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Properties

Label 18.18.4031470892...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 5^{4}\cdot 37^{9}\cdot 59^{4}$
Root discriminant $34.17$
Ramified primes $2, 5, 37, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\wr C_3:C_2$ (as 18T88)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 512, -1968, -1416, 13270, -8420, -22611, 25981, 9947, -21464, 893, 7717, -1544, -1333, 365, 106, -33, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 33*x^16 + 106*x^15 + 365*x^14 - 1333*x^13 - 1544*x^12 + 7717*x^11 + 893*x^10 - 21464*x^9 + 9947*x^8 + 25981*x^7 - 22611*x^6 - 8420*x^5 + 13270*x^4 - 1416*x^3 - 1968*x^2 + 512*x - 4)
 
gp: K = bnfinit(x^18 - 3*x^17 - 33*x^16 + 106*x^15 + 365*x^14 - 1333*x^13 - 1544*x^12 + 7717*x^11 + 893*x^10 - 21464*x^9 + 9947*x^8 + 25981*x^7 - 22611*x^6 - 8420*x^5 + 13270*x^4 - 1416*x^3 - 1968*x^2 + 512*x - 4, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 33 x^{16} + 106 x^{15} + 365 x^{14} - 1333 x^{13} - 1544 x^{12} + 7717 x^{11} + 893 x^{10} - 21464 x^{9} + 9947 x^{8} + 25981 x^{7} - 22611 x^{6} - 8420 x^{5} + 13270 x^{4} - 1416 x^{3} - 1968 x^{2} + 512 x - 4 \)

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:  \(4031470892249635921400320000=2^{12}\cdot 5^{4}\cdot 37^{9}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.17$
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:  $2, 5, 37, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{55796} a^{16} - \frac{2895}{13949} a^{15} + \frac{10253}{55796} a^{14} + \frac{6667}{55796} a^{13} - \frac{385}{27898} a^{12} - \frac{4441}{55796} a^{11} - \frac{403}{4292} a^{10} + \frac{3971}{13949} a^{9} + \frac{6575}{55796} a^{8} - \frac{4643}{55796} a^{7} - \frac{317}{962} a^{6} + \frac{1413}{4292} a^{5} + \frac{3602}{13949} a^{4} - \frac{2736}{13949} a^{3} - \frac{6181}{27898} a^{2} - \frac{6730}{13949} a + \frac{4634}{13949}$, $\frac{1}{24682772083654256156} a^{17} + \frac{4652782105583}{649546633780375162} a^{16} + \frac{1519746036638586195}{24682772083654256156} a^{15} + \frac{5115948928708401629}{24682772083654256156} a^{14} - \frac{1506965544079534255}{6170693020913564039} a^{13} + \frac{46259803530756463}{667101948206871788} a^{12} - \frac{8609500160753239729}{24682772083654256156} a^{11} - \frac{3147559110801100763}{12341386041827128078} a^{10} + \frac{88501370991760669}{1073164003637141572} a^{9} - \frac{9501862397499881893}{24682772083654256156} a^{8} + \frac{2636941079887016670}{6170693020913564039} a^{7} - \frac{5362881216131102791}{24682772083654256156} a^{6} - \frac{4029389939103123619}{12341386041827128078} a^{5} + \frac{2954499564671039700}{6170693020913564039} a^{4} - \frac{2529666338283586377}{6170693020913564039} a^{3} + \frac{2274689991717850905}{6170693020913564039} a^{2} + \frac{1365916722403618674}{6170693020913564039} a - \frac{2665723084763908115}{6170693020913564039}$

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:  $17$
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:  \( 156226776.16 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\wr C_3:C_2$ (as 18T88):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 162
The 13 conjugacy class representatives for $C_3\wr C_3:C_2$
Character table for $C_3\wr C_3:C_2$

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 6.6.810448.1, 9.9.10438327105600.1 x3

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

Sibling fields

Degree 9 siblings: data not computed
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 R ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ R

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
37Data not computed
$59$59.6.4.1$x^{6} + 295 x^{3} + 27848$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
59.6.0.1$x^{6} - x + 23$$1$$6$$0$$C_6$$[\ ]^{6}$
59.6.0.1$x^{6} - x + 23$$1$$6$$0$$C_6$$[\ ]^{6}$