}\cdot 3^{24}\cdot 7^{9}\cdot 79^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $543.66$
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:  $2, 3, 7, 79$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{5} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{3}{32} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{64} a^{9} - \frac{1}{64} a^{8} - \frac{1}{32} a^{7} - \frac{1}{32} a^{6} + \frac{1}{64} a^{5} + \frac{7}{64} a^{4} - \frac{1}{4} a^{3} - \frac{1}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{256} a^{10} - \frac{3}{256} a^{8} + \frac{1}{64} a^{7} + \frac{15}{256} a^{6} + \frac{1}{32} a^{5} + \frac{7}{256} a^{4} - \frac{7}{64} a^{3} - \frac{5}{64} a^{2} - \frac{7}{16} a - \frac{1}{2}$, $\frac{1}{256} a^{11} + \frac{1}{256} a^{9} + \frac{7}{256} a^{7} + \frac{11}{256} a^{5} + \frac{11}{64} a^{3} - \frac{1}{4} a$, $\frac{1}{40448} a^{12} + \frac{73}{40448} a^{11} + \frac{27}{40448} a^{10} - \frac{159}{40448} a^{9} + \frac{353}{40448} a^{8} + \frac{503}{40448} a^{7} - \frac{1631}{40448} a^{6} + \frac{2507}{40448} a^{5} + \frac{1733}{20224} a^{4} + \frac{77}{10112} a^{3} - \frac{341}{5056} a^{2} + \frac{547}{1264} a - \frac{59}{158}$, $\frac{1}{80896} a^{13} - \frac{1}{80896} a^{12} + \frac{155}{80896} a^{11} + \frac{55}{80896} a^{10} - \frac{47}{80896} a^{9} + \frac{609}{80896} a^{8} - \frac{1407}{80896} a^{7} + \frac{4701}{80896} a^{6} - \frac{2467}{40448} a^{5} + \frac{2395}{20224} a^{4} - \frac{2005}{10112} a^{3} + \frac{101}{2528} a^{2} - \frac{61}{158} a + \frac{25}{79}$, $\frac{1}{323584} a^{14} + \frac{1}{323584} a^{13} - \frac{3}{323584} a^{12} + \frac{37}{323584} a^{11} + \frac{275}{323584} a^{10} - \frac{277}{323584} a^{9} - \frac{1537}{323584} a^{8} - \frac{9273}{323584} a^{7} - \frac{1001}{20224} a^{6} - \frac{4801}{40448} a^{5} + \frac{399}{5056} a^{4} - \frac{4253}{20224} a^{3} + \frac{219}{5056} a^{2} + \frac{13}{316} a + \frac{15}{79}$, $\frac{1}{323584} a^{15} - \frac{1}{80896} a^{12} + \frac{233}{161792} a^{11} - \frac{37}{80896} a^{10} + \frac{149}{20224} a^{9} + \frac{517}{80896} a^{8} + \frac{373}{323584} a^{7} + \frac{2457}{80896} a^{6} + \frac{2155}{40448} a^{5} + \frac{63}{10112} a^{4} - \frac{2495}{20224} a^{3} - \frac{9}{64} a^{2} - \frac{7}{79} a + \frac{39}{79}$, $\frac{1}{10354688} a^{16} - \frac{3}{5177344} a^{15} - \frac{5}{5177344} a^{14} + \frac{13}{5177344} a^{13} - \frac{1}{161792} a^{12} + \frac{10027}{5177344} a^{11} + \frac{6657}{5177344} a^{10} + \frac{7871}{5177344} a^{9} + \frac{37903}{10354688} a^{8} - \frac{38337}{1294336} a^{7} - \frac{72615}{1294336} a^{6} + \frac{18941}{323584} a^{5} - \frac{74553}{647168} a^{4} + \frac{19905}{80896} a^{3} + \frac{2759}{40448} a^{2} - \frac{19}{316} a + \frac{73}{632}$, $\frac{1}{1983751983917498862801125841006695560268961479433519104} a^{17} - \frac{171543802030591815649958440085795984376708773}{29172823292904395041193027073627875886308257050492928} a^{16} - \frac{438036016625330940437274943785796505722631431147}{991875991958749431400562920503347780134480739716759552} a^{15} - \frac{1414083396344180703113269528385659265094112297245}{991875991958749431400562920503347780134480739716759552} a^{14} - \frac{1865003814136201271106554167511298542693569037667}{495937995979374715700281460251673890067240369858379776} a^{13} - \frac{534052834056642440221835709870431222917473758581}{58345646585808790082386054147255751772616514100985856} a^{12} + \frac{25748146253364070996398328354759737759134593487079}{58345646585808790082386054147255751772616514100985856} a^{11} - \frac{1088814166994851853589790373571652914670077923613663}{991875991958749431400562920503347780134480739716759552} a^{10} - \frac{36990293355701415098568138125274347654845620682903}{9401668170225113093844198298609931565255741608689664} a^{9} + \frac{3542092393800864799193275252443641334266942483940323}{991875991958749431400562920503347780134480739716759552} a^{8} - \frac{420963341654672441361533180878717016906954136196321}{14586411646452197520596513536813937943154128525246464} a^{7} - \frac{4105988784018335217459133204357635869129802323383021}{123984498994843678925070365062918472516810092464594944} a^{6} - \frac{14758082934984957576128554371005457219587957359718453}{123984498994843678925070365062918472516810092464594944} a^{5} - \frac{336742852559165885988763914935079727101261012884397}{61992249497421839462535182531459236258405046232297472} a^{4} + \frac{70921536585273463007762263650694354591576763172651}{484314449198608120801056113527025283268789423689824} a^{3} + \frac{877140130917741672574149411263187337380843670341439}{3874515593588864966408448908216202266150315389518592} a^{2} + \frac{4029870260425710061427484304696322350481296715989}{121078612299652030200264028381756320817197355922456} a + \frac{121189821146470686822749062970494414835517176755}{286916142890170687678350778155820665443595630148}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{12}\times C_{84}\times C_{1009932}$, which has order $16288183296$ (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:  \( 32038743174.66701 \) (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

\(\Q(\sqrt{-7}) \), 3.3.505521.1, 3.3.316.1, 6.0.87654158134263.3, 6.0.34250608.1, 9.9.653167654112852807616.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$\Q_{2}$$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$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$79$79.3.2.3$x^{3} - 316$$3$$1$$2$$C_3$$[\ ]_{3}$
79.3.2.3$x^{3} - 316$$3$$1$$2$$C_3$$[\ ]_{3}$
79.6.5.2$x^{6} - 316$$6$$1$$5$$C_6$$[\ ]_{6}$
79.6.5.2$x^{6} - 316$$6$$1$$5$$C_6$$[\ ]_{6}$