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Properties

Label 18.6.69157220416...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 5^{9}\cdot 11^{6}\cdot 47^{4}$
Root discriminant $18.57$
Ramified primes $2, 5, 11, 47$
Class number $1$
Class group Trivial
Galois group $C_3\wr S_3:C_2$ (as 18T137)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 4, -52, -108, -6, 126, 233, -32, -181, 62, -52, -8, 125, -44, -52, 30, 5, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 5*x^16 + 30*x^15 - 52*x^14 - 44*x^13 + 125*x^12 - 8*x^11 - 52*x^10 + 62*x^9 - 181*x^8 - 32*x^7 + 233*x^6 + 126*x^5 - 6*x^4 - 108*x^3 - 52*x^2 + 4*x + 4)
 
gp: K = bnfinit(x^18 - 6*x^17 + 5*x^16 + 30*x^15 - 52*x^14 - 44*x^13 + 125*x^12 - 8*x^11 - 52*x^10 + 62*x^9 - 181*x^8 - 32*x^7 + 233*x^6 + 126*x^5 - 6*x^4 - 108*x^3 - 52*x^2 + 4*x + 4, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 5 x^{16} + 30 x^{15} - 52 x^{14} - 44 x^{13} + 125 x^{12} - 8 x^{11} - 52 x^{10} + 62 x^{9} - 181 x^{8} - 32 x^{7} + 233 x^{6} + 126 x^{5} - 6 x^{4} - 108 x^{3} - 52 x^{2} + 4 x + 4 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(69157220416328000000000=2^{12}\cdot 5^{9}\cdot 11^{6}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.57$
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:  $2, 5, 11, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{28} a^{16} - \frac{3}{28} a^{15} + \frac{1}{28} a^{14} + \frac{1}{7} a^{13} - \frac{1}{4} a^{12} - \frac{3}{28} a^{11} - \frac{3}{28} a^{10} - \frac{1}{7} a^{9} + \frac{5}{28} a^{8} - \frac{13}{28} a^{7} + \frac{1}{28} a^{6} - \frac{5}{14} a^{5} + \frac{3}{7} a^{4} - \frac{1}{2} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{103349958680948} a^{17} - \frac{894695923479}{103349958680948} a^{16} - \frac{9975413270989}{103349958680948} a^{15} + \frac{6223588722608}{25837489670237} a^{14} + \frac{15020274783921}{103349958680948} a^{13} - \frac{21599627070059}{103349958680948} a^{12} - \frac{6314113857963}{103349958680948} a^{11} + \frac{7671704411191}{51674979340474} a^{10} + \frac{14908581106395}{103349958680948} a^{9} + \frac{46514500293867}{103349958680948} a^{8} - \frac{46598564868153}{103349958680948} a^{7} + \frac{221568757035}{51674979340474} a^{6} - \frac{21482996937651}{51674979340474} a^{5} + \frac{2531185331107}{25837489670237} a^{4} + \frac{6737819892271}{51674979340474} a^{3} - \frac{2311799593339}{25837489670237} a^{2} - \frac{8439726213524}{25837489670237} a - \frac{9789888099298}{25837489670237}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 47054.460979 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\wr S_3:C_2$ (as 18T137):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 324
The 17 conjugacy class representatives for $C_3\wr S_3:C_2$
Character table for $C_3\wr S_3:C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.44.1, 6.2.242000.1, 9.3.23521432000.1 x3

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

Sibling fields

Degree 9 siblings: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$$[\ ]_{3}^{2}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$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}$
$11$11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.6.4.1$x^{6} + 1786 x^{3} + 4853173$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$