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Properties

Label 18.0.94830820568...5568.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 19^{12}$
Root discriminant $40.72$
Ramified primes $2, 3, 19$
Class number $108$ (GRH)
Class group $[6, 18]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9075, 27720, 32844, 14112, 810, 384, -773, -1968, 1956, 864, -12, -168, 181, -156, 24, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 + 24*x^14 - 156*x^13 + 181*x^12 - 168*x^11 - 12*x^10 + 864*x^9 + 1956*x^8 - 1968*x^7 - 773*x^6 + 384*x^5 + 810*x^4 + 14112*x^3 + 32844*x^2 + 27720*x + 9075)
 
gp: K = bnfinit(x^18 + 6*x^16 + 24*x^14 - 156*x^13 + 181*x^12 - 168*x^11 - 12*x^10 + 864*x^9 + 1956*x^8 - 1968*x^7 - 773*x^6 + 384*x^5 + 810*x^4 + 14112*x^3 + 32844*x^2 + 27720*x + 9075, 1)
 

Normalized defining polynomial

\( x^{18} + 6 x^{16} + 24 x^{14} - 156 x^{13} + 181 x^{12} - 168 x^{11} - 12 x^{10} + 864 x^{9} + 1956 x^{8} - 1968 x^{7} - 773 x^{6} + 384 x^{5} + 810 x^{4} + 14112 x^{3} + 32844 x^{2} + 27720 x + 9075 \)

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:  \(-94830820568684146627640045568=-\,2^{12}\cdot 3^{21}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.72$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{10} a^{14} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{3}{10} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{3}{10} a^{4} - \frac{1}{5} a^{3} - \frac{1}{2} a^{2} + \frac{1}{10} a$, $\frac{1}{20} a^{15} + \frac{1}{5} a^{13} - \frac{3}{20} a^{12} - \frac{1}{5} a^{10} + \frac{1}{10} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{9}{20} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{4}$, $\frac{1}{78501203841940} a^{16} - \frac{273071899921}{15700240768388} a^{15} - \frac{162409903527}{19625300960485} a^{14} - \frac{15220741988417}{78501203841940} a^{13} + \frac{10293465664799}{78501203841940} a^{12} - \frac{965849825392}{3925060192097} a^{11} + \frac{61227926479}{3925060192097} a^{10} + \frac{5960614421743}{39250601920970} a^{9} + \frac{1601241985051}{19625300960485} a^{8} + \frac{259537937913}{7850120384194} a^{7} + \frac{87547591636}{356823653827} a^{6} + \frac{7354901410244}{19625300960485} a^{5} - \frac{4337912052677}{78501203841940} a^{4} - \frac{3087902127449}{15700240768388} a^{3} + \frac{831266302737}{1784118269135} a^{2} + \frac{20804985256483}{78501203841940} a + \frac{274820197227}{1427294615308}$, $\frac{1}{2289460998142077682340} a^{17} - \frac{11507581}{2289460998142077682340} a^{16} + \frac{1568223270207341563}{208132818012916152940} a^{15} - \frac{50128555647454210787}{2289460998142077682340} a^{14} - \frac{384595299649538697481}{2289460998142077682340} a^{13} - \frac{2859639534037299139}{15365510054644816660} a^{12} - \frac{9200144419367126751}{52033204503229038235} a^{11} + \frac{5171395277196414501}{52033204503229038235} a^{10} - \frac{134590880341000714591}{572365249535519420585} a^{9} - \frac{6487420930878749493}{104066409006458076470} a^{8} + \frac{102233754694970646113}{1144730499071038841170} a^{7} - \frac{177911834169091043543}{1144730499071038841170} a^{6} + \frac{267058969870286077}{41626563602583230588} a^{5} + \frac{158974280008670612397}{457892199628415536468} a^{4} - \frac{927899565238683684657}{2289460998142077682340} a^{3} - \frac{419188275277900950999}{2289460998142077682340} a^{2} - \frac{630068795801758598653}{2289460998142077682340} a - \frac{784526973216853145}{41626563602583230588}$

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

Class group and class number

$C_{6}\times C_{18}$, which has order $108$ (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:  \( -\frac{13451885127}{104205917046220} a^{17} + \frac{3030524337}{52102958523110} a^{16} - \frac{43482507691}{52102958523110} a^{15} + \frac{48066856707}{104205917046220} a^{14} - \frac{8452519282}{2368316296505} a^{13} + \frac{10629500391}{473663259301} a^{12} - \frac{363570315287}{10420591704622} a^{11} + \frac{475190782455}{10420591704622} a^{10} - \frac{428370516369}{10420591704622} a^{9} - \frac{310785167499}{5210295852311} a^{8} - \frac{2968108603855}{10420591704622} a^{7} + \frac{4386363388461}{10420591704622} a^{6} - \frac{13430139288823}{104205917046220} a^{5} + \frac{3328990547169}{26051479261555} a^{4} - \frac{4714210064152}{26051479261555} a^{3} - \frac{164780039744607}{104205917046220} a^{2} - \frac{207543289461261}{52102958523110} a - \frac{1527742921487}{947326518602} \) (order $6$)
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:  \( 2038760.66718 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.38988.2 x3, 3.1.1083.1 x3, 3.1.38988.1 x3, 3.1.108.1 x3, 6.0.4560192432.2, 6.0.3518667.2, 6.0.4560192432.1, 6.0.34992.1, 9.1.177792782538816.5 x9

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$