↑

Properties

Label 18.0.25901729635...2123.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 8353^{6}$
Root discriminant $105.43$
Ramified primes $3, 8353$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2\times S_3$ (as 18T17)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31554496, 0, 0, -9161192, 0, 0, 1234873, 0, 0, -95314, 0, 0, 4291, 0, 0, -102, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 102*x^15 + 4291*x^12 - 95314*x^9 + 1234873*x^6 - 9161192*x^3 + 31554496)
 
gp: K = bnfinit(x^18 - 102*x^15 + 4291*x^12 - 95314*x^9 + 1234873*x^6 - 9161192*x^3 + 31554496, 1)
 

Normalized defining polynomial

\( x^{18} - 102 x^{15} + 4291 x^{12} - 95314 x^{9} + 1234873 x^{6} - 9161192 x^{3} + 31554496 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2590172963585338621461271780963152123=-\,3^{27}\cdot 8353^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.43$
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:  $3, 8353$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{6} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6734964706792} a^{15} + \frac{198911212119}{3367482353396} a^{12} + \frac{170821876391}{6734964706792} a^{9} + \frac{1207467533917}{3367482353396} a^{6} - \frac{2767337002767}{6734964706792} a^{3} + \frac{394095185880}{841870588349}$, $\frac{1}{1064124423673136} a^{16} - \frac{63783253502405}{532062211836568} a^{13} - \frac{20034072243985}{1064124423673136} a^{10} + \frac{11309914594105}{532062211836568} a^{7} - \frac{332780607635575}{1064124423673136} a^{4} + \frac{8615753476430}{66507776479571} a$, $\frac{1}{168131658940355488} a^{17} + \frac{10178414324351529}{84065829470177744} a^{14} - \frac{10927309414893629}{168131658940355488} a^{11} - \frac{7570576604076989}{84065829470177744} a^{8} - \frac{33320637741502791}{168131658940355488} a^{5} + \frac{740201294751711}{10508228683772218} a^{2}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{147880970057}{168131658940355488} a^{17} - \frac{6709146409867}{84065829470177744} a^{14} + \frac{483620125697355}{168131658940355488} a^{11} - \frac{4291589818102949}{84065829470177744} a^{8} + \frac{81421634206003817}{168131658940355488} a^{5} - \frac{21263883853044549}{10508228683772218} a^{2} \) (order $18$)
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:  \( 359949824121.8046 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2\times S_3$ (as 18T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_3^2\times S_3$
Character table for $C_3^2\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 6.0.1373334262947.1, \(\Q(\zeta_{9})\), 6.0.1883860443.1, Deg 6

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
8353Data not computed