↑

Properties

Label 18.0.10834375376...0896.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{30}\cdot 3^{6}\cdot 7^{12}$
Root discriminant $16.76$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 13, 0, -8, 0, 1, 0, 7, 0, 45, 0, 27, 0, 18, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 8*x^16 + 18*x^14 + 27*x^12 + 45*x^10 + 7*x^8 + x^6 - 8*x^4 + 13*x^2 + 1)
 
gp: K = bnfinit(x^18 + 8*x^16 + 18*x^14 + 27*x^12 + 45*x^10 + 7*x^8 + x^6 - 8*x^4 + 13*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 8 x^{16} + 18 x^{14} + 27 x^{12} + 45 x^{10} + 7 x^{8} + x^{6} - 8 x^{4} + 13 x^{2} + 1 \)

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:  \(-10834375376002294480896=-\,2^{30}\cdot 3^{6}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.76$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4568876} a^{16} + \frac{110930}{1142219} a^{14} - \frac{1}{4} a^{13} + \frac{741847}{4568876} a^{12} - \frac{1}{4} a^{11} + \frac{1744671}{4568876} a^{10} + \frac{1}{4} a^{9} + \frac{1669299}{4568876} a^{8} - \frac{1}{4} a^{7} - \frac{2188159}{4568876} a^{6} - \frac{1}{4} a^{5} + \frac{1157049}{4568876} a^{4} - \frac{1}{4} a^{3} + \frac{266878}{1142219} a^{2} - \frac{1}{4} a - \frac{1196991}{4568876}$, $\frac{1}{4568876} a^{17} + \frac{110930}{1142219} a^{15} - \frac{1}{4} a^{14} + \frac{741847}{4568876} a^{13} - \frac{1}{4} a^{12} + \frac{1744671}{4568876} a^{11} + \frac{1}{4} a^{10} + \frac{1669299}{4568876} a^{9} - \frac{1}{4} a^{8} - \frac{2188159}{4568876} a^{7} - \frac{1}{4} a^{6} + \frac{1157049}{4568876} a^{5} - \frac{1}{4} a^{4} + \frac{266878}{1142219} a^{3} - \frac{1}{4} a^{2} - \frac{1196991}{4568876} a$

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:  \( -\frac{8083}{27859} a^{17} - \frac{125777}{55718} a^{15} - \frac{262731}{55718} a^{13} - \frac{373389}{55718} a^{11} - \frac{629451}{55718} a^{9} + \frac{63875}{55718} a^{7} + \frac{40633}{55718} a^{5} + \frac{165007}{55718} a^{3} - \frac{96729}{27859} a \) (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:  \( 8510.538851231004 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.1176.1, \(\Q(\zeta_{7})^+\), 6.0.22127616.1, 6.0.153664.1, 9.3.1626379776.1

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 12 sibling: 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 R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.12.24.315$x^{12} + 32 x^{11} - 10 x^{10} + 8 x^{9} - 18 x^{8} + 32 x^{7} + 20 x^{6} + 24 x^{5} - 24 x^{4} + 32 x^{3} + 16 x^{2} - 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed