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Properties

Label 18.18.1275032635...8096.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 19^{17}\cdot 31^{6}$
Root discriminant $101.36$
Ramified primes $2, 19, 31$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2\times C_2^2:C_9$ (as 18T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16862569939, 0, 40252586306, 0, -16125600181, 0, 2904860828, 0, -292417885, 0, 17820195, 0, -670985, 0, 15219, 0, -190, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 190*x^16 + 15219*x^14 - 670985*x^12 + 17820195*x^10 - 292417885*x^8 + 2904860828*x^6 - 16125600181*x^4 + 40252586306*x^2 - 16862569939)
 
gp: K = bnfinit(x^18 - 190*x^16 + 15219*x^14 - 670985*x^12 + 17820195*x^10 - 292417885*x^8 + 2904860828*x^6 - 16125600181*x^4 + 40252586306*x^2 - 16862569939, 1)
 

Normalized defining polynomial

\( x^{18} - 190 x^{16} + 15219 x^{14} - 670985 x^{12} + 17820195 x^{10} - 292417885 x^{8} + 2904860828 x^{6} - 16125600181 x^{4} + 40252586306 x^{2} - 16862569939 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1275032635857445963924166132749828096=2^{18}\cdot 19^{17}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.36$
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:  $2, 19, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{31} a^{8} - \frac{4}{31} a^{6} - \frac{2}{31} a^{4} + \frac{10}{31} a^{2}$, $\frac{1}{31} a^{9} - \frac{4}{31} a^{7} - \frac{2}{31} a^{5} + \frac{10}{31} a^{3}$, $\frac{1}{961} a^{10} - \frac{4}{961} a^{8} + \frac{60}{961} a^{6} + \frac{382}{961} a^{4} + \frac{10}{31} a^{2}$, $\frac{1}{961} a^{11} - \frac{4}{961} a^{9} + \frac{60}{961} a^{7} + \frac{382}{961} a^{5} + \frac{10}{31} a^{3}$, $\frac{1}{29791} a^{12} - \frac{4}{29791} a^{10} + \frac{60}{29791} a^{8} - \frac{6345}{29791} a^{6} - \frac{455}{961} a^{4} + \frac{14}{31} a^{2}$, $\frac{1}{29791} a^{13} - \frac{4}{29791} a^{11} + \frac{60}{29791} a^{9} - \frac{6345}{29791} a^{7} - \frac{455}{961} a^{5} + \frac{14}{31} a^{3}$, $\frac{1}{923521} a^{14} - \frac{4}{923521} a^{12} + \frac{60}{923521} a^{10} - \frac{6345}{923521} a^{8} - \frac{14870}{29791} a^{6} + \frac{262}{961} a^{4} - \frac{8}{31} a^{2}$, $\frac{1}{923521} a^{15} - \frac{4}{923521} a^{13} + \frac{60}{923521} a^{11} - \frac{6345}{923521} a^{9} - \frac{14870}{29791} a^{7} + \frac{262}{961} a^{5} - \frac{8}{31} a^{3}$, $\frac{1}{76061903956134197} a^{16} - \frac{22215006197}{76061903956134197} a^{14} - \frac{535691093073}{76061903956134197} a^{12} - \frac{27922767191947}{76061903956134197} a^{10} + \frac{15071427809196}{2453609805036587} a^{8} + \frac{33650395962851}{79148703388277} a^{6} + \frac{314351971226}{2553183980267} a^{4} + \frac{20218176838}{82360773557} a^{2} + \frac{849938195}{2656799147}$, $\frac{1}{76061903956134197} a^{17} - \frac{22215006197}{76061903956134197} a^{15} - \frac{535691093073}{76061903956134197} a^{13} - \frac{27922767191947}{76061903956134197} a^{11} + \frac{15071427809196}{2453609805036587} a^{9} + \frac{33650395962851}{79148703388277} a^{7} + \frac{314351971226}{2553183980267} a^{5} + \frac{20218176838}{82360773557} a^{3} + \frac{849938195}{2656799147} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $17$
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:  \( 402218304851 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:C_9$ (as 18T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 24 conjugacy class representatives for $C_2\times C_2^2:C_9$
Character table for $C_2\times C_2^2:C_9$ is not computed

Intermediate fields

3.3.361.1, 6.6.152289992896.1, \(\Q(\zeta_{19})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ $18$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R $18$ $18$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ $18$ $18$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
19Data not computed
$31$31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.0.1$x^{6} - 2 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$