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Properties

Label 18.18.6772007805...0737.1
Degree $18$
Signature $[18, 0]$
Discriminant $61^{4}\cdot 257^{9}$
Root discriminant $39.97$
Ramified primes $61, 257$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\wr S_3$ (as 18T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25, -225, 4352, -11870, -10013, 69407, -60791, -38195, 68371, -4533, -26546, 7669, 4422, -2020, -249, 213, -9, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 9*x^16 + 213*x^15 - 249*x^14 - 2020*x^13 + 4422*x^12 + 7669*x^11 - 26546*x^10 - 4533*x^9 + 68371*x^8 - 38195*x^7 - 60791*x^6 + 69407*x^5 - 10013*x^4 - 11870*x^3 + 4352*x^2 - 225*x - 25)
 
gp: K = bnfinit(x^18 - 8*x^17 - 9*x^16 + 213*x^15 - 249*x^14 - 2020*x^13 + 4422*x^12 + 7669*x^11 - 26546*x^10 - 4533*x^9 + 68371*x^8 - 38195*x^7 - 60791*x^6 + 69407*x^5 - 10013*x^4 - 11870*x^3 + 4352*x^2 - 225*x - 25, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 9 x^{16} + 213 x^{15} - 249 x^{14} - 2020 x^{13} + 4422 x^{12} + 7669 x^{11} - 26546 x^{10} - 4533 x^{9} + 68371 x^{8} - 38195 x^{7} - 60791 x^{6} + 69407 x^{5} - 10013 x^{4} - 11870 x^{3} + 4352 x^{2} - 225 x - 25 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(67720078057235840614713990737=61^{4}\cdot 257^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.97$
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:  $61, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{35} a^{15} - \frac{1}{35} a^{14} + \frac{9}{35} a^{13} + \frac{2}{7} a^{12} - \frac{2}{5} a^{11} - \frac{12}{35} a^{10} - \frac{11}{35} a^{9} + \frac{12}{35} a^{8} - \frac{2}{7} a^{7} + \frac{17}{35} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{4}{35} a^{2} + \frac{8}{35} a + \frac{3}{7}$, $\frac{1}{1505} a^{16} - \frac{1}{301} a^{15} + \frac{111}{1505} a^{14} - \frac{12}{35} a^{13} - \frac{572}{1505} a^{12} - \frac{124}{1505} a^{11} + \frac{317}{1505} a^{10} - \frac{17}{43} a^{9} - \frac{457}{1505} a^{8} + \frac{502}{1505} a^{7} - \frac{676}{1505} a^{6} + \frac{414}{1505} a^{5} + \frac{293}{1505} a^{4} - \frac{201}{1505} a^{3} - \frac{11}{35} a^{2} + \frac{39}{1505} a + \frac{51}{301}$, $\frac{1}{653919922293895900685} a^{17} + \frac{79932507341759406}{653919922293895900685} a^{16} - \frac{300166089844509213}{653919922293895900685} a^{15} - \frac{1953286797160049443}{93417131756270842955} a^{14} + \frac{12495751178070635236}{653919922293895900685} a^{13} - \frac{4491246934869042231}{653919922293895900685} a^{12} + \frac{288786734697997490494}{653919922293895900685} a^{11} - \frac{21351232662627313252}{130783984458779180137} a^{10} - \frac{104578617649997965283}{653919922293895900685} a^{9} + \frac{311569129704260842982}{653919922293895900685} a^{8} - \frac{20433502487176979602}{93417131756270842955} a^{7} - \frac{158640924510283337532}{653919922293895900685} a^{6} + \frac{240688513818841325999}{653919922293895900685} a^{5} + \frac{31526771778320046866}{93417131756270842955} a^{4} + \frac{5779879465634849873}{93417131756270842955} a^{3} - \frac{91079142601744119078}{653919922293895900685} a^{2} - \frac{40774836509852722599}{93417131756270842955} a - \frac{20864314041324405208}{130783984458779180137}$

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:  $17$
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:  \( 362512473.892 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\wr S_3$ (as 18T86):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 162
The 22 conjugacy class representatives for $C_3\wr S_3$
Character table for $C_3\wr S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{257}) \), 3.3.257.1 x3, 6.6.16974593.1, 9.9.63162460553.1

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

Sibling fields

Degree 9 siblings: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
61Data not computed
257Data not computed