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Properties

Label 18.18.5991575557...5941.1
Degree $18$
Signature $[18, 0]$
Discriminant $61^{3}\cdot 1129^{8}$
Root discriminant $45.11$
Ramified primes $61, 1129$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^2:D_9$ (as 18T67)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, 2068, 4744, -7325, -21331, 8522, 35097, -3415, -28904, -703, 13098, 1058, -3342, -340, 470, 44, -34, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 34*x^16 + 44*x^15 + 470*x^14 - 340*x^13 - 3342*x^12 + 1058*x^11 + 13098*x^10 - 703*x^9 - 28904*x^8 - 3415*x^7 + 35097*x^6 + 8522*x^5 - 21331*x^4 - 7325*x^3 + 4744*x^2 + 2068*x + 121)
 
gp: K = bnfinit(x^18 - 2*x^17 - 34*x^16 + 44*x^15 + 470*x^14 - 340*x^13 - 3342*x^12 + 1058*x^11 + 13098*x^10 - 703*x^9 - 28904*x^8 - 3415*x^7 + 35097*x^6 + 8522*x^5 - 21331*x^4 - 7325*x^3 + 4744*x^2 + 2068*x + 121, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 34 x^{16} + 44 x^{15} + 470 x^{14} - 340 x^{13} - 3342 x^{12} + 1058 x^{11} + 13098 x^{10} - 703 x^{9} - 28904 x^{8} - 3415 x^{7} + 35097 x^{6} + 8522 x^{5} - 21331 x^{4} - 7325 x^{3} + 4744 x^{2} + 2068 x + 121 \)

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:  \(599157555778187342751949715941=61^{3}\cdot 1129^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.11$
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:  $61, 1129$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{39} a^{16} - \frac{1}{39} a^{15} - \frac{1}{13} a^{14} - \frac{4}{39} a^{13} - \frac{7}{39} a^{12} - \frac{7}{39} a^{11} - \frac{11}{39} a^{10} + \frac{17}{39} a^{9} - \frac{16}{39} a^{8} + \frac{7}{39} a^{7} + \frac{10}{39} a^{6} - \frac{3}{13} a^{5} - \frac{17}{39} a^{4} + \frac{14}{39} a^{3} + \frac{6}{13} a^{2} - \frac{8}{39} a - \frac{6}{13}$, $\frac{1}{652683681012222717} a^{17} + \frac{7758315428646715}{652683681012222717} a^{16} - \frac{19922734867743137}{652683681012222717} a^{15} + \frac{1424572015512190}{59334880092020247} a^{14} - \frac{17482063566499874}{217561227004074239} a^{13} - \frac{53686693989195087}{217561227004074239} a^{12} - \frac{162334268998131877}{652683681012222717} a^{11} + \frac{21940037701188091}{50206437000940209} a^{10} - \frac{90869426401594303}{217561227004074239} a^{9} - \frac{19196965208514373}{50206437000940209} a^{8} + \frac{31210926993805146}{217561227004074239} a^{7} - \frac{68116003083639670}{652683681012222717} a^{6} + \frac{157132026959219638}{652683681012222717} a^{5} - \frac{102155784031724921}{652683681012222717} a^{4} - \frac{237726747857131634}{652683681012222717} a^{3} + \frac{171543183614753413}{652683681012222717} a^{2} + \frac{248385884746001390}{652683681012222717} a - \frac{3472842182596598}{19778293364006749}$

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:  \( 1222677739.82 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:D_9$ (as 18T67):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times C_2^2:D_9$
Character table for $C_2\times C_2^2:D_9$

Intermediate fields

3.3.1129.1, 6.6.77753101.1, 9.9.1624709678881.1

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

Sibling fields

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 $18$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$61$61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
1129Data not computed