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Properties

Label 18.0.90727313329...0368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{6}\cdot 7^{15}$
Root discriminant $14.60$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
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, -2, 17, -52, 127, -234, 373, -488, 551, -542, 465, -348, 228, -140, 76, -36, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 17*x^16 - 36*x^15 + 76*x^14 - 140*x^13 + 228*x^12 - 348*x^11 + 465*x^10 - 542*x^9 + 551*x^8 - 488*x^7 + 373*x^6 - 234*x^5 + 127*x^4 - 52*x^3 + 17*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 17*x^16 - 36*x^15 + 76*x^14 - 140*x^13 + 228*x^12 - 348*x^11 + 465*x^10 - 542*x^9 + 551*x^8 - 488*x^7 + 373*x^6 - 234*x^5 + 127*x^4 - 52*x^3 + 17*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 17 x^{16} - 36 x^{15} + 76 x^{14} - 140 x^{13} + 228 x^{12} - 348 x^{11} + 465 x^{10} - 542 x^{9} + 551 x^{8} - 488 x^{7} + 373 x^{6} - 234 x^{5} + 127 x^{4} - 52 x^{3} + 17 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-907273133293160890368=-\,2^{18}\cdot 3^{6}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.60$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{104} a^{15} - \frac{5}{104} a^{14} - \frac{1}{13} a^{13} - \frac{21}{104} a^{12} + \frac{1}{52} a^{11} - \frac{3}{26} a^{10} - \frac{1}{104} a^{9} - \frac{3}{104} a^{8} - \frac{3}{13} a^{7} - \frac{1}{8} a^{6} + \frac{1}{104} a^{5} - \frac{17}{104} a^{4} + \frac{37}{104} a^{3} - \frac{19}{52} a^{2} + \frac{1}{13} a - \frac{27}{104}$, $\frac{1}{104} a^{16} - \frac{7}{104} a^{14} - \frac{9}{104} a^{13} + \frac{1}{104} a^{12} + \frac{3}{13} a^{11} + \frac{17}{104} a^{10} + \frac{9}{52} a^{9} + \frac{1}{8} a^{8} - \frac{3}{104} a^{7} + \frac{7}{52} a^{6} + \frac{5}{13} a^{5} - \frac{11}{52} a^{4} + \frac{43}{104} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a - \frac{5}{104}$, $\frac{1}{48152} a^{17} + \frac{183}{48152} a^{16} - \frac{121}{48152} a^{15} + \frac{2669}{24076} a^{14} + \frac{1661}{24076} a^{13} + \frac{4057}{48152} a^{12} + \frac{6755}{48152} a^{11} + \frac{5875}{48152} a^{10} - \frac{11009}{48152} a^{9} - \frac{955}{24076} a^{8} - \frac{5781}{48152} a^{7} + \frac{9283}{24076} a^{6} - \frac{545}{6019} a^{5} - \frac{18191}{48152} a^{4} + \frac{23879}{48152} a^{3} - \frac{15987}{48152} a^{2} - \frac{6939}{24076} a + \frac{4581}{48152}$

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:  \( -\frac{1173}{48152} a^{17} - \frac{2605}{48152} a^{16} + \frac{25257}{48152} a^{15} - \frac{24007}{24076} a^{14} + \frac{15949}{12038} a^{13} - \frac{194607}{48152} a^{12} + \frac{315007}{48152} a^{11} - \frac{405671}{48152} a^{10} + \frac{680171}{48152} a^{9} - \frac{14309}{926} a^{8} + \frac{675069}{48152} a^{7} - \frac{83487}{6019} a^{6} + \frac{80907}{6019} a^{5} - \frac{544263}{48152} a^{4} + \frac{25967}{3704} a^{3} - \frac{323775}{48152} a^{2} + \frac{4979}{1852} a - \frac{45309}{48152} \) (order $14$)
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:  \( 7362.5561247660435 \)
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

\(\Q(\sqrt{-7}) \), 3.1.1176.1, \(\Q(\zeta_{7})^+\), 6.0.9680832.1, \(\Q(\zeta_{7})\), 9.3.1626379776.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed