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Properties

Label 18.2.23606268931...0912.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{24}\cdot 3^{20}\cdot 7^{9}$
Root discriminant $22.60$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, 0, 0, 108, 0, 0, 113, 0, 0, 106, 0, 0, 63, 0, 0, -10, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 10*x^15 + 63*x^12 + 106*x^9 + 113*x^6 + 108*x^3 - 27)
 
gp: K = bnfinit(x^18 - 10*x^15 + 63*x^12 + 106*x^9 + 113*x^6 + 108*x^3 - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 10 x^{15} + 63 x^{12} + 106 x^{9} + 113 x^{6} + 108 x^{3} - 27 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2360626893120550220070912=2^{24}\cdot 3^{20}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.60$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{2}{9} a^{6} - \frac{2}{9} a^{4}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{2}{9} a^{7} + \frac{2}{9} a^{4}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{2}{27} a^{8} + \frac{1}{3} a^{7} + \frac{11}{27} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{322191} a^{15} + \frac{1342}{322191} a^{12} - \frac{1}{9} a^{10} - \frac{11302}{322191} a^{9} - \frac{137221}{322191} a^{6} - \frac{2}{9} a^{4} - \frac{1780}{11933} a^{3} + \frac{3910}{11933}$, $\frac{1}{966573} a^{16} + \frac{1}{966573} a^{15} - \frac{34457}{966573} a^{13} - \frac{34457}{966573} a^{12} - \frac{118699}{966573} a^{10} - \frac{11302}{966573} a^{9} + \frac{113372}{966573} a^{7} + \frac{435563}{966573} a^{6} + \frac{1}{3} a^{5} + \frac{42392}{107397} a^{4} + \frac{10153}{35799} a^{3} - \frac{1}{3} a^{2} - \frac{8023}{35799} a - \frac{8023}{35799}$, $\frac{1}{966573} a^{17} - \frac{1}{966573} a^{15} + \frac{1342}{966573} a^{14} + \frac{34457}{966573} a^{12} + \frac{24497}{966573} a^{11} - \frac{96095}{966573} a^{9} - \frac{137221}{966573} a^{8} - \frac{1}{3} a^{7} - \frac{113372}{966573} a^{6} - \frac{99551}{322191} a^{5} - \frac{1}{3} a^{4} + \frac{17273}{107397} a^{3} - \frac{8023}{35799} a^{2} - \frac{1}{3} a + \frac{8023}{35799}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 631116.3194500087 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{7}) \), 3.1.756.1, 3.1.108.1, 6.2.7112448.1 x2, 6.2.16003008.1, 6.2.64012032.1, 9.1.20739898368.1

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 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$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}$
$3$3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$