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Properties

Label 18.6.21422803359...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{24}\cdot 3^{21}\cdot 5^{13}$
Root discriminant $29.03$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5569, -5988, 7347, 1826, -30129, 10854, 20030, -11010, -6792, 5640, 1293, -1656, 39, 162, 27, -32, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 32*x^15 + 27*x^14 + 162*x^13 + 39*x^12 - 1656*x^11 + 1293*x^10 + 5640*x^9 - 6792*x^8 - 11010*x^7 + 20030*x^6 + 10854*x^5 - 30129*x^4 + 1826*x^3 + 7347*x^2 - 5988*x + 5569)
 
gp: K = bnfinit(x^18 - 32*x^15 + 27*x^14 + 162*x^13 + 39*x^12 - 1656*x^11 + 1293*x^10 + 5640*x^9 - 6792*x^8 - 11010*x^7 + 20030*x^6 + 10854*x^5 - 30129*x^4 + 1826*x^3 + 7347*x^2 - 5988*x + 5569, 1)
 

Normalized defining polynomial

\( x^{18} - 32 x^{15} + 27 x^{14} + 162 x^{13} + 39 x^{12} - 1656 x^{11} + 1293 x^{10} + 5640 x^{9} - 6792 x^{8} - 11010 x^{7} + 20030 x^{6} + 10854 x^{5} - 30129 x^{4} + 1826 x^{3} + 7347 x^{2} - 5988 x + 5569 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(214228033597440000000000000=2^{24}\cdot 3^{21}\cdot 5^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.03$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{41} a^{16} + \frac{14}{41} a^{15} + \frac{13}{41} a^{14} + \frac{7}{41} a^{13} + \frac{1}{41} a^{12} + \frac{2}{41} a^{11} + \frac{7}{41} a^{10} + \frac{3}{41} a^{9} + \frac{13}{41} a^{8} - \frac{16}{41} a^{7} - \frac{6}{41} a^{6} - \frac{7}{41} a^{5} - \frac{3}{41} a^{4} - \frac{2}{41} a^{3} - \frac{6}{41} a^{2} + \frac{17}{41} a - \frac{9}{41}$, $\frac{1}{136930807300662631042530361511740224793} a^{17} + \frac{1400078036312575150026303133940912568}{136930807300662631042530361511740224793} a^{16} - \frac{7386418831475919281741811599635537621}{136930807300662631042530361511740224793} a^{15} + \frac{588889392489152827576586189656434564}{3339775787821039781525130768579029873} a^{14} + \frac{6428162490606010237275422621604292996}{136930807300662631042530361511740224793} a^{13} - \frac{30008058855659424054859501369577943006}{136930807300662631042530361511740224793} a^{12} - \frac{43217939853000322534671268810789154955}{136930807300662631042530361511740224793} a^{11} - \frac{27809879617675192574891402252514216810}{136930807300662631042530361511740224793} a^{10} - \frac{25283571342777010695590076450781057835}{136930807300662631042530361511740224793} a^{9} - \frac{66184446815408773986428241976219858992}{136930807300662631042530361511740224793} a^{8} + \frac{5989660098694243729731024724524056124}{136930807300662631042530361511740224793} a^{7} - \frac{572584980657309733174437682253458397}{136930807300662631042530361511740224793} a^{6} + \frac{35278646387873681914888555116668347041}{136930807300662631042530361511740224793} a^{5} - \frac{66821622004834126358944033007553182742}{136930807300662631042530361511740224793} a^{4} - \frac{28462516821108170383469651597647566387}{136930807300662631042530361511740224793} a^{3} + \frac{41959819169546056411391534575265928862}{136930807300662631042530361511740224793} a^{2} - \frac{31288739176759727226428181560046466751}{136930807300662631042530361511740224793} a - \frac{38104706603446655418064121621307703054}{136930807300662631042530361511740224793}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 649167.1234615253 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{15}) \), 3.1.108.1, 6.2.69984000.1, 9.3.787320000.1

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 R R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.10.2$x^{12} + 15 x^{6} + 100$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$