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Properties

Label 18.0.10490638424...4432.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{8}\cdot 3^{9}\cdot 113^{6}$
Root discriminant $11.39$
Ramified primes $2, 3, 113$
Class number $1$
Class group Trivial
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -2, -5, 17, -5, 28, -48, 49, -59, 64, -69, 56, -41, 31, -20, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 11*x^16 - 20*x^15 + 31*x^14 - 41*x^13 + 56*x^12 - 69*x^11 + 64*x^10 - 59*x^9 + 49*x^8 - 48*x^7 + 28*x^6 - 5*x^5 + 17*x^4 - 5*x^3 - 2*x^2 - x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + 11*x^16 - 20*x^15 + 31*x^14 - 41*x^13 + 56*x^12 - 69*x^11 + 64*x^10 - 59*x^9 + 49*x^8 - 48*x^7 + 28*x^6 - 5*x^5 + 17*x^4 - 5*x^3 - 2*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 11 x^{16} - 20 x^{15} + 31 x^{14} - 41 x^{13} + 56 x^{12} - 69 x^{11} + 64 x^{10} - 59 x^{9} + 49 x^{8} - 48 x^{7} + 28 x^{6} - 5 x^{5} + 17 x^{4} - 5 x^{3} - 2 x^{2} - x + 1 \)

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:  \(-10490638424730354432=-\,2^{8}\cdot 3^{9}\cdot 113^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.39$
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:  $2, 3, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{42} a^{15} + \frac{1}{14} a^{14} - \frac{1}{14} a^{13} + \frac{1}{14} a^{12} - \frac{2}{7} a^{11} + \frac{3}{14} a^{10} - \frac{4}{21} a^{9} - \frac{1}{21} a^{8} + \frac{19}{42} a^{7} + \frac{8}{21} a^{6} - \frac{3}{7} a^{5} + \frac{11}{42} a^{4} - \frac{17}{42} a^{3} + \frac{1}{42} a^{2} + \frac{3}{7} a - \frac{11}{42}$, $\frac{1}{42} a^{16} + \frac{1}{21} a^{14} - \frac{1}{21} a^{13} - \frac{1}{6} a^{12} + \frac{1}{14} a^{11} - \frac{1}{2} a^{10} - \frac{10}{21} a^{9} - \frac{1}{14} a^{8} - \frac{13}{42} a^{7} + \frac{3}{7} a^{6} - \frac{19}{42} a^{5} + \frac{1}{7} a^{4} - \frac{2}{21} a^{3} + \frac{1}{42} a^{2} - \frac{3}{14} a + \frac{19}{42}$, $\frac{1}{32298} a^{17} - \frac{3}{10766} a^{16} + \frac{4}{2307} a^{15} + \frac{719}{5383} a^{14} + \frac{1023}{10766} a^{13} + \frac{1535}{16149} a^{12} + \frac{4657}{16149} a^{11} - \frac{5113}{32298} a^{10} + \frac{853}{10766} a^{9} + \frac{2032}{16149} a^{8} + \frac{933}{10766} a^{7} + \frac{3009}{10766} a^{6} - \frac{14347}{32298} a^{5} - \frac{1887}{5383} a^{4} + \frac{10487}{32298} a^{3} + \frac{4540}{16149} a^{2} - \frac{1415}{5383} a - \frac{4819}{10766}$

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:  \( -\frac{2147}{5383} a^{17} + \frac{42883}{32298} a^{16} - \frac{108499}{32298} a^{15} + \frac{23563}{4614} a^{14} - \frac{76833}{10766} a^{13} + \frac{42873}{5383} a^{12} - \frac{356539}{32298} a^{11} + \frac{62425}{5383} a^{10} - \frac{95129}{16149} a^{9} + \frac{137185}{32298} a^{8} - \frac{12227}{16149} a^{7} + \frac{56551}{16149} a^{6} + \frac{129631}{32298} a^{5} - \frac{228467}{32298} a^{4} - \frac{15083}{4614} a^{3} - \frac{15221}{5383} a^{2} + \frac{12381}{10766} a + \frac{12400}{16149} \) (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:  \( 327.87771807 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.339.1, 6.0.344763.1, 9.3.623331504.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$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}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$113$113.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
113.12.6.1$x^{12} + 34629528 x^{6} - 18424351793 x^{2} + 299801052375696$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$