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Properties

Label 18.0.27564457068...6144.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 31^{9}\cdot 37^{14}$
Root discriminant $634.22$
Ramified primes $2, 3, 31, 37$
Class number $3175264476$ (GRH)
Class group $[3, 3, 3, 117602388]$ (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![3552952831147, 4545989325489, 2978310223176, 1131134296912, 234790211235, 4354104129, -11063580650, -2489222517, 72948210, 115321025, 14666778, -1804737, -553500, -5745, 10995, 598, -114, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 114*x^16 + 598*x^15 + 10995*x^14 - 5745*x^13 - 553500*x^12 - 1804737*x^11 + 14666778*x^10 + 115321025*x^9 + 72948210*x^8 - 2489222517*x^7 - 11063580650*x^6 + 4354104129*x^5 + 234790211235*x^4 + 1131134296912*x^3 + 2978310223176*x^2 + 4545989325489*x + 3552952831147)
 
gp: K = bnfinit(x^18 - 9*x^17 - 114*x^16 + 598*x^15 + 10995*x^14 - 5745*x^13 - 553500*x^12 - 1804737*x^11 + 14666778*x^10 + 115321025*x^9 + 72948210*x^8 - 2489222517*x^7 - 11063580650*x^6 + 4354104129*x^5 + 234790211235*x^4 + 1131134296912*x^3 + 2978310223176*x^2 + 4545989325489*x + 3552952831147, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 114 x^{16} + 598 x^{15} + 10995 x^{14} - 5745 x^{13} - 553500 x^{12} - 1804737 x^{11} + 14666778 x^{10} + 115321025 x^{9} + 72948210 x^{8} - 2489222517 x^{7} - 11063580650 x^{6} + 4354104129 x^{5} + 234790211235 x^{4} + 1131134296912 x^{3} + 2978310223176 x^{2} + 4545989325489 x + 3552952831147 \)

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:  \(-275644570682799017504631978444753999120934291206144=-\,2^{12}\cdot 3^{24}\cdot 31^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $634.22$
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:  $2, 3, 31, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{37} a^{12} - \frac{6}{37} a^{11} - \frac{11}{37} a^{10} - \frac{1}{37} a^{9} + \frac{12}{37} a^{8} + \frac{3}{37} a^{7} - \frac{2}{37} a^{6} - \frac{13}{37} a^{5} - \frac{9}{37} a^{4} + \frac{6}{37} a^{3} + \frac{10}{37} a^{2} + \frac{10}{37} a - \frac{1}{37}$, $\frac{1}{37} a^{13} - \frac{10}{37} a^{11} + \frac{7}{37} a^{10} + \frac{6}{37} a^{9} + \frac{1}{37} a^{8} + \frac{16}{37} a^{7} + \frac{12}{37} a^{6} - \frac{13}{37} a^{5} - \frac{11}{37} a^{4} + \frac{9}{37} a^{3} - \frac{4}{37} a^{2} - \frac{15}{37} a - \frac{6}{37}$, $\frac{1}{37} a^{14} - \frac{16}{37} a^{11} + \frac{7}{37} a^{10} - \frac{9}{37} a^{9} - \frac{12}{37} a^{8} + \frac{5}{37} a^{7} + \frac{4}{37} a^{6} + \frac{7}{37} a^{5} - \frac{7}{37} a^{4} - \frac{18}{37} a^{3} + \frac{11}{37} a^{2} - \frac{17}{37} a - \frac{10}{37}$, $\frac{1}{37} a^{15} - \frac{15}{37} a^{11} + \frac{9}{37} a^{9} + \frac{12}{37} a^{8} + \frac{15}{37} a^{7} + \frac{12}{37} a^{6} + \frac{7}{37} a^{5} - \frac{14}{37} a^{4} - \frac{4}{37} a^{3} - \frac{5}{37} a^{2} + \frac{2}{37} a - \frac{16}{37}$, $\frac{1}{8621} a^{16} - \frac{70}{8621} a^{15} - \frac{44}{8621} a^{14} - \frac{26}{8621} a^{13} + \frac{30}{8621} a^{12} - \frac{2992}{8621} a^{11} + \frac{467}{8621} a^{10} - \frac{2532}{8621} a^{9} - \frac{2003}{8621} a^{8} - \frac{3352}{8621} a^{7} + \frac{3399}{8621} a^{6} - \frac{2095}{8621} a^{5} + \frac{2608}{8621} a^{4} + \frac{548}{8621} a^{3} - \frac{1539}{8621} a^{2} - \frac{788}{8621} a + \frac{3891}{8621}$, $\frac{1}{121118658768186840555646329914141735295412630640178177205766532847} a^{17} - \frac{5113596805942517297821384901016048766093739364858501554668772}{121118658768186840555646329914141735295412630640178177205766532847} a^{16} - \frac{17720254401392118431580747408070684766956708420575399320568883}{121118658768186840555646329914141735295412630640178177205766532847} a^{15} + \frac{1566027240302620452571116830748871415596366818010600538870543442}{121118658768186840555646329914141735295412630640178177205766532847} a^{14} - \frac{1394932679213827851428503917035176228040329941407653182180229277}{121118658768186840555646329914141735295412630640178177205766532847} a^{13} - \frac{131435588077501887808358274024072830242042494806004299078294170}{121118658768186840555646329914141735295412630640178177205766532847} a^{12} - \frac{41491869334740214181764200397783139196557982021525917999830990177}{121118658768186840555646329914141735295412630640178177205766532847} a^{11} + \frac{51360087610169387690856768999397341523786650507958841058868222997}{121118658768186840555646329914141735295412630640178177205766532847} a^{10} + \frac{60332189796364069087746958708695526563502257576743979546965958355}{121118658768186840555646329914141735295412630640178177205766532847} a^{9} - \frac{38380352986618992403174036407890160904243437648014444008550429473}{121118658768186840555646329914141735295412630640178177205766532847} a^{8} - \frac{46514605899308366909062991154221862659306654755228600194761991924}{121118658768186840555646329914141735295412630640178177205766532847} a^{7} + \frac{46155608317631954523165995300216224288140283530011510329055897048}{121118658768186840555646329914141735295412630640178177205766532847} a^{6} - \frac{13649345129078346178991821416407029663431719218293568236396626399}{121118658768186840555646329914141735295412630640178177205766532847} a^{5} + \frac{29656722692395274176449742139955669990270480564660736334831611088}{121118658768186840555646329914141735295412630640178177205766532847} a^{4} - \frac{33294346469521572442689008679156828456850245107207901544269019197}{121118658768186840555646329914141735295412630640178177205766532847} a^{3} + \frac{444376375585480430705371900553930359753860787343192008339644382}{121118658768186840555646329914141735295412630640178177205766532847} a^{2} + \frac{57184112335512870993648100973093319448808186854513717267388151296}{121118658768186840555646329914141735295412630640178177205766532847} a + \frac{10734450706379052784706648665387298248630441008240316386700039403}{121118658768186840555646329914141735295412630640178177205766532847}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{117602388}$, which has order $3175264476$ (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

\(\Q(\sqrt{-31}) \), 3.3.148.1, 3.3.110889.1, 6.0.652542064.2, 6.0.366321168232911.1, 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$31$31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.12.6.1$x^{12} + 178746 x^{6} - 114516604 x^{2} + 7987533129$$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}$