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Properties

Label 18.0.32944359581...0848.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{30}\cdot 7^{9}\cdot 103^{14}$
Root discriminant $1214.29$
Ramified primes $2, 3, 7, 103$
Class number $12123256608$ (GRH)
Class group $[2, 6, 1010271384]$ (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![64478748483584, 110075105415168, 57382505628672, -1091786217216, -6069527174496, -605392710048, 350741017488, 79245803352, -17718435114, -4277544184, 839143659, 66139470, -15515850, -535026, 140388, 2730, -588, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 588*x^16 + 2730*x^15 + 140388*x^14 - 535026*x^13 - 15515850*x^12 + 66139470*x^11 + 839143659*x^10 - 4277544184*x^9 - 17718435114*x^8 + 79245803352*x^7 + 350741017488*x^6 - 605392710048*x^5 - 6069527174496*x^4 - 1091786217216*x^3 + 57382505628672*x^2 + 110075105415168*x + 64478748483584)
 
gp: K = bnfinit(x^18 - 6*x^17 - 588*x^16 + 2730*x^15 + 140388*x^14 - 535026*x^13 - 15515850*x^12 + 66139470*x^11 + 839143659*x^10 - 4277544184*x^9 - 17718435114*x^8 + 79245803352*x^7 + 350741017488*x^6 - 605392710048*x^5 - 6069527174496*x^4 - 1091786217216*x^3 + 57382505628672*x^2 + 110075105415168*x + 64478748483584, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 588 x^{16} + 2730 x^{15} + 140388 x^{14} - 535026 x^{13} - 15515850 x^{12} + 66139470 x^{11} + 839143659 x^{10} - 4277544184 x^{9} - 17718435114 x^{8} + 79245803352 x^{7} + 350741017488 x^{6} - 605392710048 x^{5} - 6069527174496 x^{4} - 1091786217216 x^{3} + 57382505628672 x^{2} + 110075105415168 x + 64478748483584 \)

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:  \(-32944359581476773238038574207216434174814161602181070848=-\,2^{18}\cdot 3^{30}\cdot 7^{9}\cdot 103^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1214.29$
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:  $2, 3, 7, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{48} a^{10} - \frac{1}{24} a^{9} - \frac{1}{48} a^{8} - \frac{1}{24} a^{7} - \frac{1}{16} a^{6} + \frac{5}{24} a^{5} - \frac{1}{16} a^{4} - \frac{1}{24} a^{3} - \frac{1}{24} a^{2} + \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{48} a^{11} + \frac{1}{48} a^{9} + \frac{1}{24} a^{8} + \frac{5}{48} a^{7} + \frac{1}{12} a^{6} - \frac{1}{48} a^{5} - \frac{1}{24} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{96} a^{12} - \frac{1}{96} a^{11} - \frac{1}{96} a^{10} + \frac{5}{96} a^{9} + \frac{5}{96} a^{8} + \frac{1}{32} a^{7} + \frac{1}{96} a^{6} - \frac{7}{32} a^{5} + \frac{1}{48} a^{4} - \frac{13}{48} a^{3} + \frac{1}{12} a^{2} - \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{5568} a^{13} + \frac{7}{1856} a^{12} + \frac{53}{5568} a^{11} - \frac{41}{5568} a^{10} - \frac{313}{5568} a^{9} - \frac{83}{5568} a^{8} + \frac{45}{1856} a^{7} + \frac{17}{5568} a^{6} - \frac{77}{348} a^{5} - \frac{445}{2784} a^{4} + \frac{89}{464} a^{3} + \frac{9}{116} a^{2} + \frac{10}{29} a + \frac{5}{87}$, $\frac{1}{824064} a^{14} + \frac{11}{137344} a^{13} + \frac{119}{206016} a^{12} + \frac{2419}{412032} a^{11} + \frac{593}{103008} a^{10} + \frac{16841}{412032} a^{9} - \frac{4109}{137344} a^{8} + \frac{3287}{137344} a^{7} + \frac{96335}{824064} a^{6} - \frac{17809}{206016} a^{5} + \frac{4985}{412032} a^{4} + \frac{1663}{6438} a^{3} - \frac{47123}{103008} a^{2} - \frac{385}{12876} a + \frac{823}{6438}$, $\frac{1}{11536896} a^{15} + \frac{1}{5768448} a^{14} - \frac{39}{480704} a^{13} - \frac{4747}{5768448} a^{12} + \frac{495}{137344} a^{11} - \frac{59561}{5768448} a^{10} - \frac{238337}{5768448} a^{9} - \frac{154849}{5768448} a^{8} - \frac{588821}{11536896} a^{7} + \frac{9659}{120176} a^{6} + \frac{1114597}{5768448} a^{5} + \frac{312149}{1442112} a^{4} + \frac{46099}{480704} a^{3} + \frac{71177}{360528} a^{2} - \frac{7411}{15022} a - \frac{4436}{22533}$, $\frac{1}{92295168} a^{16} - \frac{1}{23073792} a^{15} + \frac{1}{23073792} a^{14} - \frac{27}{2197504} a^{13} - \frac{157}{132608} a^{12} - \frac{375065}{46147584} a^{11} + \frac{63785}{46147584} a^{10} - \frac{650133}{15382528} a^{9} + \frac{4297567}{92295168} a^{8} + \frac{264781}{6592512} a^{7} + \frac{2276885}{46147584} a^{6} + \frac{95161}{3296256} a^{5} - \frac{958859}{3845632} a^{4} - \frac{1968161}{5768448} a^{3} + \frac{96721}{206016} a^{2} + \frac{43513}{120176} a + \frac{21113}{90132}$, $\frac{1}{1401905549032724778742814428882221971554613205344103583176548915666862638014311579648} a^{17} - \frac{30669741316396790452797100909182604377009934564418925366634168742105762431}{12517013830649328381632271686448410460309046476286639135504901032739844982270639104} a^{16} - \frac{2558828284947338657173931121233983956332208937486376041129892762394585886243}{350476387258181194685703607220555492888653301336025895794137228916715659503577894912} a^{15} - \frac{45360537984891869003923530525926629377181016397293795451842608601829969656825}{100136110645194627053058173491587283682472371810293113084039208261918759858165112832} a^{14} + \frac{3628081151417434925837628594472405299363108916631706005873236741220340962293409}{87619096814545298671425901805138873222163325334006473948534307229178914875894473728} a^{13} - \frac{1564542181131993952437863640454585208259505507887289901890490342792219449309977769}{700952774516362389371407214441110985777306602672051791588274457833431319007155789824} a^{12} - \frac{6711047541986566410783826171227491459975919777049113417726107035340726964685178635}{700952774516362389371407214441110985777306602672051791588274457833431319007155789824} a^{11} - \frac{987978888814574788464577989314447799696534178640315794040688937624907756167437707}{700952774516362389371407214441110985777306602672051791588274457833431319007155789824} a^{10} - \frac{7484395391605522530521320557412046527625769723741278025384580210764803336379275647}{200272221290389254106116346983174567364944743620586226168078416523837519716330225664} a^{9} + \frac{18182387010087436949454054475502975464378525881061291980448472785242121078826834305}{700952774516362389371407214441110985777306602672051791588274457833431319007155789824} a^{8} - \frac{20317328476827972876977283849171823341983146474756134702177628252624417833905980015}{700952774516362389371407214441110985777306602672051791588274457833431319007155789824} a^{7} + \frac{1114073678038280189112766630683157622159420782736258403752379462548681875923698685}{16689351774199104508843028915264547280412061968382185514006534710319793309694185472} a^{6} - \frac{9488001127060338255202838135478942708895489121423003740435542047026210746167130217}{58412731209696865780950601203425915481442216889337649299022871486119276583929649152} a^{5} + \frac{125153387108754374662972354212971253861499297139601906473059948550922466017730547}{789361232563471159202035151397647506505975903909968233770579344407017251134184448} a^{4} + \frac{4098195934962394528612058491675443744502992735898549512377427453010425787317095429}{10952387101818162333928237725642359152770415666750809243566788403647364359486809216} a^{3} + \frac{466861219115963153238998692931839888966536542695780687167620980080145872248810417}{1369048387727270291741029715705294894096301958343851155445848550455920544935851152} a^{2} - \frac{18001599998440828211187430633028436024690577319597195695036193603244413597981929}{85565524232954393233814357231580930881018872396490697215365534403495034058490697} a + \frac{94116541715725204947855162308853632485831573937525979776281249402931580420386231}{342262096931817572935257428926323723524075489585962788861462137613980136233962788}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{1010271384}$, which has order $12123256608$ (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:  \( 135497086932398.42 \) (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

\(\Q(\sqrt{-7}) \), 3.3.155736.1, 3.3.859329.1, 6.0.169775911872.1, 6.0.253287091272663.1, 9.9.309915626538609022216287744.1

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$0$Trivial$[\ ]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$103$103.6.4.2$x^{6} - 103 x^{3} + 53045$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
103.12.10.3$x^{12} - 3193 x^{6} + 6630625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$