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Properties

Label 18.18.6966114824...3313.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{45}\cdot 11^{9}$
Root discriminant $51.70$
Ramified primes $3, 11$
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![-40553, -99144, 531441, 110160, -1180980, -33048, 1010394, 3672, -433026, -136, 104247, 0, -14742, 0, 1215, 0, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 54*x^16 + 1215*x^14 - 14742*x^12 + 104247*x^10 - 136*x^9 - 433026*x^8 + 3672*x^7 + 1010394*x^6 - 33048*x^5 - 1180980*x^4 + 110160*x^3 + 531441*x^2 - 99144*x - 40553)
 
gp: K = bnfinit(x^18 - 54*x^16 + 1215*x^14 - 14742*x^12 + 104247*x^10 - 136*x^9 - 433026*x^8 + 3672*x^7 + 1010394*x^6 - 33048*x^5 - 1180980*x^4 + 110160*x^3 + 531441*x^2 - 99144*x - 40553, 1)
 

Normalized defining polynomial

\( x^{18} - 54 x^{16} + 1215 x^{14} - 14742 x^{12} + 104247 x^{10} - 136 x^{9} - 433026 x^{8} + 3672 x^{7} + 1010394 x^{6} - 33048 x^{5} - 1180980 x^{4} + 110160 x^{3} + 531441 x^{2} - 99144 x - 40553 \)

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:  \(6966114824903498893840251683313=3^{45}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.70$
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:  $3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(297=3^{3}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{297}(1,·)$, $\chi_{297}(67,·)$, $\chi_{297}(133,·)$, $\chi_{297}(65,·)$, $\chi_{297}(265,·)$, $\chi_{297}(34,·)$, $\chi_{297}(131,·)$, $\chi_{297}(100,·)$, $\chi_{297}(199,·)$, $\chi_{297}(197,·)$, $\chi_{297}(32,·)$, $\chi_{297}(98,·)$, $\chi_{297}(164,·)$, $\chi_{297}(166,·)$, $\chi_{297}(230,·)$, $\chi_{297}(232,·)$, $\chi_{297}(263,·)$, $\chi_{297}(296,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{74} a^{9} - \frac{27}{74} a^{7} + \frac{21}{74} a^{5} + \frac{2}{37} a^{3} - \frac{11}{74} a - \frac{31}{74}$, $\frac{1}{74} a^{10} - \frac{27}{74} a^{8} + \frac{21}{74} a^{6} + \frac{2}{37} a^{4} - \frac{11}{74} a^{2} - \frac{31}{74} a$, $\frac{1}{74} a^{11} + \frac{16}{37} a^{7} - \frac{21}{74} a^{5} + \frac{23}{74} a^{3} - \frac{31}{74} a^{2} - \frac{1}{74} a - \frac{23}{74}$, $\frac{1}{74} a^{12} + \frac{16}{37} a^{8} - \frac{21}{74} a^{6} + \frac{23}{74} a^{4} - \frac{31}{74} a^{3} - \frac{1}{74} a^{2} - \frac{23}{74} a$, $\frac{1}{74} a^{13} + \frac{29}{74} a^{7} + \frac{17}{74} a^{5} - \frac{31}{74} a^{4} + \frac{19}{74} a^{3} - \frac{23}{74} a^{2} - \frac{9}{37} a + \frac{15}{37}$, $\frac{1}{87838} a^{14} - \frac{206}{43919} a^{13} - \frac{21}{43919} a^{12} - \frac{275}{43919} a^{11} - \frac{247}{43919} a^{10} - \frac{59}{87838} a^{9} + \frac{26379}{87838} a^{8} - \frac{32099}{87838} a^{7} - \frac{1113}{87838} a^{6} - \frac{15937}{43919} a^{5} + \frac{31901}{87838} a^{4} + \frac{26445}{87838} a^{3} - \frac{4099}{43919} a^{2} + \frac{9015}{87838} a - \frac{41171}{87838}$, $\frac{1}{87838} a^{15} - \frac{45}{87838} a^{13} - \frac{49}{87838} a^{12} - \frac{377}{87838} a^{11} + \frac{577}{87838} a^{10} - \frac{303}{87838} a^{9} + \frac{8235}{87838} a^{8} - \frac{9076}{43919} a^{7} - \frac{2573}{87838} a^{6} - \frac{5203}{87838} a^{5} - \frac{35}{2374} a^{4} - \frac{32071}{87838} a^{3} + \frac{18471}{43919} a^{2} + \frac{2805}{87838} a + \frac{29453}{87838}$, $\frac{1}{87838} a^{16} + \frac{403}{87838} a^{13} + \frac{107}{87838} a^{12} - \frac{433}{87838} a^{11} + \frac{10}{43919} a^{10} - \frac{355}{87838} a^{9} + \frac{10537}{43919} a^{8} + \frac{23665}{87838} a^{7} + \frac{17119}{87838} a^{6} + \frac{10412}{43919} a^{5} - \frac{3341}{43919} a^{4} + \frac{16227}{87838} a^{3} - \frac{16279}{43919} a^{2} - \frac{2031}{43919} a + \frac{11976}{43919}$, $\frac{1}{87838} a^{17} - \frac{1}{2374} a^{13} - \frac{125}{87838} a^{12} - \frac{299}{87838} a^{11} + \frac{249}{43919} a^{10} - \frac{255}{87838} a^{9} + \frac{10643}{87838} a^{8} - \frac{17569}{43919} a^{7} - \frac{3312}{43919} a^{6} + \frac{24875}{87838} a^{5} + \frac{4106}{43919} a^{4} + \frac{1895}{43919} a^{3} + \frac{18928}{43919} a^{2} + \frac{39745}{87838} a + \frac{7729}{43919}$

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:  \( 4187524895.96 \) (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

\(\Q(\sqrt{33}) \), \(\Q(\zeta_{9})^+\), 6.6.26198073.1, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ $18$ R $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ $18$

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
3Data not computed
11Data not computed