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Properties

Label 18.0.52327541681...4083.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 29^{8}$
Root discriminant $16.09$
Ramified primes $3, 29$
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![1, 3, 27, 33, 51, 75, 173, -24, 306, -80, 237, -87, 124, -69, 36, -23, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 23*x^15 + 36*x^14 - 69*x^13 + 124*x^12 - 87*x^11 + 237*x^10 - 80*x^9 + 306*x^8 - 24*x^7 + 173*x^6 + 75*x^5 + 51*x^4 + 33*x^3 + 27*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 - 23*x^15 + 36*x^14 - 69*x^13 + 124*x^12 - 87*x^11 + 237*x^10 - 80*x^9 + 306*x^8 - 24*x^7 + 173*x^6 + 75*x^5 + 51*x^4 + 33*x^3 + 27*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 6 x^{16} - 23 x^{15} + 36 x^{14} - 69 x^{13} + 124 x^{12} - 87 x^{11} + 237 x^{10} - 80 x^{9} + 306 x^{8} - 24 x^{7} + 173 x^{6} + 75 x^{5} + 51 x^{4} + 33 x^{3} + 27 x^{2} + 3 x + 1 \)

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:  \(-5232754168105857064083=-\,3^{21}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.09$
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:  $3, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{12} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{669767780359385} a^{17} - \frac{34365238998009}{669767780359385} a^{16} - \frac{12000490149727}{133953556071877} a^{15} + \frac{61048673316056}{669767780359385} a^{14} - \frac{27691495723931}{669767780359385} a^{13} - \frac{267542551361926}{669767780359385} a^{12} - \frac{241297089902973}{669767780359385} a^{11} + \frac{35004515477188}{133953556071877} a^{10} - \frac{110622119492177}{669767780359385} a^{9} + \frac{40455428899579}{669767780359385} a^{8} + \frac{287732754828014}{669767780359385} a^{7} + \frac{243793051117334}{669767780359385} a^{6} - \frac{201500969284958}{669767780359385} a^{5} + \frac{26602579606302}{669767780359385} a^{4} + \frac{114349315073602}{669767780359385} a^{3} - \frac{76997033965542}{669767780359385} a^{2} + \frac{318332744682877}{669767780359385} a + \frac{15514862151941}{133953556071877}$

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:  \( \frac{627500896}{2754806255} a^{17} - \frac{1958616954}{2754806255} a^{16} + \frac{3790002632}{2754806255} a^{15} - \frac{2873051255}{550961251} a^{14} + \frac{23526265189}{2754806255} a^{13} - \frac{42169200547}{2754806255} a^{12} + \frac{78121277788}{2754806255} a^{11} - \frac{55644540884}{2754806255} a^{10} + \frac{138125229427}{2754806255} a^{9} - \frac{65827906823}{2754806255} a^{8} + \frac{167100714664}{2754806255} a^{7} - \frac{38369759581}{2754806255} a^{6} + \frac{79274892033}{2754806255} a^{5} + \frac{25292171228}{2754806255} a^{4} + \frac{21812017883}{2754806255} a^{3} + \frac{6851786753}{2754806255} a^{2} + \frac{12788449774}{2754806255} a + \frac{2037111233}{2754806255} \) (order $6$)
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:  \( 10792.8653088 \)
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

\(\Q(\sqrt{-3}) \), 3.1.87.1, 6.0.22707.1, 9.1.41764235769.1

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 $18$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.11$x^{12} + 6 x^{11} + 21 x^{10} + 36 x^{9} + 30 x^{8} + 36 x^{7} + 3 x^{6} + 36 x^{5} + 27 x^{4} - 9 x^{2} + 36$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$