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Properties

Label 18.0.75604221004...8944.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{31}\cdot 13^{14}\cdot 17^{9}$
Root discriminant $402.13$
Ramified primes $2, 3, 13, 17$
Class number $161165160$ (GRH)
Class group $[3, 78, 688740]$ (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![313934054923, 195259486710, 175409363589, 70641640257, 35167390251, 10073564715, 3570618183, 759000693, 215974023, 34317110, 8448546, 878361, 192669, 2517, 1605, -390, 21, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 21*x^16 - 390*x^15 + 1605*x^14 + 2517*x^13 + 192669*x^12 + 878361*x^11 + 8448546*x^10 + 34317110*x^9 + 215974023*x^8 + 759000693*x^7 + 3570618183*x^6 + 10073564715*x^5 + 35167390251*x^4 + 70641640257*x^3 + 175409363589*x^2 + 195259486710*x + 313934054923)
 
gp: K = bnfinit(x^18 - 3*x^17 + 21*x^16 - 390*x^15 + 1605*x^14 + 2517*x^13 + 192669*x^12 + 878361*x^11 + 8448546*x^10 + 34317110*x^9 + 215974023*x^8 + 759000693*x^7 + 3570618183*x^6 + 10073564715*x^5 + 35167390251*x^4 + 70641640257*x^3 + 175409363589*x^2 + 195259486710*x + 313934054923, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 21 x^{16} - 390 x^{15} + 1605 x^{14} + 2517 x^{13} + 192669 x^{12} + 878361 x^{11} + 8448546 x^{10} + 34317110 x^{9} + 215974023 x^{8} + 759000693 x^{7} + 3570618183 x^{6} + 10073564715 x^{5} + 35167390251 x^{4} + 70641640257 x^{3} + 175409363589 x^{2} + 195259486710 x + 313934054923 \)

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:  \(-75604221004961235317502546825450910936845778944=-\,2^{18}\cdot 3^{31}\cdot 13^{14}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $402.13$
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:  $2, 3, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{27} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} + \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9} a + \frac{1}{27}$, $\frac{1}{27} a^{10} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} - \frac{8}{27} a - \frac{4}{9}$, $\frac{1}{27} a^{11} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{9} a^{3} - \frac{8}{27} a^{2} - \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{54} a^{12} - \frac{1}{54} a^{10} - \frac{1}{18} a^{6} + \frac{1}{18} a^{4} + \frac{8}{27} a^{3} - \frac{8}{27} a - \frac{1}{2}$, $\frac{1}{54} a^{13} - \frac{1}{54} a^{11} - \frac{1}{18} a^{7} + \frac{1}{18} a^{5} + \frac{8}{27} a^{4} - \frac{8}{27} a^{2} - \frac{1}{2} a$, $\frac{1}{162} a^{14} - \frac{1}{162} a^{13} + \frac{1}{162} a^{12} + \frac{1}{162} a^{11} + \frac{1}{81} a^{10} - \frac{1}{81} a^{9} - \frac{7}{54} a^{8} + \frac{7}{54} a^{7} - \frac{7}{54} a^{6} - \frac{23}{162} a^{5} + \frac{7}{81} a^{4} - \frac{7}{81} a^{3} + \frac{79}{162} a^{2} - \frac{31}{162} a - \frac{25}{81}$, $\frac{1}{162} a^{15} - \frac{1}{162} a^{12} - \frac{1}{54} a^{11} - \frac{1}{54} a^{10} + \frac{1}{162} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{17}{162} a^{6} - \frac{1}{18} a^{5} - \frac{1}{6} a^{4} + \frac{35}{162} a^{3} + \frac{13}{27} a^{2} + \frac{5}{54} a + \frac{1}{162}$, $\frac{1}{162} a^{16} - \frac{1}{162} a^{13} - \frac{1}{54} a^{11} - \frac{1}{81} a^{10} - \frac{1}{9} a^{8} - \frac{17}{162} a^{7} - \frac{1}{9} a^{6} - \frac{1}{6} a^{5} + \frac{22}{81} a^{4} + \frac{1}{9} a^{3} + \frac{5}{54} a^{2} - \frac{47}{162} a - \frac{5}{18}$, $\frac{1}{2476220912473913191403236390813275847696406319672621201114} a^{17} + \frac{43154869374924559057266954375754379556729201134292179}{137567828470772955077957577267404213760911462204034511173} a^{16} + \frac{6694778327679040870722658782842076246047084493601657919}{2476220912473913191403236390813275847696406319672621201114} a^{15} - \frac{4628426148844769560600970779212127311600053310709308149}{2476220912473913191403236390813275847696406319672621201114} a^{14} - \frac{6412231506516315360271679124418675737040441439952160318}{1238110456236956595701618195406637923848203159836310600557} a^{13} - \frac{3777001153207565751543758158941253354784015582330290403}{825406970824637730467745463604425282565468773224207067038} a^{12} - \frac{59715487289368152582628757674705673949992349313702479}{30570628549060656683990572726089825280202547156452113594} a^{11} - \frac{6210560385076448992800349064566383953825504371153644112}{1238110456236956595701618195406637923848203159836310600557} a^{10} - \frac{4304019756735579516010783569905020500491552430270970661}{2476220912473913191403236390813275847696406319672621201114} a^{9} - \frac{33312983901034311275817673405672148044986696114619289639}{2476220912473913191403236390813275847696406319672621201114} a^{8} - \frac{4644050211180752294612627336125857026628197762489490683}{412703485412318865233872731802212641282734386612103533519} a^{7} - \frac{276003725520150762953595660904221844024217113098436308631}{2476220912473913191403236390813275847696406319672621201114} a^{6} + \frac{107017286655406236256092933319081958941548516333593490031}{2476220912473913191403236390813275847696406319672621201114} a^{5} + \frac{518519228735334546092144764555644520083960785388643079321}{1238110456236956595701618195406637923848203159836310600557} a^{4} + \frac{13920177005252669524556628356493185609590063681131043091}{137567828470772955077957577267404213760911462204034511173} a^{3} - \frac{114286606626892124601712468702800194188952576481067489055}{825406970824637730467745463604425282565468773224207067038} a^{2} + \frac{61701232961409803478335990793634069075551140599668039326}{1238110456236956595701618195406637923848203159836310600557} a - \frac{589107126721521312648131557864379188716647489422048493038}{1238110456236956595701618195406637923848203159836310600557}$

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

Class group and class number

$C_{3}\times C_{78}\times C_{688740}$, which has order $161165160$ (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:  \( 96135527.08384958 \) (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

\(\Q(\sqrt{-51}) \), 3.3.13689.2, 3.3.2808.1, 6.0.116215010496.4, 6.0.2761922358819.6, 9.9.460990789028310528.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
3Data not computed
$13$13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.5.6$x^{6} + 416$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.6$x^{6} + 416$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$