↑

Properties

Label 18.0.77445535014...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{45}\cdot 5^{9}$
Root discriminant $98.59$
Ramified primes $2, 3, 5$
Class number $488908$ (GRH)
Class group $[2, 244454]$ (GRH)
Galois group $C_{18}$ (as 18T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3000000000, 0, 8100000000, 0, 5400000000, 0, 1386000000, 0, 178200000, 0, 12870000, 0, 546000, 0, 13500, 0, 180, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 180*x^16 + 13500*x^14 + 546000*x^12 + 12870000*x^10 + 178200000*x^8 + 1386000000*x^6 + 5400000000*x^4 + 8100000000*x^2 + 3000000000)
 
gp: K = bnfinit(x^18 + 180*x^16 + 13500*x^14 + 546000*x^12 + 12870000*x^10 + 178200000*x^8 + 1386000000*x^6 + 5400000000*x^4 + 8100000000*x^2 + 3000000000, 1)
 

Normalized defining polynomial

\( x^{18} + 180 x^{16} + 13500 x^{14} + 546000 x^{12} + 12870000 x^{10} + 178200000 x^{8} + 1386000000 x^{6} + 5400000000 x^{4} + 8100000000 x^{2} + 3000000000 \)

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:  \(-774455350146061749097070592000000000=-\,2^{27}\cdot 3^{45}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.59$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1080=2^{3}\cdot 3^{3}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{1080}(1,·)$, $\chi_{1080}(389,·)$, $\chi_{1080}(961,·)$, $\chi_{1080}(841,·)$, $\chi_{1080}(269,·)$, $\chi_{1080}(721,·)$, $\chi_{1080}(149,·)$, $\chi_{1080}(601,·)$, $\chi_{1080}(989,·)$, $\chi_{1080}(481,·)$, $\chi_{1080}(869,·)$, $\chi_{1080}(361,·)$, $\chi_{1080}(749,·)$, $\chi_{1080}(29,·)$, $\chi_{1080}(241,·)$, $\chi_{1080}(629,·)$, $\chi_{1080}(121,·)$, $\chi_{1080}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{10} a^{2}$, $\frac{1}{10} a^{3}$, $\frac{1}{100} a^{4}$, $\frac{1}{100} a^{5}$, $\frac{1}{1000} a^{6}$, $\frac{1}{1000} a^{7}$, $\frac{1}{10000} a^{8}$, $\frac{1}{10000} a^{9}$, $\frac{1}{100000} a^{10}$, $\frac{1}{100000} a^{11}$, $\frac{1}{1000000} a^{12}$, $\frac{1}{1000000} a^{13}$, $\frac{1}{10000000} a^{14}$, $\frac{1}{10000000} a^{15}$, $\frac{1}{100000000} a^{16}$, $\frac{1}{100000000} a^{17}$

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

Class group and class number

$C_{2}\times C_{244454}$, which has order $488908$ (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:  \( -1 \) (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:  \( 40934.0329443 \) (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

\(\Q(\sqrt{-30}) \), \(\Q(\zeta_{9})^+\), 6.0.1259712000.1, \(\Q(\zeta_{27})^+\)

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 R R R $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
3Data not computed
5Data not computed