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Properties

Label 18.0.14166424145...8736.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{27}\cdot 7^{12}$
Root discriminant $53.78$
Ramified primes $2, 3, 7$
Class number $1944$ (GRH)
Class group $[2, 18, 54]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1075033, -1190916, 1625814, -775582, 651183, -380358, 321467, -196452, 121791, -58068, 30840, -12930, 5568, -1806, 609, -146, 39, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 39*x^16 - 146*x^15 + 609*x^14 - 1806*x^13 + 5568*x^12 - 12930*x^11 + 30840*x^10 - 58068*x^9 + 121791*x^8 - 196452*x^7 + 321467*x^6 - 380358*x^5 + 651183*x^4 - 775582*x^3 + 1625814*x^2 - 1190916*x + 1075033)
 
gp: K = bnfinit(x^18 - 6*x^17 + 39*x^16 - 146*x^15 + 609*x^14 - 1806*x^13 + 5568*x^12 - 12930*x^11 + 30840*x^10 - 58068*x^9 + 121791*x^8 - 196452*x^7 + 321467*x^6 - 380358*x^5 + 651183*x^4 - 775582*x^3 + 1625814*x^2 - 1190916*x + 1075033, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 39 x^{16} - 146 x^{15} + 609 x^{14} - 1806 x^{13} + 5568 x^{12} - 12930 x^{11} + 30840 x^{10} - 58068 x^{9} + 121791 x^{8} - 196452 x^{7} + 321467 x^{6} - 380358 x^{5} + 651183 x^{4} - 775582 x^{3} + 1625814 x^{2} - 1190916 x + 1075033 \)

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:  \(-14166424145858741301073711988736=-\,2^{27}\cdot 3^{27}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.78$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(197,·)$, $\chi_{504}(193,·)$, $\chi_{504}(457,·)$, $\chi_{504}(337,·)$, $\chi_{504}(149,·)$, $\chi_{504}(25,·)$, $\chi_{504}(221,·)$, $\chi_{504}(389,·)$, $\chi_{504}(289,·)$, $\chi_{504}(485,·)$, $\chi_{504}(169,·)$, $\chi_{504}(365,·)$, $\chi_{504}(29,·)$, $\chi_{504}(53,·)$, $\chi_{504}(361,·)$, $\chi_{504}(121,·)$, $\chi_{504}(317,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{12} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} + \frac{3}{20} a^{9} + \frac{3}{20} a^{8} + \frac{1}{10} a^{7} - \frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{10} a^{4} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2} - \frac{9}{20} a + \frac{3}{20}$, $\frac{1}{20} a^{14} + \frac{1}{10} a^{12} - \frac{1}{4} a^{10} - \frac{1}{5} a^{9} + \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{1}{2} a^{4} - \frac{2}{5} a^{3} + \frac{1}{4} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{18740} a^{15} + \frac{163}{18740} a^{14} + \frac{203}{18740} a^{13} - \frac{365}{3748} a^{12} + \frac{1423}{18740} a^{11} + \frac{3103}{18740} a^{10} + \frac{4551}{18740} a^{9} + \frac{743}{18740} a^{8} - \frac{1533}{9370} a^{7} + \frac{341}{1874} a^{6} + \frac{1578}{4685} a^{5} + \frac{1506}{4685} a^{4} + \frac{2229}{18740} a^{3} + \frac{2713}{18740} a^{2} + \frac{331}{18740} a + \frac{8097}{18740}$, $\frac{1}{118033625341520260} a^{16} - \frac{602999449146}{29508406335380065} a^{15} - \frac{649084387402273}{29508406335380065} a^{14} + \frac{2504658013890943}{118033625341520260} a^{13} + \frac{3928593368880753}{59016812670760130} a^{12} + \frac{1801643315745406}{29508406335380065} a^{11} + \frac{7124606767720202}{29508406335380065} a^{10} + \frac{6055190640889429}{118033625341520260} a^{9} - \frac{3149618666759783}{23606725068304052} a^{8} - \frac{481535072100687}{5901681267076013} a^{7} - \frac{401307874781767}{59016812670760130} a^{6} + \frac{22352515028828061}{59016812670760130} a^{5} - \frac{48252925218379031}{118033625341520260} a^{4} - \frac{3992029313959224}{29508406335380065} a^{3} - \frac{28500590107488547}{59016812670760130} a^{2} + \frac{58880684534433793}{118033625341520260} a - \frac{52841668072239879}{118033625341520260}$, $\frac{1}{74293664600285527258070900} a^{17} - \frac{142372719}{74293664600285527258070900} a^{16} - \frac{735340972913411377807}{37146832300142763629035450} a^{15} - \frac{656582663546626887611857}{37146832300142763629035450} a^{14} - \frac{1175548202014721818550789}{74293664600285527258070900} a^{13} + \frac{5512626802734752597999191}{74293664600285527258070900} a^{12} + \frac{918966366209552808371783}{3714683230014276362903545} a^{11} - \frac{908488431918228067629922}{3714683230014276362903545} a^{10} + \frac{1733284918033877347182969}{7429366460028552725807090} a^{9} + \frac{179908750187885865116363}{18573416150071381814517725} a^{8} - \frac{507712768249238344360287}{3714683230014276362903545} a^{7} + \frac{8374923236555981821407399}{37146832300142763629035450} a^{6} + \frac{12454308990927920875014003}{74293664600285527258070900} a^{5} + \frac{21565437121592197184959683}{74293664600285527258070900} a^{4} - \frac{6720120922811334309842204}{18573416150071381814517725} a^{3} + \frac{3364511985247038913757993}{37146832300142763629035450} a^{2} - \frac{3027688728332862009185147}{37146832300142763629035450} a + \frac{3299112272448226543074163}{37146832300142763629035450}$

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

Class group and class number

$C_{2}\times C_{18}\times C_{54}$, which has order $1944$ (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:  \( 54408.4888887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, 6.0.10077696.1, 6.0.24196548096.2, 6.0.33191424.1, 6.0.24196548096.1, 9.9.62523502209.1

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 R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$