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Properties

Label 18.2.42037384368...1744.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{18}\cdot 7^{12}\cdot 41^{5}$
Root discriminant $20.53$
Ramified primes $2, 7, 41$
Class number $1$
Class group Trivial
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 24, 24, 16, -2, -84, 81, 34, 1, 130, -71, 56, -40, -16, 10, -6, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 6*x^15 + 10*x^14 - 16*x^13 - 40*x^12 + 56*x^11 - 71*x^10 + 130*x^9 + x^8 + 34*x^7 + 81*x^6 - 84*x^5 - 2*x^4 + 16*x^3 + 24*x^2 + 24*x + 8)
 
gp: K = bnfinit(x^18 + 3*x^16 - 6*x^15 + 10*x^14 - 16*x^13 - 40*x^12 + 56*x^11 - 71*x^10 + 130*x^9 + x^8 + 34*x^7 + 81*x^6 - 84*x^5 - 2*x^4 + 16*x^3 + 24*x^2 + 24*x + 8, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} - 6 x^{15} + 10 x^{14} - 16 x^{13} - 40 x^{12} + 56 x^{11} - 71 x^{10} + 130 x^{9} + x^{8} + 34 x^{7} + 81 x^{6} - 84 x^{5} - 2 x^{4} + 16 x^{3} + 24 x^{2} + 24 x + 8 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(420373843688235571871744=2^{18}\cdot 7^{12}\cdot 41^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.53$
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:  $2, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{80183577779328950588} a^{17} + \frac{6334040587862970089}{80183577779328950588} a^{16} + \frac{14938157024135774359}{80183577779328950588} a^{15} + \frac{8591082861957966535}{80183577779328950588} a^{14} - \frac{6775192498148250774}{20045894444832237647} a^{13} - \frac{3643781750715575294}{20045894444832237647} a^{12} - \frac{8937421570482246884}{20045894444832237647} a^{11} + \frac{1149699897764295383}{20045894444832237647} a^{10} + \frac{37619200786538286209}{80183577779328950588} a^{9} + \frac{30261663739857717879}{80183577779328950588} a^{8} - \frac{31675169518390665473}{80183577779328950588} a^{7} + \frac{7343102169165772057}{80183577779328950588} a^{6} + \frac{6343412618063844635}{80183577779328950588} a^{5} + \frac{4212929868532723215}{80183577779328950588} a^{4} + \frac{16713517644834346257}{40091788889664475294} a^{3} - \frac{403974965006989122}{20045894444832237647} a^{2} - \frac{4843664753122928395}{20045894444832237647} a + \frac{8566113790370434335}{20045894444832237647}$

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:  $9$
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:  \( 30261.5446687 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 192 conjugacy class representatives for t18n839 are not computed
Character table for t18n839 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.5.12657150016.1

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

Sibling fields

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 R $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R $18$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$2$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.12.12.11$x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
7Data not computed
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
41.6.0.1$x^{6} - x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$