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Properties

Label 18.0.17996605488...5424.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{24}\cdot 7^{15}$
Root discriminant $61.94$
Ramified primes $2, 3, 7$
Class number $9072$ (GRH)
Class group $[2, 18, 252]$ (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![8256151, 5384334, 8823831, 2370846, 2503317, 233514, 623048, 50502, 163782, 5912, 28167, -36, 3941, 6, 387, -6, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 24*x^16 - 6*x^15 + 387*x^14 + 6*x^13 + 3941*x^12 - 36*x^11 + 28167*x^10 + 5912*x^9 + 163782*x^8 + 50502*x^7 + 623048*x^6 + 233514*x^5 + 2503317*x^4 + 2370846*x^3 + 8823831*x^2 + 5384334*x + 8256151)
 
gp: K = bnfinit(x^18 + 24*x^16 - 6*x^15 + 387*x^14 + 6*x^13 + 3941*x^12 - 36*x^11 + 28167*x^10 + 5912*x^9 + 163782*x^8 + 50502*x^7 + 623048*x^6 + 233514*x^5 + 2503317*x^4 + 2370846*x^3 + 8823831*x^2 + 5384334*x + 8256151, 1)
 

Normalized defining polynomial

\( x^{18} + 24 x^{16} - 6 x^{15} + 387 x^{14} + 6 x^{13} + 3941 x^{12} - 36 x^{11} + 28167 x^{10} + 5912 x^{9} + 163782 x^{8} + 50502 x^{7} + 623048 x^{6} + 233514 x^{5} + 2503317 x^{4} + 2370846 x^{3} + 8823831 x^{2} + 5384334 x + 8256151 \)

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:  \(-179966054889983269121047526375424=-\,2^{27}\cdot 3^{24}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.94$
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$$\chi_{504}(1,·)$, $\chi_{504}(325,·)$, $\chi_{504}(193,·)$, $\chi_{504}(457,·)$, $\chi_{504}(13,·)$, $\chi_{504}(397,·)$, $\chi_{504}(337,·)$, $\chi_{504}(25,·)$, $\chi_{504}(349,·)$, $\chi_{504}(289,·)$, $\chi_{504}(229,·)$, $\chi_{504}(169,·)$, $\chi_{504}(493,·)$, $\chi_{504}(157,·)$, $\chi_{504}(181,·)$, $\chi_{504}(361,·)$, $\chi_{504}(121,·)$, $\chi_{504}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{41969875493740783684} a^{16} - \frac{297456961923123423}{10492468873435195921} a^{15} - \frac{620459013753054487}{20984937746870391842} a^{14} + \frac{590875696581089586}{10492468873435195921} a^{13} + \frac{303933101644963143}{20984937746870391842} a^{12} + \frac{809532726747411465}{10492468873435195921} a^{11} - \frac{4050505943805003441}{20984937746870391842} a^{10} - \frac{655854756669851719}{10492468873435195921} a^{9} + \frac{6652459431273458843}{41969875493740783684} a^{8} + \frac{485237110132676550}{10492468873435195921} a^{7} - \frac{2124939351041494186}{10492468873435195921} a^{6} + \frac{2316488084517721914}{10492468873435195921} a^{5} + \frac{14505828403085056211}{41969875493740783684} a^{4} + \frac{1324264064964563399}{10492468873435195921} a^{3} - \frac{3449307125438538731}{10492468873435195921} a^{2} + \frac{4241254469967259423}{10492468873435195921} a + \frac{5041860582745362311}{10492468873435195921}$, $\frac{1}{45900619160664112420296050398436} a^{17} + \frac{191318174417}{45900619160664112420296050398436} a^{16} - \frac{567544080747278198660178691813}{11475154790166028105074012599609} a^{15} - \frac{16280043086820325266359808062}{161621898453042649367239614079} a^{14} + \frac{1829000370752809026438558472245}{22950309580332056210148025199218} a^{13} - \frac{2465242001594984743265784326891}{22950309580332056210148025199218} a^{12} + \frac{5232835884139846096586448879349}{22950309580332056210148025199218} a^{11} + \frac{1189785454922866881361103493119}{22950309580332056210148025199218} a^{10} - \frac{5897911344055222032216884265997}{45900619160664112420296050398436} a^{9} - \frac{4541294937546015183895035244733}{45900619160664112420296050398436} a^{8} - \frac{3426219857303767804353497953277}{22950309580332056210148025199218} a^{7} - \frac{5356870009666279703149743871519}{22950309580332056210148025199218} a^{6} - \frac{17087418435758115966395315390853}{45900619160664112420296050398436} a^{5} - \frac{8399594902862514948991535702953}{45900619160664112420296050398436} a^{4} - \frac{4601025035081512425724181559471}{22950309580332056210148025199218} a^{3} + \frac{10748280237816244629211680115139}{22950309580332056210148025199218} a^{2} - \frac{4368350171864469639073389919285}{11475154790166028105074012599609} a + \frac{4725337256143940810400250705736}{11475154790166028105074012599609}$

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

Class group and class number

$C_{2}\times C_{18}\times C_{252}$, which has order $9072$ (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

\(\Q(\sqrt{-14}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, 6.0.1152216576.2, 6.0.8605184.1, 6.0.56458612224.1, 6.0.56458612224.2, 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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed