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Properties

Label 18.0.38767561196...7232.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 37^{10}$
Root discriminant $20.44$
Ramified primes $2, 3, 37$
Class number $1$
Class group Trivial
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 14, 68, 138, 122, 23, 25, 50, 20, 70, 73, 16, 45, -20, -22, 9, 1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + x^16 + 9*x^15 - 22*x^14 - 20*x^13 + 45*x^12 + 16*x^11 + 73*x^10 + 70*x^9 + 20*x^8 + 50*x^7 + 25*x^6 + 23*x^5 + 122*x^4 + 138*x^3 + 68*x^2 + 14*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + x^16 + 9*x^15 - 22*x^14 - 20*x^13 + 45*x^12 + 16*x^11 + 73*x^10 + 70*x^9 + 20*x^8 + 50*x^7 + 25*x^6 + 23*x^5 + 122*x^4 + 138*x^3 + 68*x^2 + 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + x^{16} + 9 x^{15} - 22 x^{14} - 20 x^{13} + 45 x^{12} + 16 x^{11} + 73 x^{10} + 70 x^{9} + 20 x^{8} + 50 x^{7} + 25 x^{6} + 23 x^{5} + 122 x^{4} + 138 x^{3} + 68 x^{2} + 14 x + 1 \)

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:  \(-387675611964622937567232=-\,2^{12}\cdot 3^{9}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.44$
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:  $2, 3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} + \frac{8}{23} a^{15} + \frac{6}{23} a^{14} + \frac{9}{23} a^{13} - \frac{7}{23} a^{12} + \frac{7}{23} a^{11} - \frac{10}{23} a^{10} - \frac{8}{23} a^{9} + \frac{10}{23} a^{8} - \frac{7}{23} a^{7} + \frac{10}{23} a^{6} + \frac{5}{23} a^{5} + \frac{9}{23} a^{4} + \frac{6}{23} a^{3} - \frac{7}{23} a^{2} + \frac{5}{23}$, $\frac{1}{32307669657005069} a^{17} + \frac{453628973118091}{32307669657005069} a^{16} - \frac{2943065257200439}{32307669657005069} a^{15} - \frac{10077723684170126}{32307669657005069} a^{14} + \frac{341411235899876}{873180261000137} a^{13} + \frac{10695944235253026}{32307669657005069} a^{12} - \frac{3307354099349614}{32307669657005069} a^{11} - \frac{13505970217644552}{32307669657005069} a^{10} + \frac{9947543725003264}{32307669657005069} a^{9} - \frac{2022583162551315}{32307669657005069} a^{8} - \frac{11028542953585107}{32307669657005069} a^{7} - \frac{10078707982926039}{32307669657005069} a^{6} + \frac{9465309981508714}{32307669657005069} a^{5} - \frac{8568955508001219}{32307669657005069} a^{4} + \frac{8524423383622956}{32307669657005069} a^{3} - \frac{440321064706771}{1404681289435003} a^{2} - \frac{5485114337929051}{32307669657005069} a - \frac{587705323431761}{1404681289435003}$

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:  \( -\frac{16188770160}{3354914573} a^{17} + \frac{51329565404}{3354914573} a^{16} - \frac{24864629234}{3354914573} a^{15} - \frac{141731219309}{3354914573} a^{14} + \frac{380457598637}{3354914573} a^{13} + \frac{259733715114}{3354914573} a^{12} - \frac{775136334131}{3354914573} a^{11} - \frac{128024004474}{3354914573} a^{10} - \frac{1154654195510}{3354914573} a^{9} - \frac{936593679275}{3354914573} a^{8} - \frac{159036722350}{3354914573} a^{7} - \frac{775257469923}{3354914573} a^{6} - \frac{276265967810}{3354914573} a^{5} - \frac{320016893584}{3354914573} a^{4} - \frac{1921128564992}{3354914573} a^{3} - \frac{1906156733363}{3354914573} a^{2} - \frac{764553498117}{3354914573} a - \frac{86694574527}{3354914573} \) (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:  \( 65819.9777767664 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.148.1, 6.0.591408.1, 6.0.36963.1

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

Sibling fields

Degree 12 sibling: 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$