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Properties

Label 18.0.68552970751...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{31}\cdot 3^{21}\cdot 5^{15}$
Root discriminant $45.45$
Ramified primes $2, 3, 5$
Class number $216$ (GRH)
Class group $[2, 6, 18]$ (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![18252000, 0, 5886000, 0, 81000, 0, -168300, 0, -27000, 0, 6300, 0, 2370, 0, 360, 0, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 30*x^16 + 360*x^14 + 2370*x^12 + 6300*x^10 - 27000*x^8 - 168300*x^6 + 81000*x^4 + 5886000*x^2 + 18252000)
 
gp: K = bnfinit(x^18 + 30*x^16 + 360*x^14 + 2370*x^12 + 6300*x^10 - 27000*x^8 - 168300*x^6 + 81000*x^4 + 5886000*x^2 + 18252000, 1)
 

Normalized defining polynomial

\( x^{18} + 30 x^{16} + 360 x^{14} + 2370 x^{12} + 6300 x^{10} - 27000 x^{8} - 168300 x^{6} + 81000 x^{4} + 5886000 x^{2} + 18252000 \)

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:  \(-685529707511808000000000000000=-\,2^{31}\cdot 3^{21}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.45$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{30} a^{6}$, $\frac{1}{30} a^{7}$, $\frac{1}{30} a^{8}$, $\frac{1}{30} a^{9}$, $\frac{1}{30} a^{10}$, $\frac{1}{30} a^{11}$, $\frac{1}{900} a^{12}$, $\frac{1}{900} a^{13}$, $\frac{1}{7200} a^{14} + \frac{1}{3600} a^{12} + \frac{1}{120} a^{10} + \frac{1}{80} a^{8} - \frac{1}{120} a^{6} - \frac{1}{2} a^{4} + \frac{1}{8} a^{2} - \frac{1}{4}$, $\frac{1}{7200} a^{15} + \frac{1}{3600} a^{13} + \frac{1}{120} a^{11} + \frac{1}{80} a^{9} - \frac{1}{120} a^{7} - \frac{1}{2} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{94734116970681600} a^{16} + \frac{78978896857}{5920882310667600} a^{14} - \frac{33139826189}{157890194951136} a^{12} - \frac{1587127436279}{210520259934848} a^{10} - \frac{893433372097}{197362743688920} a^{8} + \frac{251151610753}{34323955424160} a^{6} + \frac{29504087374153}{105260129967424} a^{4} - \frac{676380963193}{26315032491856} a^{2} - \frac{7850371792879}{26315032491856}$, $\frac{1}{1231543520618860800} a^{17} + \frac{374261995371}{8552385559853200} a^{15} - \frac{33139826189}{2052572534364768} a^{13} - \frac{113195767148819}{13683816895765120} a^{11} + \frac{28710978181241}{2565715667955960} a^{9} + \frac{2539415305697}{446211420514080} a^{7} + \frac{240024347309001}{1368381689576512} a^{5} - \frac{66463962192833}{342095422394128} a^{3} - \frac{165740566744015}{342095422394128} a$

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

Class group and class number

$C_{2}\times C_{6}\times C_{18}$, which has order $216$ (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:  \( -1 \) (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:  \( 225327.942978311 \) (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

\(\Q(\sqrt{-30}) \), 3.1.675.1, 6.0.3499200000.2, 9.3.4920750000.1

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.12.22.27$x^{12} + 2 x^{6} + 4$$6$$2$$22$$C_6\times S_3$$[3]_{3}^{6}$
3Data not computed
5Data not computed