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Properties

Label 18.0.40835694247...1232.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 29^{9}\cdot 37^{14}$
Root discriminant $736.68$
Ramified primes $2, 3, 29, 37$
Class number $15158849328$ (GRH)
Class group $[3, 3, 378, 4455864]$ (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![329489799094441, 162135328484937, 68512294190292, 13611173843684, 2476828380501, 109343418069, 19476958064, -152631309, 1568694924, 104943021, 24015348, -3204009, 162726, -9861, 10995, -410, 12, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 12*x^16 - 410*x^15 + 10995*x^14 - 9861*x^13 + 162726*x^12 - 3204009*x^11 + 24015348*x^10 + 104943021*x^9 + 1568694924*x^8 - 152631309*x^7 + 19476958064*x^6 + 109343418069*x^5 + 2476828380501*x^4 + 13611173843684*x^3 + 68512294190292*x^2 + 162135328484937*x + 329489799094441)
 
gp: K = bnfinit(x^18 - 9*x^17 + 12*x^16 - 410*x^15 + 10995*x^14 - 9861*x^13 + 162726*x^12 - 3204009*x^11 + 24015348*x^10 + 104943021*x^9 + 1568694924*x^8 - 152631309*x^7 + 19476958064*x^6 + 109343418069*x^5 + 2476828380501*x^4 + 13611173843684*x^3 + 68512294190292*x^2 + 162135328484937*x + 329489799094441, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 12 x^{16} - 410 x^{15} + 10995 x^{14} - 9861 x^{13} + 162726 x^{12} - 3204009 x^{11} + 24015348 x^{10} + 104943021 x^{9} + 1568694924 x^{8} - 152631309 x^{7} + 19476958064 x^{6} + 109343418069 x^{5} + 2476828380501 x^{4} + 13611173843684 x^{3} + 68512294190292 x^{2} + 162135328484937 x + 329489799094441 \)

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:  \(-4083569424758367433043797712456360033585527698911232=-\,2^{12}\cdot 3^{27}\cdot 29^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $736.68$
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:  $2, 3, 29, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{111} a^{12} - \frac{2}{37} a^{11} - \frac{1}{111} a^{10} - \frac{14}{111} a^{9} + \frac{16}{37} a^{8} - \frac{26}{111} a^{7} - \frac{43}{111} a^{6} - \frac{18}{37} a^{5} - \frac{4}{111} a^{4} - \frac{12}{37} a^{3} - \frac{3}{37} a^{2} - \frac{4}{111} a - \frac{10}{111}$, $\frac{1}{111} a^{13} + \frac{17}{111} a^{10} + \frac{1}{111} a^{9} + \frac{1}{37} a^{8} - \frac{14}{111} a^{7} - \frac{16}{111} a^{6} + \frac{14}{37} a^{5} - \frac{23}{111} a^{4} + \frac{34}{111} a^{3} - \frac{7}{37} a^{2} + \frac{1}{37} a - \frac{23}{111}$, $\frac{1}{111} a^{14} + \frac{17}{111} a^{11} + \frac{1}{111} a^{10} + \frac{1}{37} a^{9} - \frac{14}{111} a^{8} - \frac{16}{111} a^{7} + \frac{14}{37} a^{6} - \frac{23}{111} a^{5} + \frac{34}{111} a^{4} - \frac{7}{37} a^{3} + \frac{1}{37} a^{2} - \frac{23}{111} a$, $\frac{1}{333} a^{15} + \frac{1}{333} a^{14} - \frac{1}{333} a^{12} + \frac{52}{333} a^{11} - \frac{52}{333} a^{10} - \frac{55}{333} a^{9} - \frac{43}{333} a^{8} + \frac{13}{333} a^{7} - \frac{44}{111} a^{6} + \frac{7}{111} a^{5} + \frac{11}{333} a^{4} + \frac{1}{333} a^{3} - \frac{154}{333} a^{2} + \frac{86}{333} a - \frac{116}{333}$, $\frac{1}{333} a^{16} - \frac{1}{333} a^{14} - \frac{1}{333} a^{13} - \frac{1}{333} a^{12} - \frac{2}{333} a^{11} + \frac{17}{111} a^{10} - \frac{1}{37} a^{9} + \frac{17}{333} a^{8} - \frac{73}{333} a^{7} - \frac{26}{111} a^{6} + \frac{20}{333} a^{5} - \frac{127}{333} a^{4} + \frac{13}{333} a^{3} - \frac{18}{37} a^{2} + \frac{14}{333} a - \frac{121}{333}$, $\frac{1}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{17} + \frac{390303671233046708763393578303654914963883266478907931911846042731057362}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{16} + \frac{296465186874556702827985852453054821043583557009584296548327213913775374}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{15} + \frac{1204692820188188615400592710423830692802245575058641993696874993879230556}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{14} + \frac{1186806070322637284768533793049468849654239008194979435326610166473707976}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{13} + \frac{173395540370684843188327430872302991775871312096660620016712647417806368}{102002559948633377212269351229174842541663302692720080480223094753052214367} a^{12} - \frac{14720751555893934835055204591716736003561096280859806992850350083100970629}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{11} + \frac{1569638482283284102469050290384151722871899713207518342314594362384828167}{102002559948633377212269351229174842541663302692720080480223094753052214367} a^{10} + \frac{46655600068316702028717806638650315247424003598081911021779832020977886241}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{9} - \frac{115931748977210971229092445181306992895904929779323456160359577726619721265}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{8} - \frac{152880249683174539019104419687307029133836902892399248297068273139355927736}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{7} + \frac{106467978162645780586266275354413003820861780871603132949301530070733355721}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{6} - \frac{28237400148399071337313473581215191233255938754417534045555437505546022266}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{5} - \frac{36303290029270610370738510975728826413436384871120756837628404440855058378}{102002559948633377212269351229174842541663302692720080480223094753052214367} a^{4} + \frac{108787696440466452918910791184348284037776683226541239612235879772246273224}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{3} - \frac{59795842462240040315413458528199545967255982383455217197492736868567253070}{306007679845900131636808053687524527624989908078160241440669284259156643101} a^{2} - \frac{65106308587312698532987900441568367872958117961116397363723594733579327939}{306007679845900131636808053687524527624989908078160241440669284259156643101} a - \frac{29530618164023987138367548741680313757435277959555382707084966499605310081}{306007679845900131636808053687524527624989908078160241440669284259156643101}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{378}\times C_{4455864}$, which has order $15158849328$ (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:  \( 118546543.87559307 \) (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

\(\Q(\sqrt{-87}) \), 3.3.110889.1, 3.3.148.1, 6.0.899688527276607.1, 6.0.14423849712.4, 9.9.3228844269788073792.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$29$29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.12.6.1$x^{12} + 146334 x^{6} - 20511149 x^{2} + 5353409889$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$