↑

Properties

Label 18.6.68552970751...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{24}\cdot 3^{21}\cdot 5^{8}$
Root discriminant $18.56$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, -15, -18, -57, -132, -144, -90, -126, -34, 123, 138, 45, -12, -15, -18, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^15 - 15*x^14 - 12*x^13 + 45*x^12 + 138*x^11 + 123*x^10 - 34*x^9 - 126*x^8 - 90*x^7 - 144*x^6 - 132*x^5 - 57*x^4 - 18*x^3 - 15*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^18 - 18*x^15 - 15*x^14 - 12*x^13 + 45*x^12 + 138*x^11 + 123*x^10 - 34*x^9 - 126*x^8 - 90*x^7 - 144*x^6 - 132*x^5 - 57*x^4 - 18*x^3 - 15*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{15} - 15 x^{14} - 12 x^{13} + 45 x^{12} + 138 x^{11} + 123 x^{10} - 34 x^{9} - 126 x^{8} - 90 x^{7} - 144 x^{6} - 132 x^{5} - 57 x^{4} - 18 x^{3} - 15 x^{2} + 6 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(68552970751180800000000=2^{24}\cdot 3^{21}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.56$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{1}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{11} + \frac{1}{9} a^{5} + \frac{1}{9} a^{2}$, $\frac{1}{45} a^{15} + \frac{1}{45} a^{14} - \frac{1}{45} a^{13} + \frac{2}{45} a^{12} + \frac{7}{45} a^{11} - \frac{7}{45} a^{10} - \frac{1}{9} a^{9} + \frac{1}{15} a^{8} - \frac{1}{9} a^{6} - \frac{2}{45} a^{5} + \frac{8}{45} a^{4} + \frac{8}{45} a^{3} + \frac{19}{45} a^{2} + \frac{4}{9} a - \frac{2}{45}$, $\frac{1}{45} a^{16} - \frac{2}{45} a^{14} - \frac{2}{45} a^{13} + \frac{1}{45} a^{11} - \frac{1}{15} a^{10} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} - \frac{1}{9} a^{7} + \frac{1}{15} a^{6} + \frac{2}{9} a^{5} - \frac{1}{9} a^{4} + \frac{2}{15} a^{3} + \frac{16}{45} a^{2} + \frac{2}{5} a - \frac{1}{15}$, $\frac{1}{131735370105} a^{17} + \frac{5461141}{131735370105} a^{16} - \frac{826769891}{131735370105} a^{15} + \frac{2835483797}{131735370105} a^{14} - \frac{1210997771}{43911790035} a^{13} + \frac{643222273}{131735370105} a^{12} + \frac{2533140344}{26347074021} a^{11} + \frac{19245107873}{131735370105} a^{10} + \frac{1056049699}{8782358007} a^{9} - \frac{4152986836}{26347074021} a^{8} + \frac{5886852433}{131735370105} a^{7} + \frac{11903431588}{131735370105} a^{6} + \frac{209238319}{2802880215} a^{5} + \frac{6571231681}{14637263345} a^{4} - \frac{13004973901}{26347074021} a^{3} - \frac{50986609982}{131735370105} a^{2} - \frac{3252113900}{26347074021} a - \frac{1845933839}{8782358007}$

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:  $11$
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:  \( 29545.241293507042 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{3}) \), 3.1.108.1, 6.2.559872.1, 9.3.787320000.1

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

Sibling fields

Degree 18 siblings: 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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$