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Properties

Label 18.0.21399777632...7328.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 61^{12}$
Root discriminant $42.61$
Ramified primes $2, 3, 61$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2187, -19683, 78651, -176418, 233688, -172794, 55882, -2598, 9279, -11331, 7995, -3390, 916, -336, 216, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 216*x^14 - 336*x^13 + 916*x^12 - 3390*x^11 + 7995*x^10 - 11331*x^9 + 9279*x^8 - 2598*x^7 + 55882*x^6 - 172794*x^5 + 233688*x^4 - 176418*x^3 + 78651*x^2 - 19683*x + 2187)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 216*x^14 - 336*x^13 + 916*x^12 - 3390*x^11 + 7995*x^10 - 11331*x^9 + 9279*x^8 - 2598*x^7 + 55882*x^6 - 172794*x^5 + 233688*x^4 - 176418*x^3 + 78651*x^2 - 19683*x + 2187, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 216 x^{14} - 336 x^{13} + 916 x^{12} - 3390 x^{11} + 7995 x^{10} - 11331 x^{9} + 9279 x^{8} - 2598 x^{7} + 55882 x^{6} - 172794 x^{5} + 233688 x^{4} - 176418 x^{3} + 78651 x^{2} - 19683 x + 2187 \)

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:  \(-213997776327063094698935267328=-\,2^{12}\cdot 3^{9}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.61$
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:  $2, 3, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{8} - \frac{1}{18} a^{7} + \frac{1}{18} a^{6} - \frac{2}{9} a^{5} - \frac{4}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{18} a^{9} - \frac{1}{6} a^{5} - \frac{2}{9} a^{3} - \frac{1}{6} a$, $\frac{1}{18} a^{10} - \frac{1}{2} a^{5} + \frac{4}{9} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{324} a^{12} - \frac{1}{54} a^{11} - \frac{1}{324} a^{10} + \frac{1}{54} a^{9} - \frac{1}{54} a^{8} - \frac{2}{27} a^{7} + \frac{23}{324} a^{6} - \frac{2}{9} a^{5} - \frac{14}{81} a^{4} - \frac{7}{27} a^{3} - \frac{17}{108} a^{2} - \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{324} a^{13} - \frac{1}{324} a^{11} - \frac{1}{54} a^{9} - \frac{1}{54} a^{8} - \frac{13}{324} a^{7} + \frac{1}{27} a^{6} - \frac{73}{162} a^{5} - \frac{25}{54} a^{4} - \frac{47}{108} a^{3} - \frac{4}{9} a^{2} - \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{324} a^{14} - \frac{1}{54} a^{11} - \frac{7}{324} a^{10} - \frac{1}{324} a^{8} + \frac{2}{27} a^{7} + \frac{1}{108} a^{6} + \frac{7}{27} a^{5} - \frac{71}{324} a^{4} + \frac{13}{54} a^{3} - \frac{13}{54} a^{2} + \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{324} a^{15} - \frac{7}{324} a^{11} - \frac{1}{54} a^{10} - \frac{1}{324} a^{9} + \frac{1}{54} a^{8} + \frac{1}{108} a^{7} + \frac{2}{27} a^{6} - \frac{17}{324} a^{5} - \frac{11}{27} a^{4} - \frac{2}{27} a^{3} - \frac{4}{9} a^{2} - \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{1566349812} a^{16} - \frac{2}{391587453} a^{15} + \frac{91145}{783174906} a^{14} - \frac{637945}{783174906} a^{13} + \frac{157825}{391587453} a^{12} + \frac{6399203}{783174906} a^{11} - \frac{3782464}{391587453} a^{10} + \frac{7471117}{783174906} a^{9} + \frac{21369551}{1566349812} a^{8} + \frac{8889181}{783174906} a^{7} - \frac{13346938}{391587453} a^{6} + \frac{60907513}{783174906} a^{5} - \frac{279535}{87019434} a^{4} + \frac{98680873}{261058302} a^{3} - \frac{145571}{29006478} a^{2} + \frac{260057}{4834413} a - \frac{508289}{2148628}$, $\frac{1}{913181940396} a^{17} + \frac{283}{913181940396} a^{16} + \frac{304747981}{913181940396} a^{15} - \frac{823258253}{913181940396} a^{14} - \frac{298125725}{228295485099} a^{13} - \frac{694590379}{456590970198} a^{12} + \frac{23254738049}{913181940396} a^{11} + \frac{555242251}{83016540036} a^{10} + \frac{1980713300}{228295485099} a^{9} + \frac{3909812747}{228295485099} a^{8} - \frac{19830289903}{913181940396} a^{7} - \frac{36351863509}{913181940396} a^{6} + \frac{13285653335}{304393980132} a^{5} - \frac{146655782881}{304393980132} a^{4} + \frac{16257601147}{50732330022} a^{3} + \frac{4075970429}{16910776674} a^{2} + \frac{796605433}{1878975186} a + \frac{74079185}{313162531}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{206104042}{76098495033} a^{17} - \frac{1751884357}{76098495033} a^{16} + \frac{7167893864}{76098495033} a^{15} - \frac{18721516840}{76098495033} a^{14} + \frac{35337479204}{76098495033} a^{13} - \frac{52031593744}{76098495033} a^{12} + \frac{54530476808}{25366165011} a^{11} - \frac{56186727028}{6918045003} a^{10} + \frac{1342941202150}{76098495033} a^{9} - \frac{1679321661734}{76098495033} a^{8} + \frac{1103669797048}{76098495033} a^{7} - \frac{18906779468}{76098495033} a^{6} + \frac{11526598998524}{76098495033} a^{5} - \frac{1105304398840}{2818462779} a^{4} + \frac{11188381606928}{25366165011} a^{3} - \frac{2249698025876}{8455388337} a^{2} + \frac{83520555650}{939487593} a - \frac{4021472847}{313162531} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 196727117.739 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.44652.1 x3, 3.3.3721.1, 6.0.5981403312.1, 6.0.1607472.1 x2, 6.0.373837707.1, 9.3.89027206895808.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.1607472.1
Degree 9 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$61$61.9.6.1$x^{9} + 1830 x^{6} + 1112579 x^{3} + 226981000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
61.9.6.1$x^{9} + 1830 x^{6} + 1112579 x^{3} + 226981000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$