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Properties

Label 18.0.20093016350...1152.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 7^{10}\cdot 13^{10}$
Root discriminant $42.46$
Ramified primes $2, 3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![138087, 158994, -140268, -362142, -220562, 151945, 313237, 95416, -87068, -36958, 25059, 12134, -2893, -1424, 324, 97, -25, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 25*x^16 + 97*x^15 + 324*x^14 - 1424*x^13 - 2893*x^12 + 12134*x^11 + 25059*x^10 - 36958*x^9 - 87068*x^8 + 95416*x^7 + 313237*x^6 + 151945*x^5 - 220562*x^4 - 362142*x^3 - 140268*x^2 + 158994*x + 138087)
 
gp: K = bnfinit(x^18 - 3*x^17 - 25*x^16 + 97*x^15 + 324*x^14 - 1424*x^13 - 2893*x^12 + 12134*x^11 + 25059*x^10 - 36958*x^9 - 87068*x^8 + 95416*x^7 + 313237*x^6 + 151945*x^5 - 220562*x^4 - 362142*x^3 - 140268*x^2 + 158994*x + 138087, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 25 x^{16} + 97 x^{15} + 324 x^{14} - 1424 x^{13} - 2893 x^{12} + 12134 x^{11} + 25059 x^{10} - 36958 x^{9} - 87068 x^{8} + 95416 x^{7} + 313237 x^{6} + 151945 x^{5} - 220562 x^{4} - 362142 x^{3} - 140268 x^{2} + 158994 x + 138087 \)

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:  \(-200930163501792205662554161152=-\,2^{18}\cdot 3^{9}\cdot 7^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.46$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{11} + \frac{1}{6} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{14} - \frac{1}{3} a^{11} - \frac{1}{6} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{1842} a^{15} - \frac{74}{921} a^{14} + \frac{2}{921} a^{13} - \frac{25}{1842} a^{12} - \frac{279}{614} a^{11} - \frac{141}{614} a^{10} + \frac{328}{921} a^{9} - \frac{319}{921} a^{8} - \frac{21}{307} a^{7} + \frac{91}{307} a^{6} - \frac{22}{921} a^{5} + \frac{418}{921} a^{4} - \frac{173}{1842} a^{3} + \frac{334}{921} a^{2} + \frac{35}{614} a + \frac{45}{614}$, $\frac{1}{5526} a^{16} - \frac{103}{5526} a^{14} - \frac{47}{5526} a^{13} + \frac{125}{1842} a^{12} + \frac{343}{5526} a^{11} - \frac{2083}{5526} a^{10} + \frac{640}{2763} a^{9} + \frac{308}{921} a^{8} - \frac{1376}{2763} a^{7} - \frac{1063}{2763} a^{6} + \frac{539}{2763} a^{5} + \frac{1369}{5526} a^{4} + \frac{119}{2763} a^{3} - \frac{403}{2763} a^{2} - \frac{305}{614} a - \frac{401}{1842}$, $\frac{1}{11606121869973241602190511865359034} a^{17} - \frac{265961154459828937720241720731}{5803060934986620801095255932679517} a^{16} + \frac{1355323282215320048096886862555}{5803060934986620801095255932679517} a^{15} - \frac{76160551856376802092918244405808}{1934353644995540267031751977559839} a^{14} - \frac{455115628805865183928139313013967}{11606121869973241602190511865359034} a^{13} - \frac{184350493294997208094176767452157}{11606121869973241602190511865359034} a^{12} - \frac{29732811344777828848311802020079}{1934353644995540267031751977559839} a^{11} - \frac{3307891030228020285475099875210611}{11606121869973241602190511865359034} a^{10} + \frac{1439588312933728066549194621286846}{5803060934986620801095255932679517} a^{9} + \frac{2719198314512530483979713933115932}{5803060934986620801095255932679517} a^{8} + \frac{340829942933176952338297292907320}{1934353644995540267031751977559839} a^{7} - \frac{1710966730858488795473368130877932}{5803060934986620801095255932679517} a^{6} - \frac{4709292592681868658193645630690597}{11606121869973241602190511865359034} a^{5} + \frac{2815027469635803319499875756750639}{5803060934986620801095255932679517} a^{4} - \frac{1689283312582624369954555334713261}{11606121869973241602190511865359034} a^{3} + \frac{1116792840483332658905713576010495}{5803060934986620801095255932679517} a^{2} - \frac{620887877070310604354626897390555}{1934353644995540267031751977559839} a + \frac{27182641004824436994157164668377}{57741899850613142299455282912234}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{4483978576306547}{1115926554573558198} a^{17} + \frac{3557382330649714}{185987759095593033} a^{16} + \frac{74468977801064783}{1115926554573558198} a^{15} - \frac{282798671586213547}{557963277286779099} a^{14} - \frac{25383947323679686}{61995919698531011} a^{13} + \frac{7179652498751357887}{1115926554573558198} a^{12} + \frac{172938812805192796}{557963277286779099} a^{11} - \frac{54912094179860851537}{1115926554573558198} a^{10} - \frac{884223825364639105}{61995919698531011} a^{9} + \frac{96530931712142259139}{557963277286779099} a^{8} + \frac{25941886794083057876}{557963277286779099} a^{7} - \frac{258398468744761478959}{557963277286779099} a^{6} - \frac{499002537550884123773}{1115926554573558198} a^{5} + \frac{95627786631079824050}{557963277286779099} a^{4} + \frac{328559708883832918883}{557963277286779099} a^{3} + \frac{79592079321924536408}{185987759095593033} a^{2} - \frac{34094619062556010075}{185987759095593033} a - \frac{590831116390971609}{1850624468612866} \) (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:  \( 187527338.95768923 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.2184.1, 6.0.223587.1, 6.0.14309568.3

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$