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Properties

Label 18.12.7127140159...8347.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,3^{3}\cdot 1129^{8}$
Root discriminant $27.31$
Ramified primes $3, 1129$
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![-169, -416, -81, 413, 566, 548, -204, -773, -317, 97, 252, 240, -61, -155, 8, 43, -4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 4*x^16 + 43*x^15 + 8*x^14 - 155*x^13 - 61*x^12 + 240*x^11 + 252*x^10 + 97*x^9 - 317*x^8 - 773*x^7 - 204*x^6 + 548*x^5 + 566*x^4 + 413*x^3 - 81*x^2 - 416*x - 169)
 
gp: K = bnfinit(x^18 - 5*x^17 - 4*x^16 + 43*x^15 + 8*x^14 - 155*x^13 - 61*x^12 + 240*x^11 + 252*x^10 + 97*x^9 - 317*x^8 - 773*x^7 - 204*x^6 + 548*x^5 + 566*x^4 + 413*x^3 - 81*x^2 - 416*x - 169, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 4 x^{16} + 43 x^{15} + 8 x^{14} - 155 x^{13} - 61 x^{12} + 240 x^{11} + 252 x^{10} + 97 x^{9} - 317 x^{8} - 773 x^{7} - 204 x^{6} + 548 x^{5} + 566 x^{4} + 413 x^{3} - 81 x^{2} - 416 x - 169 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-71271401597539257710128347=-\,3^{3}\cdot 1129^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.31$
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:  $3, 1129$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{3}{13} a^{13} - \frac{5}{13} a^{12} - \frac{3}{13} a^{11} + \frac{4}{13} a^{10} - \frac{3}{13} a^{9} - \frac{2}{13} a^{8} + \frac{3}{13} a^{7} + \frac{2}{13} a^{6} + \frac{4}{13} a^{5} + \frac{2}{13} a^{4} - \frac{4}{13} a^{3} + \frac{3}{13} a^{2} + \frac{4}{13} a$, $\frac{1}{13} a^{15} - \frac{1}{13} a^{13} - \frac{1}{13} a^{12} - \frac{2}{13} a^{10} - \frac{6}{13} a^{9} - \frac{4}{13} a^{8} + \frac{6}{13} a^{7} - \frac{2}{13} a^{6} + \frac{3}{13} a^{5} + \frac{3}{13} a^{4} + \frac{2}{13} a^{3} - \frac{5}{13} a^{2} + \frac{1}{13} a$, $\frac{1}{13} a^{16} + \frac{2}{13} a^{13} - \frac{5}{13} a^{12} - \frac{5}{13} a^{11} - \frac{2}{13} a^{10} + \frac{6}{13} a^{9} + \frac{4}{13} a^{8} + \frac{1}{13} a^{7} + \frac{5}{13} a^{6} - \frac{6}{13} a^{5} + \frac{4}{13} a^{4} + \frac{4}{13} a^{3} + \frac{4}{13} a^{2} + \frac{4}{13} a$, $\frac{1}{212738877871} a^{17} - \frac{8046384403}{212738877871} a^{16} - \frac{337162298}{16364529067} a^{15} - \frac{3259372677}{212738877871} a^{14} - \frac{89364849575}{212738877871} a^{13} + \frac{82749260738}{212738877871} a^{12} + \frac{78973026115}{212738877871} a^{11} + \frac{3112089670}{16364529067} a^{10} - \frac{12070377684}{212738877871} a^{9} + \frac{28119608571}{212738877871} a^{8} - \frac{57694808573}{212738877871} a^{7} + \frac{7647057825}{16364529067} a^{6} + \frac{80055969973}{212738877871} a^{5} + \frac{33211355829}{212738877871} a^{4} - \frac{120715517}{16364529067} a^{3} + \frac{10005555896}{212738877871} a^{2} + \frac{36496110940}{212738877871} a - \frac{3563929054}{16364529067}$

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:  $14$
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:  \( 3718494.22814 \) (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, 6.4.3823923.1, 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 $18$ R $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
1129Data not computed