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Properties

Label 18.6.65881655515...7641.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 163^{2}\cdot 937^{2}$
Root discriminant $16.30$
Ramified primes $3, 163, 937$
Class number $1$
Class group Trivial
Galois group 18T703

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -18, -6, 102, -15, -218, 72, 291, -152, -234, 180, 88, -108, -3, 30, -6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 6*x^16 + 30*x^15 - 3*x^14 - 108*x^13 + 88*x^12 + 180*x^11 - 234*x^10 - 152*x^9 + 291*x^8 + 72*x^7 - 218*x^6 - 15*x^5 + 102*x^4 - 6*x^3 - 18*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 6*x^16 + 30*x^15 - 3*x^14 - 108*x^13 + 88*x^12 + 180*x^11 - 234*x^10 - 152*x^9 + 291*x^8 + 72*x^7 - 218*x^6 - 15*x^5 + 102*x^4 - 6*x^3 - 18*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 6 x^{16} + 30 x^{15} - 3 x^{14} - 108 x^{13} + 88 x^{12} + 180 x^{11} - 234 x^{10} - 152 x^{9} + 291 x^{8} + 72 x^{7} - 218 x^{6} - 15 x^{5} + 102 x^{4} - 6 x^{3} - 18 x^{2} - 3 x + 1 \)

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:  \(6588165551501521267641=3^{24}\cdot 163^{2}\cdot 937^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.30$
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:  $3, 163, 937$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{9} a^{15} + \frac{1}{3} a^{13} + \frac{2}{9} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{4}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{9} a^{16} + \frac{1}{3} a^{14} + \frac{2}{9} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{4}{9} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{4}{9} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{9} a$, $\frac{1}{698391} a^{17} + \frac{11452}{698391} a^{16} - \frac{37255}{698391} a^{15} + \frac{193703}{698391} a^{14} + \frac{312452}{698391} a^{13} + \frac{38875}{698391} a^{12} - \frac{3565}{9567} a^{11} + \frac{92087}{698391} a^{10} - \frac{335}{9567} a^{9} - \frac{77386}{698391} a^{8} + \frac{34637}{698391} a^{7} - \frac{229577}{698391} a^{6} - \frac{129841}{698391} a^{5} + \frac{11363}{698391} a^{4} - \frac{280652}{698391} a^{3} - \frac{174893}{698391} a^{2} + \frac{51649}{698391} a + \frac{179714}{698391}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6959.59409822 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T703:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 55 conjugacy class representatives for t18n703 are not computed
Character table for t18n703 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.3.81167515371.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
163Data not computed
937Data not computed