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Properties

Label 18.0.50078803235...8288.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 43^{8}$
Root discriminant $30.43$
Ramified primes $2, 3, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -729, 3618, -5193, 3897, 1149, -3449, -2358, 3279, 2735, -3414, 24, 936, -309, 0, 2, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 + 2*x^15 - 309*x^13 + 936*x^12 + 24*x^11 - 3414*x^10 + 2735*x^9 + 3279*x^8 - 2358*x^7 - 3449*x^6 + 1149*x^5 + 3897*x^4 - 5193*x^3 + 3618*x^2 - 729*x + 81)
 
gp: K = bnfinit(x^18 - 6*x^17 + 15*x^16 + 2*x^15 - 309*x^13 + 936*x^12 + 24*x^11 - 3414*x^10 + 2735*x^9 + 3279*x^8 - 2358*x^7 - 3449*x^6 + 1149*x^5 + 3897*x^4 - 5193*x^3 + 3618*x^2 - 729*x + 81, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 15 x^{16} + 2 x^{15} - 309 x^{13} + 936 x^{12} + 24 x^{11} - 3414 x^{10} + 2735 x^{9} + 3279 x^{8} - 2358 x^{7} - 3449 x^{6} + 1149 x^{5} + 3897 x^{4} - 5193 x^{3} + 3618 x^{2} - 729 x + 81 \)

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:  \(-500788032352702912536588288=-\,2^{12}\cdot 3^{21}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.43$
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:  $2, 3, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{2}{9} a^{7} - \frac{4}{9} a^{6} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{14} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} + \frac{1}{3} a^{7} + \frac{2}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{13} + \frac{1}{9} a^{9} + \frac{8}{27} a^{7} - \frac{2}{9} a^{6} + \frac{10}{27} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{893604591026368149362123853} a^{17} + \frac{1473883005655347620189894}{99289399002929794373569317} a^{16} + \frac{14916466084725253093520873}{297868197008789383120707951} a^{15} - \frac{46866820612476589388556334}{893604591026368149362123853} a^{14} + \frac{2917603763933239095219733}{297868197008789383120707951} a^{13} - \frac{12235563140462885111242327}{297868197008789383120707951} a^{12} + \frac{4313413323203946486950468}{99289399002929794373569317} a^{11} - \frac{4107270598370950292945854}{297868197008789383120707951} a^{10} + \frac{39175564254974143350639887}{297868197008789383120707951} a^{9} + \frac{45529167283689519175760018}{893604591026368149362123853} a^{8} - \frac{125649335019315868158577312}{297868197008789383120707951} a^{7} + \frac{309582240262199914054877}{99289399002929794373569317} a^{6} - \frac{379380632974052914392788252}{893604591026368149362123853} a^{5} - \frac{114905833473164803253098943}{297868197008789383120707951} a^{4} - \frac{30500955264714177076995032}{99289399002929794373569317} a^{3} - \frac{4422161125493933315146265}{99289399002929794373569317} a^{2} + \frac{665561371078957973916852}{11032155444769977152618813} a + \frac{1650899977864096289120641}{11032155444769977152618813}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{401196644787568}{202869765251160561} a^{17} + \frac{2780176272386123}{202869765251160561} a^{16} - \frac{2553594855311497}{67623255083720187} a^{15} + \frac{2035820831082682}{202869765251160561} a^{14} + \frac{5928647494972861}{202869765251160561} a^{13} + \frac{44065502312559695}{67623255083720187} a^{12} - \frac{159914923874812517}{67623255083720187} a^{11} + \frac{57445775621800477}{67623255083720187} a^{10} + \frac{568893680401900241}{67623255083720187} a^{9} - \frac{1913259580062469769}{202869765251160561} a^{8} - \frac{1712124673517601539}{202869765251160561} a^{7} + \frac{217265238031621886}{22541085027906729} a^{6} + \frac{2051058203705617694}{202869765251160561} a^{5} - \frac{1172908021814389636}{202869765251160561} a^{4} - \frac{249385173333187448}{22541085027906729} a^{3} + \frac{104254428662278865}{7513695009302243} a^{2} - \frac{212096296342039069}{22541085027906729} a + \frac{12186545520228041}{7513695009302243} \) (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:  \( 5044619.947255855 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.4306704645312.1

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

Sibling fields

Degree 9 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$43$43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.2.3$x^{3} - 3483$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.3$x^{3} - 3483$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.2.2$x^{3} + 387$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.2$x^{3} + 387$$3$$1$$2$$C_3$$[\ ]_{3}$