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Properties

Label 18.6.27634239965...7472.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{27}\cdot 3^{30}$
Root discriminant $17.65$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -12, 24, 96, 9, -246, -372, -210, 39, 160, 99, -6, -24, -6, 6, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 + 6*x^14 - 6*x^13 - 24*x^12 - 6*x^11 + 99*x^10 + 160*x^9 + 39*x^8 - 210*x^7 - 372*x^6 - 246*x^5 + 9*x^4 + 96*x^3 + 24*x^2 - 12*x - 1)
 
gp: K = bnfinit(x^18 - 6*x^16 + 6*x^14 - 6*x^13 - 24*x^12 - 6*x^11 + 99*x^10 + 160*x^9 + 39*x^8 - 210*x^7 - 372*x^6 - 246*x^5 + 9*x^4 + 96*x^3 + 24*x^2 - 12*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} + 6 x^{14} - 6 x^{13} - 24 x^{12} - 6 x^{11} + 99 x^{10} + 160 x^{9} + 39 x^{8} - 210 x^{7} - 372 x^{6} - 246 x^{5} + 9 x^{4} + 96 x^{3} + 24 x^{2} - 12 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(27634239965091669737472=2^{27}\cdot 3^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.65$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17} a^{16} - \frac{1}{17} a^{15} - \frac{2}{17} a^{14} - \frac{1}{17} a^{13} + \frac{1}{17} a^{12} + \frac{7}{17} a^{11} + \frac{6}{17} a^{10} - \frac{8}{17} a^{9} + \frac{6}{17} a^{8} - \frac{6}{17} a^{7} - \frac{5}{17} a^{6} - \frac{2}{17} a^{5} + \frac{6}{17} a^{4} - \frac{3}{17} a^{3} - \frac{4}{17} a^{2} + \frac{6}{17} a + \frac{6}{17}$, $\frac{1}{4362207736306807} a^{17} + \frac{80692141212263}{4362207736306807} a^{16} + \frac{76001386410230}{256600455076871} a^{15} - \frac{567648909894815}{4362207736306807} a^{14} - \frac{1695844799290447}{4362207736306807} a^{13} + \frac{21125086359053}{49013570070863} a^{12} + \frac{80828187006522}{229589880858253} a^{11} - \frac{385513806737083}{4362207736306807} a^{10} + \frac{946728877501633}{4362207736306807} a^{9} - \frac{1498952157196544}{4362207736306807} a^{8} - \frac{1474761262543326}{4362207736306807} a^{7} - \frac{526523486389685}{4362207736306807} a^{6} + \frac{462248748018397}{4362207736306807} a^{5} - \frac{723779715573949}{4362207736306807} a^{4} + \frac{695086197047208}{4362207736306807} a^{3} - \frac{616238443193028}{4362207736306807} a^{2} - \frac{1558787412245323}{4362207736306807} a - \frac{1515308213030440}{4362207736306807}$

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:  $11$
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:  \( 19815.232273990656 \) (assuming GRH)
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

\(\Q(\sqrt{2}) \), 3.1.216.1, \(\Q(\zeta_{9})^+\), 6.2.373248.1, 6.6.3359232.1, 9.3.7346640384.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
3Data not computed