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Properties

Label 18.0.37907050706...5168.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{24}$
Root discriminant $12.24$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
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![1, 0, 6, -4, 15, -12, 38, -54, 84, -84, 84, -54, 38, -12, 15, -4, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 - 4*x^15 + 15*x^14 - 12*x^13 + 38*x^12 - 54*x^11 + 84*x^10 - 84*x^9 + 84*x^8 - 54*x^7 + 38*x^6 - 12*x^5 + 15*x^4 - 4*x^3 + 6*x^2 + 1)
 
gp: K = bnfinit(x^18 + 6*x^16 - 4*x^15 + 15*x^14 - 12*x^13 + 38*x^12 - 54*x^11 + 84*x^10 - 84*x^9 + 84*x^8 - 54*x^7 + 38*x^6 - 12*x^5 + 15*x^4 - 4*x^3 + 6*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 6 x^{16} - 4 x^{15} + 15 x^{14} - 12 x^{13} + 38 x^{12} - 54 x^{11} + 84 x^{10} - 84 x^{9} + 84 x^{8} - 54 x^{7} + 38 x^{6} - 12 x^{5} + 15 x^{4} - 4 x^{3} + 6 x^{2} + 1 \)

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:  \(-37907050706572935168=-\,2^{27}\cdot 3^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.24$
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 Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} + \frac{3}{8} a^{11} + \frac{1}{8} a^{10} - \frac{3}{8} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{3}{16} a - \frac{5}{16}$, $\frac{1}{33472} a^{16} - \frac{19}{16736} a^{15} - \frac{643}{33472} a^{14} + \frac{887}{8368} a^{13} + \frac{2999}{16736} a^{12} + \frac{2797}{8368} a^{11} + \frac{1027}{4184} a^{10} + \frac{1451}{16736} a^{9} - \frac{157}{4184} a^{8} - \frac{641}{16736} a^{7} - \frac{19}{4184} a^{6} + \frac{2797}{8368} a^{5} - \frac{3277}{16736} a^{4} - \frac{1205}{8368} a^{3} + \frac{9817}{33472} a^{2} + \frac{4165}{16736} a - \frac{10459}{33472}$, $\frac{1}{66944} a^{17} - \frac{1}{66944} a^{16} - \frac{2049}{66944} a^{15} - \frac{3507}{66944} a^{14} + \frac{1693}{33472} a^{13} - \frac{595}{33472} a^{12} - \frac{3241}{16736} a^{11} + \frac{2823}{33472} a^{10} + \frac{2851}{33472} a^{9} + \frac{9595}{33472} a^{8} + \frac{9679}{33472} a^{7} + \frac{1391}{16736} a^{6} + \frac{2869}{33472} a^{5} - \frac{6507}{33472} a^{4} - \frac{1163}{66944} a^{3} + \frac{3367}{66944} a^{2} + \frac{29975}{66944} a + \frac{31417}{66944}$

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:  \( -1 \) (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:  \( 182.9899199 \)
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

\(\Q(\sqrt{-2}) \), 3.1.648.1 x3, \(\Q(\zeta_{9})^+\), 6.0.3359232.4, 6.0.3359232.1, 6.0.41472.1 x2, 9.3.272097792.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.41472.1
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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$