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Properties

Label 18.6.34625722497...3041.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 41^{4}\cdot 97^{4}$
Root discriminant $23.08$
Ramified primes $7, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T838

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, -108, -1332, -755, 1703, -2743, 880, 2568, -1493, -30, 417, -305, 56, -31, -17, 37, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 6*x^16 + 37*x^15 - 17*x^14 - 31*x^13 + 56*x^12 - 305*x^11 + 417*x^10 - 30*x^9 - 1493*x^8 + 2568*x^7 + 880*x^6 - 2743*x^5 + 1703*x^4 - 755*x^3 - 1332*x^2 - 108*x - 27)
 
gp: K = bnfinit(x^18 - 4*x^17 - 6*x^16 + 37*x^15 - 17*x^14 - 31*x^13 + 56*x^12 - 305*x^11 + 417*x^10 - 30*x^9 - 1493*x^8 + 2568*x^7 + 880*x^6 - 2743*x^5 + 1703*x^4 - 755*x^3 - 1332*x^2 - 108*x - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 6 x^{16} + 37 x^{15} - 17 x^{14} - 31 x^{13} + 56 x^{12} - 305 x^{11} + 417 x^{10} - 30 x^{9} - 1493 x^{8} + 2568 x^{7} + 880 x^{6} - 2743 x^{5} + 1703 x^{4} - 755 x^{3} - 1332 x^{2} - 108 x - 27 \)

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:  \(3462572249708025242543041=7^{12}\cdot 41^{4}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.08$
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:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{27} a^{16} - \frac{7}{27} a^{15} + \frac{2}{9} a^{14} + \frac{1}{27} a^{13} + \frac{7}{27} a^{12} - \frac{7}{27} a^{11} - \frac{13}{27} a^{10} + \frac{13}{27} a^{9} + \frac{1}{3} a^{8} - \frac{4}{9} a^{7} + \frac{1}{27} a^{6} + \frac{7}{27} a^{4} - \frac{10}{27} a^{3} - \frac{4}{27} a^{2} - \frac{5}{27} a - \frac{4}{9}$, $\frac{1}{892437857573869200697868590713} a^{17} + \frac{3988813978922466021677227826}{892437857573869200697868590713} a^{16} + \frac{25532453786997429265005930793}{99159761952652133410874287857} a^{15} + \frac{182816139656645235874173265912}{892437857573869200697868590713} a^{14} + \frac{289209536016910236033125120707}{892437857573869200697868590713} a^{13} - \frac{5631488903998006554866961754}{892437857573869200697868590713} a^{12} - \frac{386699490179712371451910234843}{892437857573869200697868590713} a^{11} + \frac{215701336095732705795079679341}{892437857573869200697868590713} a^{10} - \frac{4616059977554913850488584116}{297479285857956400232622863571} a^{9} + \frac{147463978308588258710504220815}{297479285857956400232622863571} a^{8} - \frac{385178107877724625574457562604}{892437857573869200697868590713} a^{7} + \frac{52829641159355863964066623412}{297479285857956400232622863571} a^{6} - \frac{153027390790479737040501133766}{892437857573869200697868590713} a^{5} - \frac{161889093063373586709141253066}{892437857573869200697868590713} a^{4} + \frac{173137732423125797011688660420}{892437857573869200697868590713} a^{3} + \frac{300484723146913678395688203373}{892437857573869200697868590713} a^{2} - \frac{70213505557891712136064583627}{297479285857956400232622863571} a + \frac{15353293467253346150931530701}{99159761952652133410874287857}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 195039.272071 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T838:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 180 conjugacy class representatives for t18n838 are not computed
Character table for t18n838 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.5.467890073.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
41Data not computed
97Data not computed