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Properties

Label 18.18.4145498517...0013.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{24}\cdot 7^{12}\cdot 13^{9}$
Root discriminant $57.09$
Ramified primes $3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-953, -26043, -61668, 107821, 299925, -153138, -413984, 159384, 245598, -100088, -68550, 32592, 8330, -5196, -228, 364, -21, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 21*x^16 + 364*x^15 - 228*x^14 - 5196*x^13 + 8330*x^12 + 32592*x^11 - 68550*x^10 - 100088*x^9 + 245598*x^8 + 159384*x^7 - 413984*x^6 - 153138*x^5 + 299925*x^4 + 107821*x^3 - 61668*x^2 - 26043*x - 953)
 
gp: K = bnfinit(x^18 - 9*x^17 - 21*x^16 + 364*x^15 - 228*x^14 - 5196*x^13 + 8330*x^12 + 32592*x^11 - 68550*x^10 - 100088*x^9 + 245598*x^8 + 159384*x^7 - 413984*x^6 - 153138*x^5 + 299925*x^4 + 107821*x^3 - 61668*x^2 - 26043*x - 953, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 21 x^{16} + 364 x^{15} - 228 x^{14} - 5196 x^{13} + 8330 x^{12} + 32592 x^{11} - 68550 x^{10} - 100088 x^{9} + 245598 x^{8} + 159384 x^{7} - 413984 x^{6} - 153138 x^{5} + 299925 x^{4} + 107821 x^{3} - 61668 x^{2} - 26043 x - 953 \)

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:  \(41454985178292648293852083940013=3^{24}\cdot 7^{12}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.09$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(819=3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{819}(64,·)$, $\chi_{819}(1,·)$, $\chi_{819}(781,·)$, $\chi_{819}(142,·)$, $\chi_{819}(79,·)$, $\chi_{819}(337,·)$, $\chi_{819}(274,·)$, $\chi_{819}(25,·)$, $\chi_{819}(415,·)$, $\chi_{819}(352,·)$, $\chi_{819}(610,·)$, $\chi_{819}(547,·)$, $\chi_{819}(298,·)$, $\chi_{819}(235,·)$, $\chi_{819}(688,·)$, $\chi_{819}(625,·)$, $\chi_{819}(571,·)$, $\chi_{819}(508,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{508} a^{16} + \frac{27}{127} a^{15} + \frac{13}{127} a^{14} - \frac{23}{127} a^{13} - \frac{29}{254} a^{12} + \frac{13}{127} a^{11} + \frac{59}{254} a^{10} - \frac{18}{127} a^{9} - \frac{51}{254} a^{8} - \frac{55}{127} a^{7} + \frac{36}{127} a^{6} + \frac{21}{254} a^{5} - \frac{107}{254} a^{4} - \frac{105}{254} a^{3} + \frac{165}{508} a^{2} - \frac{49}{127} a - \frac{19}{508}$, $\frac{1}{10556685455623739727340116202772} a^{17} - \frac{7905849164269073548381371983}{10556685455623739727340116202772} a^{16} + \frac{184918018828532758225175882750}{2639171363905934931835029050693} a^{15} - \frac{2564519194214487661249617786}{20780876881149093951456921659} a^{14} + \frac{154603046545358581790340556637}{5278342727811869863670058101386} a^{13} + \frac{575988992917059733781909076553}{5278342727811869863670058101386} a^{12} + \frac{1146548085682980485282648891791}{5278342727811869863670058101386} a^{11} - \frac{10343609772102234220996513771}{41561753762298187902913843318} a^{10} + \frac{739776112483777617573319084795}{5278342727811869863670058101386} a^{9} + \frac{526732492083434000339768071639}{5278342727811869863670058101386} a^{8} + \frac{93275959277827962336938592838}{2639171363905934931835029050693} a^{7} + \frac{972327273535368198802797588763}{5278342727811869863670058101386} a^{6} + \frac{985945354912656922940869996559}{2639171363905934931835029050693} a^{5} - \frac{1029424784175405787807406157573}{2639171363905934931835029050693} a^{4} - \frac{610526930109915911610399751217}{10556685455623739727340116202772} a^{3} + \frac{1024972483284113358567263536453}{10556685455623739727340116202772} a^{2} + \frac{939436731316089436558241492357}{10556685455623739727340116202772} a - \frac{1174531929808668257980712360307}{10556685455623739727340116202772}$

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:  \( 7477366325.71 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, 6.6.14414517.1, 6.6.5274997.1, 6.6.34609255317.1, 6.6.34609255317.2, 9.9.62523502209.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$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}$
7Data not computed
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$