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Properties

Label 18.18.4261736528...8125.1
Degree $18$
Signature $[18, 0]$
Discriminant $5^{9}\cdot 139^{4}\cdot 197^{6}$
Root discriminant $38.95$
Ramified primes $5, 139, 197$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![-1445, 0, 11454, 0, -33085, 0, 45712, 0, -33434, 0, 13735, 0, -3276, 0, 450, 0, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 33*x^16 + 450*x^14 - 3276*x^12 + 13735*x^10 - 33434*x^8 + 45712*x^6 - 33085*x^4 + 11454*x^2 - 1445)
 
gp: K = bnfinit(x^18 - 33*x^16 + 450*x^14 - 3276*x^12 + 13735*x^10 - 33434*x^8 + 45712*x^6 - 33085*x^4 + 11454*x^2 - 1445, 1)
 

Normalized defining polynomial

\( x^{18} - 33 x^{16} + 450 x^{14} - 3276 x^{12} + 13735 x^{10} - 33434 x^{8} + 45712 x^{6} - 33085 x^{4} + 11454 x^{2} - 1445 \)

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:  \(42617365285248096686111328125=5^{9}\cdot 139^{4}\cdot 197^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.95$
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:  $5, 139, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{10} + \frac{3}{10} a^{8} - \frac{3}{10} a^{6} - \frac{1}{2} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{10} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{9} - \frac{1}{2} a^{8} - \frac{3}{10} a^{7} - \frac{1}{2} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{10} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{8} - \frac{1}{2} a^{7} + \frac{3}{10} a^{6} - \frac{3}{10} a^{4} - \frac{1}{5} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{9} - \frac{1}{2} a^{8} + \frac{3}{10} a^{7} - \frac{3}{10} a^{5} - \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{550} a^{14} + \frac{1}{110} a^{12} - \frac{13}{275} a^{10} + \frac{53}{275} a^{8} - \frac{1}{2} a^{7} + \frac{7}{25} a^{6} - \frac{1}{2} a^{5} - \frac{3}{550} a^{4} - \frac{1}{2} a^{3} + \frac{117}{275} a^{2} - \frac{1}{2} a + \frac{8}{55}$, $\frac{1}{550} a^{15} + \frac{1}{110} a^{13} - \frac{13}{275} a^{11} + \frac{53}{275} a^{9} - \frac{1}{2} a^{8} + \frac{7}{25} a^{7} - \frac{1}{2} a^{6} - \frac{3}{550} a^{5} - \frac{1}{2} a^{4} + \frac{117}{275} a^{3} - \frac{1}{2} a^{2} + \frac{8}{55} a$, $\frac{1}{6050} a^{16} + \frac{1}{3025} a^{14} + \frac{179}{6050} a^{12} + \frac{37}{3025} a^{10} + \frac{991}{6050} a^{8} - \frac{89}{242} a^{6} - \frac{1}{2} a^{5} + \frac{2443}{6050} a^{4} + \frac{2623}{6050} a^{2} - \frac{1}{2} a + \frac{337}{1210}$, $\frac{1}{102850} a^{17} + \frac{7}{20570} a^{15} - \frac{1471}{102850} a^{13} - \frac{1389}{102850} a^{11} + \frac{127}{51425} a^{9} + \frac{11327}{102850} a^{7} - \frac{1}{2} a^{6} - \frac{681}{102850} a^{5} - \frac{4592}{10285} a^{3} - \frac{553}{4114} a$

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:  $17$
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:  \( 463890141.975 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 3.3.985.1, 6.6.4851125.1, 9.9.92322657333125.1 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139Data not computed
$197$197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$