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Properties

Label 18.0.45757273303...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{30}\cdot 5^{9}\cdot 7^{14}$
Root discriminant $159.72$
Ramified primes $2, 3, 5, 7$
Class number $1959552$ (GRH)
Class group $[6, 6, 6, 12, 756]$ (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![163257381, 100403604, 164145177, 70378146, 64200627, 19084950, 13116918, 2445570, 1595346, 135622, 139035, 2220, 11127, -264, 597, -102, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 12*x^16 - 102*x^15 + 597*x^14 - 264*x^13 + 11127*x^12 + 2220*x^11 + 139035*x^10 + 135622*x^9 + 1595346*x^8 + 2445570*x^7 + 13116918*x^6 + 19084950*x^5 + 64200627*x^4 + 70378146*x^3 + 164145177*x^2 + 100403604*x + 163257381)
 
gp: K = bnfinit(x^18 - 6*x^17 + 12*x^16 - 102*x^15 + 597*x^14 - 264*x^13 + 11127*x^12 + 2220*x^11 + 139035*x^10 + 135622*x^9 + 1595346*x^8 + 2445570*x^7 + 13116918*x^6 + 19084950*x^5 + 64200627*x^4 + 70378146*x^3 + 164145177*x^2 + 100403604*x + 163257381, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 12 x^{16} - 102 x^{15} + 597 x^{14} - 264 x^{13} + 11127 x^{12} + 2220 x^{11} + 139035 x^{10} + 135622 x^{9} + 1595346 x^{8} + 2445570 x^{7} + 13116918 x^{6} + 19084950 x^{5} + 64200627 x^{4} + 70378146 x^{3} + 164145177 x^{2} + 100403604 x + 163257381 \)

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:  \(-4575727330315213559892705312768000000000=-\,2^{24}\cdot 3^{30}\cdot 5^{9}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $159.72$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{14} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{15} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{6666} a^{16} - \frac{307}{6666} a^{15} + \frac{119}{3333} a^{14} - \frac{461}{6666} a^{13} + \frac{161}{3333} a^{12} + \frac{91}{2222} a^{11} + \frac{11}{202} a^{10} - \frac{49}{606} a^{9} - \frac{1523}{6666} a^{8} + \frac{433}{2222} a^{7} + \frac{2393}{6666} a^{6} - \frac{1867}{6666} a^{5} - \frac{342}{1111} a^{4} - \frac{985}{2222} a^{3} - \frac{402}{1111} a^{2} - \frac{50}{101} a + \frac{189}{1111}$, $\frac{1}{4566075907205809600656647616732509179345855933746} a^{17} + \frac{113135895298003483704323973666459318780559225}{2283037953602904800328323808366254589672927966873} a^{16} - \frac{198787211098916209446614610548350134326688182407}{4566075907205809600656647616732509179345855933746} a^{15} - \frac{45910100263288273034087838957556583359590097069}{1522025302401936533552215872244169726448618644582} a^{14} + \frac{9343119020733415038414973270604927413975186165}{4566075907205809600656647616732509179345855933746} a^{13} + \frac{71723548038169288244476566783089403356932349369}{1522025302401936533552215872244169726448618644582} a^{12} + \frac{702657083751505848856546846775935670658468199}{138365936581994230322928715658560884222601694962} a^{11} + \frac{11362128248336926739155319030905184063606351837}{415097809745982690968786146975682652667805084886} a^{10} - \frac{319809380051585013734547336888827352792660073313}{4566075907205809600656647616732509179345855933746} a^{9} - \frac{340745173198396407144526129600050637108519699055}{761012651200968266776107936122084863224309322291} a^{8} - \frac{725596257498949868429766727313752815238165319966}{2283037953602904800328323808366254589672927966873} a^{7} + \frac{632577003452618501948747615984122114581421662573}{1522025302401936533552215872244169726448618644582} a^{6} - \frac{934779345813553570919324384362377319204376890189}{4566075907205809600656647616732509179345855933746} a^{5} + \frac{9622806295191142847789970455116224051181086750}{761012651200968266776107936122084863224309322291} a^{4} + \frac{11728112181394046326828874981222529869853518433}{1522025302401936533552215872244169726448618644582} a^{3} - \frac{20032787749268814760551039195413949278689763920}{69182968290997115161464357829280442111300847481} a^{2} + \frac{27790432496588853718925250601534318949426583941}{1522025302401936533552215872244169726448618644582} a + \frac{7745432617403879957475593850509454582263363459}{69182968290997115161464357829280442111300847481}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{6}\times C_{12}\times C_{756}$, which has order $1959552$ (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:  \( 4695974.091249611 \) (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

\(\Q(\sqrt{-5}) \), 3.3.3969.2, 3.3.756.1, 6.0.126023688000.1, 6.0.1143072000.8, 9.9.756284282720064.2

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 R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.9.15.14$x^{9} + 3 x^{7} + 3 x^{6} + 6 x^{3} + 15$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
3.9.15.14$x^{9} + 3 x^{7} + 3 x^{6} + 6 x^{3} + 15$$9$$1$$15$$S_3\times C_3$$[3/2, 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$$[\ ]_{2}^{3}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$