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Properties

Label 18.0.69237686076...6323.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 163^{12}$
Root discriminant $51.68$
Ramified primes $3, 163$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16875, 0, -38349, 0, 22089, 0, 5140, 0, -6021, 0, -258, 0, 493, 0, 54, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 + 54*x^14 + 493*x^12 - 258*x^10 - 6021*x^8 + 5140*x^6 + 22089*x^4 - 38349*x^2 + 16875)
 
gp: K = bnfinit(x^18 + 3*x^16 + 54*x^14 + 493*x^12 - 258*x^10 - 6021*x^8 + 5140*x^6 + 22089*x^4 - 38349*x^2 + 16875, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} + 54 x^{14} + 493 x^{12} - 258 x^{10} - 6021 x^{8} + 5140 x^{6} + 22089 x^{4} - 38349 x^{2} + 16875 \)

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:  \(-6923768607655667252896641526323=-\,3^{9}\cdot 163^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.68$
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:  $3, 163$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{10} - \frac{1}{6} a^{7} + \frac{1}{9} a^{6} - \frac{1}{6} a^{5} - \frac{5}{18} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{450} a^{13} - \frac{31}{450} a^{11} - \frac{1}{75} a^{9} - \frac{1}{6} a^{8} + \frac{28}{225} a^{7} - \frac{1}{6} a^{6} + \frac{97}{450} a^{5} + \frac{1}{3} a^{4} + \frac{4}{25} a^{3} - \frac{1}{2} a^{2} + \frac{1}{50} a$, $\frac{1}{13950} a^{14} - \frac{1}{2325} a^{12} + \frac{719}{13950} a^{10} + \frac{431}{13950} a^{8} - \frac{21}{775} a^{6} - \frac{1}{2} a^{5} + \frac{2486}{6975} a^{4} - \frac{1}{2} a^{3} - \frac{661}{2325} a^{2} - \frac{1}{31}$, $\frac{1}{13950} a^{15} - \frac{1}{2325} a^{13} + \frac{719}{13950} a^{11} + \frac{431}{13950} a^{9} - \frac{21}{775} a^{7} - \frac{1}{6} a^{6} + \frac{2486}{6975} a^{5} - \frac{1}{6} a^{4} - \frac{661}{2325} a^{3} + \frac{1}{3} a^{2} - \frac{1}{31} a$, $\frac{1}{132176250} a^{16} - \frac{3239}{132176250} a^{14} + \frac{2210267}{132176250} a^{12} + \frac{6889879}{132176250} a^{10} - \frac{9786488}{66088125} a^{8} - \frac{15547829}{132176250} a^{6} - \frac{1}{2} a^{5} + \frac{1198633}{132176250} a^{4} + \frac{2761588}{22029375} a^{2} + \frac{8181}{23498}$, $\frac{1}{132176250} a^{17} - \frac{3239}{132176250} a^{15} - \frac{46511}{44058750} a^{13} - \frac{2794607}{44058750} a^{11} - \frac{2737088}{66088125} a^{9} - \frac{4986793}{44058750} a^{7} - \frac{1}{6} a^{6} - \frac{6438217}{132176250} a^{5} + \frac{1}{3} a^{4} - \frac{3406637}{22029375} a^{3} + \frac{1}{3} a^{2} + \frac{110533}{587450} a$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{22997}{5287050} a^{17} - \frac{46358}{2643525} a^{15} - \frac{671519}{2643525} a^{13} - \frac{4241107}{1762350} a^{11} - \frac{3766087}{2643525} a^{9} + \frac{128948287}{5287050} a^{7} + \frac{1895972}{528705} a^{5} - \frac{26696956}{293725} a^{3} + \frac{40301567}{587450} a + \frac{1}{2} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28381401.6524 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.79707.1 x3, 3.3.26569.1, 6.0.19059617547.1, 6.0.717363.1 x2, 6.0.19059617547.2, 9.3.506394978606243.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.717363.1
Degree 9 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$163$163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$