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Properties

Label 18.6.10982017456...3408.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{24}\cdot 37^{7}$
Root discriminant $27.97$
Ramified primes $2, 3, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times A_4$ (as 18T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -18, -225, 2332, -5283, 5268, -4144, 2748, -1548, 738, -153, -168, 203, -150, 87, -44, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 18*x^16 - 44*x^15 + 87*x^14 - 150*x^13 + 203*x^12 - 168*x^11 - 153*x^10 + 738*x^9 - 1548*x^8 + 2748*x^7 - 4144*x^6 + 5268*x^5 - 5283*x^4 + 2332*x^3 - 225*x^2 - 18*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 18*x^16 - 44*x^15 + 87*x^14 - 150*x^13 + 203*x^12 - 168*x^11 - 153*x^10 + 738*x^9 - 1548*x^8 + 2748*x^7 - 4144*x^6 + 5268*x^5 - 5283*x^4 + 2332*x^3 - 225*x^2 - 18*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 18 x^{16} - 44 x^{15} + 87 x^{14} - 150 x^{13} + 203 x^{12} - 168 x^{11} - 153 x^{10} + 738 x^{9} - 1548 x^{8} + 2748 x^{7} - 4144 x^{6} + 5268 x^{5} - 5283 x^{4} + 2332 x^{3} - 225 x^{2} - 18 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(109820174565148398342033408=2^{12}\cdot 3^{24}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{308} a^{16} - \frac{8}{77} a^{15} + \frac{65}{308} a^{14} + \frac{5}{28} a^{13} - \frac{2}{77} a^{12} - \frac{37}{154} a^{11} + \frac{1}{22} a^{10} - \frac{19}{154} a^{9} - \frac{145}{308} a^{8} + \frac{75}{154} a^{7} - \frac{39}{308} a^{6} + \frac{7}{44} a^{5} + \frac{95}{308} a^{4} - \frac{4}{77} a^{3} - \frac{5}{28} a^{2} - \frac{79}{308} a - \frac{59}{154}$, $\frac{1}{20314404060517344713012} a^{17} + \frac{407858987567625007}{10157202030258672356506} a^{16} - \frac{438622212068272395501}{5078601015129336178253} a^{15} - \frac{2398902558442073623117}{20314404060517344713012} a^{14} + \frac{336606914781007608883}{2902057722931049244716} a^{13} - \frac{1876667198690869015917}{20314404060517344713012} a^{12} - \frac{1259256549863874532675}{10157202030258672356506} a^{11} - \frac{1173435143291265039611}{5078601015129336178253} a^{10} + \frac{2499765560128675836975}{20314404060517344713012} a^{9} - \frac{1892990712961057332229}{5078601015129336178253} a^{8} + \frac{305798451162548166465}{725514430732762311179} a^{7} - \frac{5703786868607391651853}{20314404060517344713012} a^{6} - \frac{304979048678501230697}{5078601015129336178253} a^{5} - \frac{8743850782786390712069}{20314404060517344713012} a^{4} + \frac{1985276602738949130595}{10157202030258672356506} a^{3} - \frac{9713026188685686925621}{20314404060517344713012} a^{2} - \frac{8646802780488214643137}{20314404060517344713012} a - \frac{5275467544530218756869}{20314404060517344713012}$

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:  $11$
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:  \( 1206805.78255 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times A_4$ (as 18T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $S_3\times A_4$
Character table for $S_3\times A_4$

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.148.1, 6.2.242757.1, 9.9.1722821182272.1

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 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
37Data not computed