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Properties

Label 18.18.6523520634...1216.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{20}\cdot 3^{33}\cdot 47^{9}$
Root discriminant $110.98$
Ramified primes $2, 3, 47$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_3:S_4$ (as 18T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12, 2088, 26424, 1605, -264699, -375174, 153052, 452259, 8409, -226761, -20253, 59949, 2989, -8343, 231, 498, -42, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 42*x^16 + 498*x^15 + 231*x^14 - 8343*x^13 + 2989*x^12 + 59949*x^11 - 20253*x^10 - 226761*x^9 + 8409*x^8 + 452259*x^7 + 153052*x^6 - 375174*x^5 - 264699*x^4 + 1605*x^3 + 26424*x^2 + 2088*x + 12)
 
gp: K = bnfinit(x^18 - 9*x^17 - 42*x^16 + 498*x^15 + 231*x^14 - 8343*x^13 + 2989*x^12 + 59949*x^11 - 20253*x^10 - 226761*x^9 + 8409*x^8 + 452259*x^7 + 153052*x^6 - 375174*x^5 - 264699*x^4 + 1605*x^3 + 26424*x^2 + 2088*x + 12, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 42 x^{16} + 498 x^{15} + 231 x^{14} - 8343 x^{13} + 2989 x^{12} + 59949 x^{11} - 20253 x^{10} - 226761 x^{9} + 8409 x^{8} + 452259 x^{7} + 153052 x^{6} - 375174 x^{5} - 264699 x^{4} + 1605 x^{3} + 26424 x^{2} + 2088 x + 12 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6523520634696466078625264232972681216=2^{20}\cdot 3^{33}\cdot 47^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.98$
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:  $2, 3, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{38} a^{16} - \frac{7}{19} a^{15} - \frac{9}{19} a^{14} + \frac{8}{19} a^{13} - \frac{9}{38} a^{12} + \frac{5}{19} a^{11} + \frac{9}{38} a^{10} + \frac{6}{19} a^{9} - \frac{17}{38} a^{8} + \frac{6}{19} a^{7} + \frac{11}{38} a^{6} - \frac{8}{19} a^{5} + \frac{9}{19} a^{4} + \frac{17}{38} a^{2} - \frac{4}{19}$, $\frac{1}{1583986859149165483659369813438161582} a^{17} - \frac{5435432679565093249648713016925010}{791993429574582741829684906719080791} a^{16} - \frac{237717458283921924827696075513889306}{791993429574582741829684906719080791} a^{15} - \frac{69990710054213539418691831831811247}{791993429574582741829684906719080791} a^{14} + \frac{133904624032419481947387726112604463}{1583986859149165483659369813438161582} a^{13} - \frac{343955218755545533691314302594845491}{791993429574582741829684906719080791} a^{12} + \frac{782854512150961781212373333202058033}{1583986859149165483659369813438161582} a^{11} - \frac{292997963425903109189662971069044139}{791993429574582741829684906719080791} a^{10} - \frac{530743445181635202135032282438004529}{1583986859149165483659369813438161582} a^{9} - \frac{3048504754732896314916348487641897}{41683864714451723254193942458898989} a^{8} - \frac{263879060371033443763622395412625911}{1583986859149165483659369813438161582} a^{7} + \frac{150153971558037224418034503888420623}{791993429574582741829684906719080791} a^{6} + \frac{8510809461502842885344450168291454}{41683864714451723254193942458898989} a^{5} - \frac{146162589484457578327801623829515403}{791993429574582741829684906719080791} a^{4} - \frac{108101298001967522551013465486092143}{1583986859149165483659369813438161582} a^{3} - \frac{178115648163093459326134137991100530}{791993429574582741829684906719080791} a^{2} + \frac{205052864976792151678610843186921729}{791993429574582741829684906719080791} a - \frac{11425537682426427795984560200588503}{791993429574582741829684906719080791}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  $17$
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:  \( 5751301540210 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_4$ (as 18T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3:S_4$
Character table for $C_3:S_4$

Intermediate fields

3.3.564.1, 3.3.45684.2, 3.3.45684.1, 3.3.11421.1, 6.6.294270927696.3, 9.9.13443473060864064.2

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$3$3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
47Data not computed