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Properties

Label 18.0.28916314870...7536.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{32}\cdot 47^{8}$
Root discriminant $72.27$
Ramified primes $2, 3, 47$
Class number $144$ (GRH)
Class group $[6, 24]$ (GRH)
Galois group $C_2\times C_3:S_4$ (as 18T66)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![173889, 0, 593001, 0, 843588, 0, 651252, 0, 297342, 0, 82062, 0, 13444, 0, 1236, 0, 57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 57*x^16 + 1236*x^14 + 13444*x^12 + 82062*x^10 + 297342*x^8 + 651252*x^6 + 843588*x^4 + 593001*x^2 + 173889)
 
gp: K = bnfinit(x^18 + 57*x^16 + 1236*x^14 + 13444*x^12 + 82062*x^10 + 297342*x^8 + 651252*x^6 + 843588*x^4 + 593001*x^2 + 173889, 1)
 

Normalized defining polynomial

\( x^{18} + 57 x^{16} + 1236 x^{14} + 13444 x^{12} + 82062 x^{10} + 297342 x^{8} + 651252 x^{6} + 843588 x^{4} + 593001 x^{2} + 173889 \)

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:  \(-2891631487010844893007652585537536=-\,2^{16}\cdot 3^{32}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.27$
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 a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{3}{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{4} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{7}{16} a^{2} - \frac{5}{16}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{16} a^{3} - \frac{1}{2} a^{2} + \frac{3}{16} a - \frac{1}{2}$, $\frac{1}{96} a^{12} - \frac{1}{32} a^{8} - \frac{1}{8} a^{7} - \frac{1}{12} a^{6} - \frac{1}{8} a^{5} - \frac{7}{32} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{3}{32}$, $\frac{1}{96} a^{13} - \frac{1}{32} a^{9} - \frac{1}{12} a^{7} - \frac{1}{8} a^{6} + \frac{1}{32} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{11}{32} a - \frac{1}{8}$, $\frac{1}{192} a^{14} - \frac{1}{192} a^{12} - \frac{1}{32} a^{11} + \frac{1}{64} a^{10} + \frac{1}{32} a^{9} + \frac{1}{192} a^{8} + \frac{1}{16} a^{7} - \frac{1}{192} a^{6} + \frac{1}{16} a^{5} - \frac{5}{64} a^{4} + \frac{7}{32} a^{3} - \frac{17}{64} a^{2} + \frac{5}{32} a - \frac{11}{64}$, $\frac{1}{576} a^{15} - \frac{1}{192} a^{13} + \frac{1}{192} a^{11} - \frac{1}{32} a^{10} - \frac{17}{576} a^{9} + \frac{1}{32} a^{8} + \frac{13}{192} a^{7} + \frac{1}{16} a^{6} - \frac{5}{64} a^{5} + \frac{1}{16} a^{4} + \frac{23}{192} a^{3} - \frac{9}{32} a^{2} - \frac{23}{64} a - \frac{11}{32}$, $\frac{1}{645696} a^{16} + \frac{383}{215232} a^{14} + \frac{515}{215232} a^{12} - \frac{1}{32} a^{11} - \frac{17909}{645696} a^{10} + \frac{1}{32} a^{9} - \frac{11021}{215232} a^{8} - \frac{1}{16} a^{7} - \frac{85}{3648} a^{6} - \frac{1}{16} a^{5} - \frac{34315}{215232} a^{4} - \frac{5}{32} a^{3} + \frac{20389}{71744} a^{2} - \frac{7}{32} a + \frac{14191}{35872}$, $\frac{1}{179503488} a^{17} - \frac{1}{1291392} a^{16} - \frac{4607}{9972416} a^{15} + \frac{123}{71744} a^{14} + \frac{26803}{9972416} a^{13} + \frac{101}{71744} a^{12} - \frac{1224679}{89751744} a^{11} + \frac{13999}{645696} a^{10} + \frac{46405}{7479312} a^{9} - \frac{1343}{26904} a^{8} - \frac{35677}{507072} a^{7} - \frac{119}{3648} a^{6} + \frac{1494511}{29917248} a^{5} + \frac{12113}{215232} a^{4} + \frac{405347}{9972416} a^{3} + \frac{7181}{71744} a^{2} - \frac{6068737}{19944832} a + \frac{15337}{143488}$

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

Class group and class number

$C_{6}\times C_{24}$, which has order $144$ (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:  \( -1 \) (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:  \( 22177194.2243 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_4$ (as 18T66):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times C_3:S_4$
Character table for $C_2\times C_3:S_4$

Intermediate fields

3.3.45684.2, 3.3.45684.1, 3.3.564.1, 3.3.11421.1, 6.0.521756964.1, 9.9.13443473060864064.2

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

Sibling fields

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.6.10.2$x^{6} + 9$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.12.22.67$x^{12} + 99 x^{11} + 117 x^{10} - 114 x^{9} - 81 x^{8} + 9 x^{7} - 15 x^{6} + 54 x^{5} - 108 x^{4} + 45 x^{3} - 81 x^{2} - 108 x - 72$$6$$2$$22$$D_6$$[5/2]_{2}^{2}$
47Data not computed