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Properties

Label 18.0.31465827661...0304.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 13^{12}\cdot 61^{6}$
Root discriminant $43.53$
Ramified primes $2, 13, 61$
Class number $108$ (GRH)
Class group $[3, 3, 12]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 120, 0, 2128, 0, 7946, 0, 11256, 0, 7216, 0, 2337, 0, 392, 0, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 32*x^16 + 392*x^14 + 2337*x^12 + 7216*x^10 + 11256*x^8 + 7946*x^6 + 2128*x^4 + 120*x^2 + 1)
 
gp: K = bnfinit(x^18 + 32*x^16 + 392*x^14 + 2337*x^12 + 7216*x^10 + 11256*x^8 + 7946*x^6 + 2128*x^4 + 120*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 32 x^{16} + 392 x^{14} + 2337 x^{12} + 7216 x^{10} + 11256 x^{8} + 7946 x^{6} + 2128 x^{4} + 120 x^{2} + 1 \)

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:  \(-314658276613728674017349730304=-\,2^{18}\cdot 13^{12}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.53$
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:  $2, 13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{47} a^{14} - \frac{6}{47} a^{12} + \frac{14}{47} a^{10} - \frac{11}{47} a^{8} - \frac{4}{47} a^{6} - \frac{21}{47} a^{4} - \frac{18}{47} a^{2} - \frac{19}{47}$, $\frac{1}{47} a^{15} - \frac{6}{47} a^{13} + \frac{14}{47} a^{11} - \frac{11}{47} a^{9} - \frac{4}{47} a^{7} - \frac{21}{47} a^{5} - \frac{18}{47} a^{3} - \frac{19}{47} a$, $\frac{1}{222937027} a^{16} + \frac{961061}{222937027} a^{14} + \frac{88486143}{222937027} a^{12} - \frac{4636454}{222937027} a^{10} + \frac{42411312}{222937027} a^{8} - \frac{46570109}{222937027} a^{6} + \frac{61193798}{222937027} a^{4} - \frac{51389547}{222937027} a^{2} - \frac{8556418}{222937027}$, $\frac{1}{222937027} a^{17} + \frac{961061}{222937027} a^{15} + \frac{88486143}{222937027} a^{13} - \frac{4636454}{222937027} a^{11} + \frac{42411312}{222937027} a^{9} - \frac{46570109}{222937027} a^{7} + \frac{61193798}{222937027} a^{5} - \frac{51389547}{222937027} a^{3} - \frac{8556418}{222937027} a$

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

Class group and class number

$C_{3}\times C_{3}\times C_{12}$, which has order $108$ (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:  \( -\frac{476684}{7191517} a^{17} - \frac{15271285}{7191517} a^{15} - \frac{187385408}{7191517} a^{13} - \frac{1119924316}{7191517} a^{11} - \frac{3470936444}{7191517} a^{9} - \frac{5447105072}{7191517} a^{7} - \frac{3892443380}{7191517} a^{5} - \frac{1081643653}{7191517} a^{3} - \frac{82187328}{7191517} a \) (order $4$)
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:  \( 230602.64598601736 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.3.10309.1, 3.3.169.1, 6.0.6801630784.1, 6.0.1827904.1, 9.9.1095593933629.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$61$61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.6.3.1$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
61.6.3.1$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$