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Properties

Label 18.18.1044300241...3613.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{44}\cdot 13^{9}$
Root discriminant $52.88$
Ramified primes $3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-159, -297, 8559, -23004, -10584, 107991, -104295, -56754, 118251, -12218, -44460, 13338, 6864, -3024, -360, 276, -9, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 9*x^16 + 276*x^15 - 360*x^14 - 3024*x^13 + 6864*x^12 + 13338*x^11 - 44460*x^10 - 12218*x^9 + 118251*x^8 - 56754*x^7 - 104295*x^6 + 107991*x^5 - 10584*x^4 - 23004*x^3 + 8559*x^2 - 297*x - 159)
 
gp: K = bnfinit(x^18 - 9*x^17 - 9*x^16 + 276*x^15 - 360*x^14 - 3024*x^13 + 6864*x^12 + 13338*x^11 - 44460*x^10 - 12218*x^9 + 118251*x^8 - 56754*x^7 - 104295*x^6 + 107991*x^5 - 10584*x^4 - 23004*x^3 + 8559*x^2 - 297*x - 159, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 9 x^{16} + 276 x^{15} - 360 x^{14} - 3024 x^{13} + 6864 x^{12} + 13338 x^{11} - 44460 x^{10} - 12218 x^{9} + 118251 x^{8} - 56754 x^{7} - 104295 x^{6} + 107991 x^{5} - 10584 x^{4} - 23004 x^{3} + 8559 x^{2} - 297 x - 159 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10443002414754749649962321483613=3^{44}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.88$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(351=3^{3}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{351}(64,·)$, $\chi_{351}(1,·)$, $\chi_{351}(259,·)$, $\chi_{351}(196,·)$, $\chi_{351}(142,·)$, $\chi_{351}(79,·)$, $\chi_{351}(337,·)$, $\chi_{351}(274,·)$, $\chi_{351}(25,·)$, $\chi_{351}(220,·)$, $\chi_{351}(157,·)$, $\chi_{351}(103,·)$, $\chi_{351}(40,·)$, $\chi_{351}(298,·)$, $\chi_{351}(235,·)$, $\chi_{351}(181,·)$, $\chi_{351}(118,·)$, $\chi_{351}(313,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{475841472328900505091739568831} a^{17} + \frac{237034030310632616109384031477}{475841472328900505091739568831} a^{16} + \frac{187270903909199858009483579929}{475841472328900505091739568831} a^{15} + \frac{24341380056272874695412324780}{475841472328900505091739568831} a^{14} + \frac{70835545735500653554426117314}{475841472328900505091739568831} a^{13} + \frac{24184210412881534780267346753}{475841472328900505091739568831} a^{12} + \frac{198101714925313460732624596701}{475841472328900505091739568831} a^{11} - \frac{227517229316689588553534659327}{475841472328900505091739568831} a^{10} + \frac{23964182464906176753334852854}{475841472328900505091739568831} a^{9} + \frac{168153873620399331930785280101}{475841472328900505091739568831} a^{8} + \frac{133385832147417720946508847963}{475841472328900505091739568831} a^{7} - \frac{20322045221946619730526950065}{475841472328900505091739568831} a^{6} - \frac{182881977952094186987025620716}{475841472328900505091739568831} a^{5} - \frac{142680769883609378811110671537}{475841472328900505091739568831} a^{4} + \frac{116481737738676652518414470275}{475841472328900505091739568831} a^{3} + \frac{119434844828192048182955617720}{475841472328900505091739568831} a^{2} - \frac{170741573424693243264312422128}{475841472328900505091739568831} a - \frac{3304169787961732981598450548}{8978140987337745379089425827}$

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:  $17$
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:  \( 9384873349.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 6.6.14414517.1, \(\Q(\zeta_{27})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ R $18$ $18$ $18$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ $18$

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
$3$3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
13Data not computed