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Properties

Label 18.0.76610513899...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{30}\cdot 3^{6}\cdot 5^{9}\cdot 7^{14}\cdot 43^{14}$
Root discriminant $866.99$
Ramified primes $2, 3, 5, 7, 43$
Class number $34731099648$ (GRH)
Class group $[3, 3, 6, 12, 252, 212688]$ (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![299674652229, 342514005968, 256578470140, 108856477778, 28187586667, 1527209132, -1210841585, -223146292, 101924407, 49750964, 4445026, -1847422, -396092, 26048, 12163, 122, -161, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 161*x^16 + 122*x^15 + 12163*x^14 + 26048*x^13 - 396092*x^12 - 1847422*x^11 + 4445026*x^10 + 49750964*x^9 + 101924407*x^8 - 223146292*x^7 - 1210841585*x^6 + 1527209132*x^5 + 28187586667*x^4 + 108856477778*x^3 + 256578470140*x^2 + 342514005968*x + 299674652229)
 
gp: K = bnfinit(x^18 - 4*x^17 - 161*x^16 + 122*x^15 + 12163*x^14 + 26048*x^13 - 396092*x^12 - 1847422*x^11 + 4445026*x^10 + 49750964*x^9 + 101924407*x^8 - 223146292*x^7 - 1210841585*x^6 + 1527209132*x^5 + 28187586667*x^4 + 108856477778*x^3 + 256578470140*x^2 + 342514005968*x + 299674652229, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 161 x^{16} + 122 x^{15} + 12163 x^{14} + 26048 x^{13} - 396092 x^{12} - 1847422 x^{11} + 4445026 x^{10} + 49750964 x^{9} + 101924407 x^{8} - 223146292 x^{7} - 1210841585 x^{6} + 1527209132 x^{5} + 28187586667 x^{4} + 108856477778 x^{3} + 256578470140 x^{2} + 342514005968 x + 299674652229 \)

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:  \(-76610513899544453304430517636008399400337408000000000=-\,2^{30}\cdot 3^{6}\cdot 5^{9}\cdot 7^{14}\cdot 43^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $866.99$
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:  $2, 3, 5, 7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{7} - \frac{1}{6} a^{5} + \frac{1}{3} a^{3} + \frac{5}{18} a$, $\frac{1}{18} a^{8} - \frac{1}{2} a^{4} - \frac{1}{18} a^{2} - \frac{1}{2}$, $\frac{1}{54} a^{9} - \frac{1}{54} a^{7} - \frac{1}{3} a^{4} - \frac{19}{54} a^{3} - \frac{1}{3} a^{2} - \frac{4}{27} a - \frac{1}{3}$, $\frac{1}{54} a^{10} - \frac{1}{54} a^{8} - \frac{19}{54} a^{4} - \frac{4}{27} a^{2}$, $\frac{1}{162} a^{11} + \frac{1}{162} a^{9} - \frac{1}{81} a^{7} + \frac{1}{18} a^{6} - \frac{19}{162} a^{5} - \frac{1}{6} a^{4} - \frac{23}{81} a^{3} + \frac{1}{3} a^{2} - \frac{8}{81} a + \frac{5}{18}$, $\frac{1}{324} a^{12} - \frac{1}{162} a^{10} + \frac{1}{324} a^{8} + \frac{2}{81} a^{6} - \frac{31}{81} a^{4} + \frac{29}{81} a^{2} - \frac{1}{4}$, $\frac{1}{2916} a^{13} + \frac{1}{972} a^{12} - \frac{2}{243} a^{10} - \frac{7}{972} a^{9} - \frac{17}{972} a^{8} - \frac{13}{1458} a^{7} - \frac{1}{243} a^{6} + \frac{2}{27} a^{5} - \frac{5}{486} a^{4} - \frac{104}{243} a^{3} + \frac{74}{243} a^{2} - \frac{623}{2916} a + \frac{49}{324}$, $\frac{1}{2916} a^{14} - \frac{1}{486} a^{11} - \frac{7}{972} a^{10} - \frac{2}{243} a^{9} + \frac{7}{729} a^{8} - \frac{13}{486} a^{7} + \frac{73}{486} a^{5} + \frac{35}{486} a^{4} - \frac{167}{486} a^{3} + \frac{1267}{2916} a^{2} - \frac{25}{243} a - \frac{5}{54}$, $\frac{1}{122472} a^{15} + \frac{1}{13608} a^{14} + \frac{1}{8748} a^{13} + \frac{11}{13608} a^{12} - \frac{73}{40824} a^{11} + \frac{23}{4536} a^{10} - \frac{475}{61236} a^{9} + \frac{19}{13608} a^{8} + \frac{4}{15309} a^{7} + \frac{67}{1701} a^{6} - \frac{169}{2916} a^{5} - \frac{907}{2268} a^{4} - \frac{48407}{122472} a^{3} + \frac{5515}{13608} a^{2} + \frac{1705}{61236} a - \frac{1033}{13608}$, $\frac{1}{122472} a^{16} + \frac{17}{122472} a^{14} + \frac{5}{40824} a^{13} + \frac{25}{20412} a^{12} - \frac{5}{3402} a^{11} - \frac{239}{122472} a^{10} - \frac{109}{13608} a^{9} + \frac{2105}{122472} a^{8} - \frac{5}{729} a^{7} - \frac{187}{20412} a^{6} + \frac{25}{1701} a^{5} + \frac{11875}{122472} a^{4} + \frac{473}{6804} a^{3} + \frac{54227}{122472} a^{2} - \frac{17599}{40824} a + \frac{1055}{4536}$, $\frac{1}{9306416510904507861103639636627356873969563792831103027768} a^{17} - \frac{1685362338816774864645421453875294928341124101848305}{1034046278989389762344848848514150763774395976981233669752} a^{16} - \frac{11790542554059088101124863859812786279437869713744737}{9306416510904507861103639636627356873969563792831103027768} a^{15} + \frac{18844880957802319128162832482155636628440494502412055}{172341046498231627057474808085691793962399329496872278292} a^{14} + \frac{377896839789940118831047359445624696273603490362909963}{9306416510904507861103639636627356873969563792831103027768} a^{13} + \frac{619382404245668080454892745628420492261218392037129793}{517023139494694881172424424257075381887197988490616834876} a^{12} + \frac{8971775831363275334959927687260056517586225196092734283}{9306416510904507861103639636627356873969563792831103027768} a^{11} + \frac{1208921518582346712156475087450998845394986071896234697}{517023139494694881172424424257075381887197988490616834876} a^{10} - \frac{34068752793694288581408491506196033565844624492475610583}{4653208255452253930551819818313678436984781896415551513884} a^{9} - \frac{880753984145206975051613062726082071223177994425431205}{114894030998821084704983205390461195974932886331248185528} a^{8} + \frac{70017830860559625687171935266884329647713038682710891315}{4653208255452253930551819818313678436984781896415551513884} a^{7} + \frac{35077671795749977892326190620168635425682798827879612705}{517023139494694881172424424257075381887197988490616834876} a^{6} + \frac{1297571197018537784495298758682946333669619950055226722959}{9306416510904507861103639636627356873969563792831103027768} a^{5} - \frac{240380814735485503148499061005218084858425242866105438497}{1034046278989389762344848848514150763774395976981233669752} a^{4} - \frac{288358885405639743156954566946808426239141880611229216271}{1329488072986358265871948519518193839138509113261586146824} a^{3} + \frac{5210058886660212356764511990354820030897273849592513280}{43085261624557906764368702021422948490599832374218069573} a^{2} - \frac{1756095845582540587683897870015873709074026281606567335133}{4653208255452253930551819818313678436984781896415551513884} a + \frac{4830520570244258511182269804110612868523547201235015093}{10039284262032910314027658723438356929848504630885763784}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{6}\times C_{12}\times C_{252}\times C_{212688}$, which has order $34731099648$ (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:  \( 58418500973.81414 \) (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

\(\Q(\sqrt{-5}) \), 3.3.90601.1, 3.3.7224.1, 6.0.104372352000.1, 6.0.65668329608000.1, 9.9.3094562042913836404224.1

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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.12.24.307$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
$43$43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.6.5.5$x^{6} + 31347$$6$$1$$5$$C_6$$[\ ]_{6}$
43.6.5.5$x^{6} + 31347$$6$$1$$5$$C_6$$[\ ]_{6}$