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Properties

Label 18.14.4161912503...1857.1
Degree $18$
Signature $[14, 2]$
Discriminant $19^{16}\cdot 113^{3}$
Root discriminant $30.12$
Ramified primes $19, 113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T264

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -34, 170, 24, -506, 127, 558, -271, -422, 241, 403, -123, -261, 45, 69, -12, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 12*x^16 + 69*x^15 + 45*x^14 - 261*x^13 - 123*x^12 + 403*x^11 + 241*x^10 - 422*x^9 - 271*x^8 + 558*x^7 + 127*x^6 - 506*x^5 + 24*x^4 + 170*x^3 - 34*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 5*x^17 - 12*x^16 + 69*x^15 + 45*x^14 - 261*x^13 - 123*x^12 + 403*x^11 + 241*x^10 - 422*x^9 - 271*x^8 + 558*x^7 + 127*x^6 - 506*x^5 + 24*x^4 + 170*x^3 - 34*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 12 x^{16} + 69 x^{15} + 45 x^{14} - 261 x^{13} - 123 x^{12} + 403 x^{11} + 241 x^{10} - 422 x^{9} - 271 x^{8} + 558 x^{7} + 127 x^{6} - 506 x^{5} + 24 x^{4} + 170 x^{3} - 34 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(416191250312479879983411857=19^{16}\cdot 113^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.12$
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:  $19, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{10185409643151258749} a^{17} + \frac{4241924082880040214}{10185409643151258749} a^{16} - \frac{2301324669647692296}{10185409643151258749} a^{15} - \frac{4506366034543620994}{10185409643151258749} a^{14} - \frac{2600447003254253491}{10185409643151258749} a^{13} - \frac{1301396342384291874}{10185409643151258749} a^{12} - \frac{1457709548120244464}{10185409643151258749} a^{11} - \frac{209577798216454416}{10185409643151258749} a^{10} - \frac{3809226873938093131}{10185409643151258749} a^{9} - \frac{3999256757407697862}{10185409643151258749} a^{8} - \frac{3986446613768752369}{10185409643151258749} a^{7} - \frac{873167866457609944}{10185409643151258749} a^{6} + \frac{86900636637332621}{275281341706790777} a^{5} - \frac{3444507595678570891}{10185409643151258749} a^{4} - \frac{18525814995745870}{10185409643151258749} a^{3} - \frac{3997387787013422505}{10185409643151258749} a^{2} - \frac{3257684408426293130}{10185409643151258749} a - \frac{978230718678161634}{10185409643151258749}$

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:  $15$
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:  \( 8874898.10477 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T264:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 32 conjugacy class representatives for t18n264
Character table for t18n264 is not computed

Intermediate fields

3.3.361.1, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
19Data not computed
113Data not computed