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Properties

Label 18.0.24634059670...0336.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 59^{8}$
Root discriminant $15.43$
Ramified primes $2, 59$
Class number $1$
Class group Trivial
Galois group $D_{18}$ (as 18T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, 16, 0, 20, 0, 13, 0, 33, 0, 10, 0, -7, 0, -2, 0, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + x^16 - 2*x^14 - 7*x^12 + 10*x^10 + 33*x^8 + 13*x^6 + 20*x^4 + 16*x^2 + 4)
 
gp: K = bnfinit(x^18 + x^16 - 2*x^14 - 7*x^12 + 10*x^10 + 33*x^8 + 13*x^6 + 20*x^4 + 16*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{18} + x^{16} - 2 x^{14} - 7 x^{12} + 10 x^{10} + 33 x^{8} + 13 x^{6} + 20 x^{4} + 16 x^{2} + 4 \)

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:  \(-2463405967062215950336=-\,2^{24}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.43$
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:  $2, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{274298} a^{16} + \frac{3709}{137149} a^{14} + \frac{22555}{274298} a^{12} - \frac{15676}{137149} a^{10} - \frac{3053}{11926} a^{8} + \frac{38806}{137149} a^{6} + \frac{16932}{137149} a^{4} - \frac{43830}{137149} a^{2} - \frac{43972}{137149}$, $\frac{1}{548596} a^{17} - \frac{129731}{548596} a^{15} - \frac{57297}{274298} a^{13} + \frac{105797}{548596} a^{11} + \frac{1455}{11926} a^{9} - \frac{1}{2} a^{8} - \frac{59537}{548596} a^{7} - \frac{103285}{548596} a^{5} + \frac{93319}{274298} a^{3} - \frac{1}{2} a^{2} - \frac{21986}{137149} a$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{1497}{11926} a^{17} - \frac{820}{5963} a^{15} + \frac{1817}{5963} a^{13} + \frac{11097}{11926} a^{11} - \frac{7905}{5963} a^{9} - \frac{55739}{11926} a^{7} - \frac{14871}{11926} a^{5} - \frac{12647}{11926} a^{3} - \frac{11399}{5963} a \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6192.4028326 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{18}$ (as 18T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $D_{18}$
Character table for $D_{18}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.59.1, 6.0.222784.1, 9.1.775511104.1

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

Sibling fields

Galois closure: data not computed
Degree 18 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ R

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
$2$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$