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Properties

Label 18.0.45305484158...5968.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{15}\cdot 13^{12}$
Root discriminant $44.42$
Ramified primes $2, 7, 13$
Class number $156$ (GRH)
Class group $[156]$ (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![7168, 2688, 4256, 7728, 2212, 2562, 4810, 51, -771, 1339, 653, -118, 90, 68, 8, 19, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 5*x^16 + 19*x^15 + 8*x^14 + 68*x^13 + 90*x^12 - 118*x^11 + 653*x^10 + 1339*x^9 - 771*x^8 + 51*x^7 + 4810*x^6 + 2562*x^5 + 2212*x^4 + 7728*x^3 + 4256*x^2 + 2688*x + 7168)
 
gp: K = bnfinit(x^18 - x^17 + 5*x^16 + 19*x^15 + 8*x^14 + 68*x^13 + 90*x^12 - 118*x^11 + 653*x^10 + 1339*x^9 - 771*x^8 + 51*x^7 + 4810*x^6 + 2562*x^5 + 2212*x^4 + 7728*x^3 + 4256*x^2 + 2688*x + 7168, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 5 x^{16} + 19 x^{15} + 8 x^{14} + 68 x^{13} + 90 x^{12} - 118 x^{11} + 653 x^{10} + 1339 x^{9} - 771 x^{8} + 51 x^{7} + 4810 x^{6} + 2562 x^{5} + 2212 x^{4} + 7728 x^{3} + 4256 x^{2} + 2688 x + 7168 \)

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:  \(-453054841581020940100929875968=-\,2^{12}\cdot 7^{15}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.42$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{10} + \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{7} + \frac{3}{32} a^{6} + \frac{7}{32} a^{5} - \frac{3}{16} a^{4} - \frac{5}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{11} + \frac{1}{32} a^{9} + \frac{1}{32} a^{7} - \frac{1}{8} a^{6} - \frac{1}{32} a^{5} + \frac{7}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{10} - \frac{1}{32} a^{9} - \frac{3}{64} a^{8} - \frac{1}{8} a^{7} - \frac{7}{64} a^{6} - \frac{7}{32} a^{5} - \frac{7}{32} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{1}{64} a^{9} - \frac{1}{16} a^{8} - \frac{3}{64} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{3}{16} a^{4} - \frac{3}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{14} - \frac{1}{32} a^{9} - \frac{1}{64} a^{8} + \frac{1}{32} a^{7} + \frac{15}{128} a^{6} + \frac{1}{32} a^{5} - \frac{15}{64} a^{4} + \frac{7}{32} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{15} + \frac{1}{64} a^{9} - \frac{1}{32} a^{8} - \frac{9}{128} a^{7} - \frac{1}{8} a^{6} + \frac{15}{64} a^{5} - \frac{7}{32} a^{4} + \frac{5}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{23552} a^{16} + \frac{33}{23552} a^{15} + \frac{5}{11776} a^{14} + \frac{57}{11776} a^{13} - \frac{55}{11776} a^{12} + \frac{183}{11776} a^{11} + \frac{87}{5888} a^{10} - \frac{193}{5888} a^{9} - \frac{1047}{23552} a^{8} - \frac{2143}{23552} a^{7} + \frac{1085}{11776} a^{6} - \frac{951}{11776} a^{5} - \frac{1087}{5888} a^{4} + \frac{677}{1472} a^{3} + \frac{85}{736} a^{2} - \frac{1}{184} a - \frac{3}{23}$, $\frac{1}{995290788988928} a^{17} - \frac{1768382067}{248822697247232} a^{16} + \frac{429990890485}{995290788988928} a^{15} + \frac{16396525239}{124411348623616} a^{14} + \frac{938362896811}{124411348623616} a^{13} - \frac{1725157227395}{248822697247232} a^{12} + \frac{3827267667019}{497645394494464} a^{11} - \frac{221364042135}{31102837155904} a^{10} + \frac{48171283218717}{995290788988928} a^{9} + \frac{7154149862027}{248822697247232} a^{8} + \frac{85013756011333}{995290788988928} a^{7} - \frac{6732072977687}{124411348623616} a^{6} + \frac{6604719855237}{497645394494464} a^{5} + \frac{8183223583403}{248822697247232} a^{4} - \frac{185765591515}{2704594535296} a^{3} + \frac{541111987137}{1352297267648} a^{2} - \frac{2016996073725}{7775709288976} a + \frac{268219213763}{971963661122}$

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

Class group and class number

$C_{156}$, which has order $156$ (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:  \( 392092503.6861413 \) (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

\(\Q(\sqrt{-7}) \), 3.3.8281.1, 3.1.676.1, 6.0.480024727.1, 6.0.156742768.2, 9.3.36343632130624.4

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$7$7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13Data not computed