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Properties

Label 18.0.11568313814...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 5^{6}$
Root discriminant $14.80$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
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, -6, 15, -18, 12, 18, -24, 30, 51, -118, 201, -210, 186, -144, 93, -50, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 50*x^15 + 93*x^14 - 144*x^13 + 186*x^12 - 210*x^11 + 201*x^10 - 118*x^9 + 51*x^8 + 30*x^7 - 24*x^6 + 18*x^5 + 12*x^4 - 18*x^3 + 15*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 - 50*x^15 + 93*x^14 - 144*x^13 + 186*x^12 - 210*x^11 + 201*x^10 - 118*x^9 + 51*x^8 + 30*x^7 - 24*x^6 + 18*x^5 + 12*x^4 - 18*x^3 + 15*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} - 50 x^{15} + 93 x^{14} - 144 x^{13} + 186 x^{12} - 210 x^{11} + 201 x^{10} - 118 x^{9} + 51 x^{8} + 30 x^{7} - 24 x^{6} + 18 x^{5} + 12 x^{4} - 18 x^{3} + 15 x^{2} - 6 x + 1 \)

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:  \(-1156831381426176000000=-\,2^{18}\cdot 3^{24}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.80$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{53} a^{16} - \frac{20}{53} a^{15} - \frac{13}{53} a^{14} - \frac{1}{53} a^{13} + \frac{2}{53} a^{12} - \frac{17}{53} a^{11} + \frac{8}{53} a^{10} - \frac{19}{53} a^{9} + \frac{22}{53} a^{8} - \frac{25}{53} a^{7} + \frac{12}{53} a^{6} - \frac{26}{53} a^{5} + \frac{17}{53} a^{4} - \frac{6}{53} a^{3} + \frac{5}{53} a^{2} - \frac{6}{53} a + \frac{13}{53}$, $\frac{1}{879929679817} a^{17} + \frac{7050043504}{879929679817} a^{16} + \frac{388913353353}{879929679817} a^{15} - \frac{176261253311}{879929679817} a^{14} - \frac{235905705814}{879929679817} a^{13} + \frac{183133968697}{879929679817} a^{12} + \frac{329128805799}{879929679817} a^{11} + \frac{226873664100}{879929679817} a^{10} + \frac{14924223620}{879929679817} a^{9} + \frac{316677688607}{879929679817} a^{8} + \frac{402973940696}{879929679817} a^{7} + \frac{230245754724}{879929679817} a^{6} - \frac{99810147655}{879929679817} a^{5} + \frac{351681317793}{879929679817} a^{4} - \frac{186115573731}{879929679817} a^{3} + \frac{108225238472}{879929679817} a^{2} - \frac{92881695542}{879929679817} a + \frac{117962250823}{879929679817}$

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:  \( \frac{140520117}{139295501} a^{17} - \frac{768423177}{139295501} a^{16} + \frac{2534897291}{139295501} a^{15} - \frac{5648205735}{139295501} a^{14} + \frac{9978823686}{139295501} a^{13} - \frac{14761222914}{139295501} a^{12} + \frac{17986356723}{139295501} a^{11} - \frac{19477615296}{139295501} a^{10} + \frac{17309550539}{139295501} a^{9} - \frac{6653384991}{139295501} a^{8} + \frac{2950317459}{139295501} a^{7} + \frac{6091755754}{139295501} a^{6} - \frac{913509207}{139295501} a^{5} + \frac{1794879330}{139295501} a^{4} + \frac{2278867945}{139295501} a^{3} - \frac{1516597125}{139295501} a^{2} + \frac{1073826486}{139295501} a - \frac{230127401}{139295501} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2329.0641692193603 \)
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

\(\Q(\sqrt{-1}) \), 3.1.1620.1, \(\Q(\zeta_{9})^+\), 6.0.10497600.1, 6.0.419904.1, 9.3.4251528000.1

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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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}$
3Data not computed
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$