↑

Properties

Label 18.0.18261252046...3072.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{45}\cdot 11^{9}$
Root discriminant $103.40$
Ramified primes $2, 3, 11$
Class number $694564$ (GRH)
Class group $[2, 347282]$ (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![7073843073, 0, 17363069361, 0, 10523072340, 0, 2455383546, 0, 286992882, 0, 18842967, 0, 726726, 0, 16335, 0, 198, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 198*x^16 + 16335*x^14 + 726726*x^12 + 18842967*x^10 + 286992882*x^8 + 2455383546*x^6 + 10523072340*x^4 + 17363069361*x^2 + 7073843073)
 
gp: K = bnfinit(x^18 + 198*x^16 + 16335*x^14 + 726726*x^12 + 18842967*x^10 + 286992882*x^8 + 2455383546*x^6 + 10523072340*x^4 + 17363069361*x^2 + 7073843073, 1)
 

Normalized defining polynomial

\( x^{18} + 198 x^{16} + 16335 x^{14} + 726726 x^{12} + 18842967 x^{10} + 286992882 x^{8} + 2455383546 x^{6} + 10523072340 x^{4} + 17363069361 x^{2} + 7073843073 \)

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:  \(-1826125204659502814026858937270403072=-\,2^{18}\cdot 3^{45}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.40$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1188=2^{2}\cdot 3^{3}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1188}(1,·)$, $\chi_{1188}(131,·)$, $\chi_{1188}(133,·)$, $\chi_{1188}(263,·)$, $\chi_{1188}(265,·)$, $\chi_{1188}(395,·)$, $\chi_{1188}(397,·)$, $\chi_{1188}(527,·)$, $\chi_{1188}(529,·)$, $\chi_{1188}(659,·)$, $\chi_{1188}(661,·)$, $\chi_{1188}(791,·)$, $\chi_{1188}(793,·)$, $\chi_{1188}(923,·)$, $\chi_{1188}(925,·)$, $\chi_{1188}(1055,·)$, $\chi_{1188}(1057,·)$, $\chi_{1188}(1187,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{11} a^{2}$, $\frac{1}{11} a^{3}$, $\frac{1}{121} a^{4}$, $\frac{1}{121} a^{5}$, $\frac{1}{1331} a^{6}$, $\frac{1}{1331} a^{7}$, $\frac{1}{14641} a^{8}$, $\frac{1}{14641} a^{9}$, $\frac{1}{161051} a^{10}$, $\frac{1}{161051} a^{11}$, $\frac{1}{1771561} a^{12}$, $\frac{1}{1771561} a^{13}$, $\frac{1}{19487171} a^{14}$, $\frac{1}{19487171} a^{15}$, $\frac{1}{214358881} a^{16}$, $\frac{1}{214358881} a^{17}$

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

Class group and class number

$C_{2}\times C_{347282}$, which has order $694564$ (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:  \( 40934.03294431194 \) (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

\(\Q(\sqrt{-33}) \), \(\Q(\zeta_{9})^+\), 6.0.1676676672.2, \(\Q(\zeta_{27})^+\)

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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ R $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
11Data not computed