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Properties

Label 18.2.39346408075...5424.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{18}\cdot 3^{36}$
Root discriminant $18.00$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3:S_4$ (as 18T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 0, 0, 0, 0, 327, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^12 + 327*x^6 - 1)
 
gp: K = bnfinit(x^18 - 3*x^12 + 327*x^6 - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{12} + 327 x^{6} - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(39346408075296537575424=2^{18}\cdot 3^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.00$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{6} - \frac{1}{6} a^{3} + \frac{1}{6}$, $\frac{1}{18} a^{10} - \frac{1}{9} a^{8} - \frac{1}{6} a^{7} - \frac{1}{9} a^{6} - \frac{1}{18} a^{4} + \frac{1}{9} a^{2} + \frac{1}{6} a + \frac{1}{9}$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{9} a^{7} - \frac{1}{6} a^{6} - \frac{1}{18} a^{5} - \frac{1}{18} a^{3} + \frac{1}{6} a^{2} + \frac{1}{9} a + \frac{1}{6}$, $\frac{1}{126} a^{12} - \frac{1}{9} a^{6} - \frac{23}{126}$, $\frac{1}{126} a^{13} - \frac{1}{9} a^{7} - \frac{23}{126} a$, $\frac{1}{378} a^{14} - \frac{1}{378} a^{12} - \frac{4}{27} a^{8} + \frac{4}{27} a^{6} - \frac{107}{378} a^{2} + \frac{107}{378}$, $\frac{1}{756} a^{15} + \frac{1}{378} a^{13} - \frac{1}{252} a^{12} - \frac{2}{27} a^{9} - \frac{4}{27} a^{7} + \frac{1}{18} a^{6} + \frac{271}{756} a^{3} - \frac{107}{378} a + \frac{23}{252}$, $\frac{1}{756} a^{16} - \frac{1}{252} a^{13} + \frac{1}{378} a^{12} - \frac{1}{54} a^{10} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{2}{27} a^{6} + \frac{229}{756} a^{4} + \frac{1}{9} a^{2} + \frac{65}{252} a + \frac{187}{378}$, $\frac{1}{756} a^{17} - \frac{1}{756} a^{14} + \frac{1}{378} a^{13} - \frac{1}{378} a^{12} - \frac{1}{54} a^{11} + \frac{1}{18} a^{9} + \frac{2}{27} a^{8} + \frac{2}{27} a^{7} - \frac{1}{54} a^{6} + \frac{229}{756} a^{5} - \frac{1}{18} a^{3} - \frac{271}{756} a^{2} + \frac{187}{378} a + \frac{85}{189}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 39507.1018471 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_4$ (as 18T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3:S_4$
Character table for $C_3:S_4$

Intermediate fields

3.1.972.2, 3.1.243.1, 3.1.108.1, 3.1.972.1, 6.2.3779136.3, 9.1.24794911296.1

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.6.6.7$x^{6} + 2 x^{2} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.12.12.28$x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1$$6$$2$$12$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$3.9.18.2$x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$$9$$1$$18$$C_3^2:C_2$$[3/2, 5/2]_{2}$
3.9.18.2$x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$$9$$1$$18$$C_3^2:C_2$$[3/2, 5/2]_{2}$