↑

Properties

Label 18.12.7127140159...8347.2
Degree $18$
Signature $[12, 3]$
Discriminant $-\,3^{3}\cdot 1129^{8}$
Root discriminant $27.31$
Ramified primes $3, 1129$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^2:D_9$ (as 18T67)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -13, -76, 49, -258, -261, 733, 236, -238, 31, -381, 73, 292, -141, -53, 50, -3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 3*x^16 + 50*x^15 - 53*x^14 - 141*x^13 + 292*x^12 + 73*x^11 - 381*x^10 + 31*x^9 - 238*x^8 + 236*x^7 + 733*x^6 - 261*x^5 - 258*x^4 + 49*x^3 - 76*x^2 - 13*x + 1)
 
gp: K = bnfinit(x^18 - 5*x^17 - 3*x^16 + 50*x^15 - 53*x^14 - 141*x^13 + 292*x^12 + 73*x^11 - 381*x^10 + 31*x^9 - 238*x^8 + 236*x^7 + 733*x^6 - 261*x^5 - 258*x^4 + 49*x^3 - 76*x^2 - 13*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 3 x^{16} + 50 x^{15} - 53 x^{14} - 141 x^{13} + 292 x^{12} + 73 x^{11} - 381 x^{10} + 31 x^{9} - 238 x^{8} + 236 x^{7} + 733 x^{6} - 261 x^{5} - 258 x^{4} + 49 x^{3} - 76 x^{2} - 13 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{1}{11} a^{15} + \frac{2}{11} a^{14} - \frac{5}{11} a^{13} + \frac{3}{11} a^{12} + \frac{3}{11} a^{11} - \frac{1}{11} a^{10} - \frac{2}{11} a^{9} + \frac{4}{11} a^{8} + \frac{2}{11} a^{7} + \frac{1}{11} a^{6} - \frac{2}{11} a^{5} + \frac{5}{11} a^{4} + \frac{2}{11} a^{3} + \frac{2}{11} a^{2} + \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{109627307848826559583} a^{17} - \frac{3608479001458995854}{109627307848826559583} a^{16} - \frac{17117271820658198950}{109627307848826559583} a^{15} + \frac{35969874620146939866}{109627307848826559583} a^{14} + \frac{30485430490842935070}{109627307848826559583} a^{13} - \frac{34625336578528156558}{109627307848826559583} a^{12} - \frac{34005213999147791985}{109627307848826559583} a^{11} + \frac{52529287803783920307}{109627307848826559583} a^{10} - \frac{27523035028419050218}{109627307848826559583} a^{9} + \frac{23523170295463791312}{109627307848826559583} a^{8} - \frac{31820716837693798714}{109627307848826559583} a^{7} - \frac{6594141666963451915}{109627307848826559583} a^{6} + \frac{43874571275993927845}{109627307848826559583} a^{5} + \frac{5981429879800721405}{109627307848826559583} a^{4} + \frac{17047459323331792492}{109627307848826559583} a^{3} + \frac{44667773029690666984}{109627307848826559583} a^{2} + \frac{52488033981227812942}{109627307848826559583} a - \frac{27095685390824378750}{109627307848826559583}$

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:  $14$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3010403.34878 \)
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, 6.4.3823923.2, 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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
1129Data not computed