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Properties

Label 18.0.55590605665...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{8}\cdot 3^{33}\cdot 5^{8}$
Root discriminant $20.85$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![201, -585, 1539, -3669, 7254, -11286, 14878, -16377, 14961, -11348, 7065, -3627, 1644, -693, 252, -74, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 74*x^15 + 252*x^14 - 693*x^13 + 1644*x^12 - 3627*x^11 + 7065*x^10 - 11348*x^9 + 14961*x^8 - 16377*x^7 + 14878*x^6 - 11286*x^5 + 7254*x^4 - 3669*x^3 + 1539*x^2 - 585*x + 201)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 - 74*x^15 + 252*x^14 - 693*x^13 + 1644*x^12 - 3627*x^11 + 7065*x^10 - 11348*x^9 + 14961*x^8 - 16377*x^7 + 14878*x^6 - 11286*x^5 + 7254*x^4 - 3669*x^3 + 1539*x^2 - 585*x + 201, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} - 74 x^{15} + 252 x^{14} - 693 x^{13} + 1644 x^{12} - 3627 x^{11} + 7065 x^{10} - 11348 x^{9} + 14961 x^{8} - 16377 x^{7} + 14878 x^{6} - 11286 x^{5} + 7254 x^{4} - 3669 x^{3} + 1539 x^{2} - 585 x + 201 \)

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:  \(-555906056655552300000000=-\,2^{8}\cdot 3^{33}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.85$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{14} + \frac{1}{3} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{6} a^{9} - \frac{1}{2} a^{7} + \frac{1}{18} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{1854} a^{16} - \frac{14}{927} a^{15} - \frac{35}{618} a^{14} - \frac{10}{309} a^{13} + \frac{23}{618} a^{12} + \frac{95}{618} a^{11} + \frac{23}{103} a^{10} + \frac{83}{206} a^{9} + \frac{131}{309} a^{8} + \frac{787}{1854} a^{7} + \frac{304}{927} a^{6} + \frac{65}{206} a^{5} + \frac{29}{206} a^{4} + \frac{39}{206} a^{3} - \frac{43}{206} a^{2} + \frac{85}{618} a - \frac{193}{618}$, $\frac{1}{36599866026053876982} a^{17} - \frac{1390465759647608}{6099977671008979497} a^{16} - \frac{313757534012801}{60696295233920194} a^{15} + \frac{94709866072896865}{2033325890336326499} a^{14} + \frac{58717888587353845}{4066651780672652998} a^{13} + \frac{482787957766928771}{12199955342017958994} a^{12} - \frac{304344628539086071}{6099977671008979497} a^{11} - \frac{1805351298401364779}{4066651780672652998} a^{10} - \frac{2568610531524918172}{6099977671008979497} a^{9} - \frac{5873144756285501927}{36599866026053876982} a^{8} + \frac{2269784126701989826}{6099977671008979497} a^{7} + \frac{1980355473297206261}{12199955342017958994} a^{6} + \frac{40367565221951677}{140229371747332862} a^{5} - \frac{1253623229817242171}{4066651780672652998} a^{4} + \frac{3289567634819021}{66666422633977918} a^{3} + \frac{5234796126272296585}{12199955342017958994} a^{2} + \frac{461741558694405053}{4066651780672652998} a + \frac{13359853068230204}{30348147616960097}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1153093228}{270538584909} a^{17} - \frac{1917199105}{90179528303} a^{16} + \frac{85342087}{1345963109} a^{15} - \frac{21031982145}{90179528303} a^{14} + \frac{70965726527}{90179528303} a^{13} - \frac{530576492120}{270538584909} a^{12} + \frac{396278301354}{90179528303} a^{11} - \frac{863504943345}{90179528303} a^{10} + \frac{4683645737975}{270538584909} a^{9} - \frac{6597579467435}{270538584909} a^{8} + \frac{2582600641584}{90179528303} a^{7} - \frac{7793753274995}{270538584909} a^{6} + \frac{75339314883}{3109638907} a^{5} - \frac{1675842562455}{90179528303} a^{4} + \frac{18592352103}{1478352923} a^{3} - \frac{535346193650}{90179528303} a^{2} + \frac{281548993374}{90179528303} a - \frac{923498848}{1345963109} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 295029.2522758723 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 6.0.177147.2, 9.3.143489070000.1

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 R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[\ ]^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.9$x^{6} + 3$$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}$
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$