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Properties

Label 18.0.10335746235...5951.6
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{12}\cdot 31^{9}$
Root discriminant $88.15$
Ramified primes $3, 7, 31$
Class number $240084$ (GRH)
Class group $[3, 6, 13338]$ (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![1875384008, -1247429208, 1476438822, -727075369, 478893783, -186727650, 87960240, -28265604, 10478436, -2838208, 869706, -200712, 52678, -10500, 2424, -412, 78, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 78*x^16 - 412*x^15 + 2424*x^14 - 10500*x^13 + 52678*x^12 - 200712*x^11 + 869706*x^10 - 2838208*x^9 + 10478436*x^8 - 28265604*x^7 + 87960240*x^6 - 186727650*x^5 + 478893783*x^4 - 727075369*x^3 + 1476438822*x^2 - 1247429208*x + 1875384008)
 
gp: K = bnfinit(x^18 - 9*x^17 + 78*x^16 - 412*x^15 + 2424*x^14 - 10500*x^13 + 52678*x^12 - 200712*x^11 + 869706*x^10 - 2838208*x^9 + 10478436*x^8 - 28265604*x^7 + 87960240*x^6 - 186727650*x^5 + 478893783*x^4 - 727075369*x^3 + 1476438822*x^2 - 1247429208*x + 1875384008, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 78 x^{16} - 412 x^{15} + 2424 x^{14} - 10500 x^{13} + 52678 x^{12} - 200712 x^{11} + 869706 x^{10} - 2838208 x^{9} + 10478436 x^{8} - 28265604 x^{7} + 87960240 x^{6} - 186727650 x^{5} + 478893783 x^{4} - 727075369 x^{3} + 1476438822 x^{2} - 1247429208 x + 1875384008 \)

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:  \(-103357462359885242066848874838225951=-\,3^{24}\cdot 7^{12}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.15$
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:  $3, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1953=3^{2}\cdot 7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1953}(1024,·)$, $\chi_{1953}(1,·)$, $\chi_{1953}(898,·)$, $\chi_{1953}(1675,·)$, $\chi_{1953}(652,·)$, $\chi_{1953}(1549,·)$, $\chi_{1953}(526,·)$, $\chi_{1953}(466,·)$, $\chi_{1953}(340,·)$, $\chi_{1953}(1303,·)$, $\chi_{1953}(1177,·)$, $\chi_{1953}(1117,·)$, $\chi_{1953}(991,·)$, $\chi_{1953}(1828,·)$, $\chi_{1953}(1768,·)$, $\chi_{1953}(1642,·)$, $\chi_{1953}(373,·)$, $\chi_{1953}(247,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7}$, $\frac{1}{142} a^{15} - \frac{16}{71} a^{14} + \frac{14}{71} a^{13} - \frac{23}{142} a^{12} + \frac{11}{142} a^{11} + \frac{16}{71} a^{10} + \frac{5}{142} a^{8} + \frac{8}{71} a^{7} + \frac{25}{71} a^{6} - \frac{17}{142} a^{5} - \frac{27}{142} a^{4} - \frac{23}{71} a^{3} + \frac{31}{71} a^{2} + \frac{1}{71} a + \frac{8}{71}$, $\frac{1}{36068} a^{16} - \frac{31}{9017} a^{15} - \frac{38}{9017} a^{14} + \frac{575}{9017} a^{13} + \frac{88}{9017} a^{12} - \frac{4111}{18034} a^{11} - \frac{975}{18034} a^{10} - \frac{1898}{9017} a^{9} + \frac{31}{9017} a^{8} - \frac{604}{9017} a^{7} + \frac{2520}{9017} a^{6} + \frac{1680}{9017} a^{5} - \frac{201}{18034} a^{4} - \frac{3391}{18034} a^{3} + \frac{8427}{36068} a^{2} + \frac{1307}{9017} a - \frac{2214}{9017}$, $\frac{1}{868110330163292037928503576823857715356459549741101372} a^{17} - \frac{3792148622125457076283842358328512662235827435591}{868110330163292037928503576823857715356459549741101372} a^{16} + \frac{180549686956598270351706869842386328718234014287203}{217027582540823009482125894205964428839114887435275343} a^{15} - \frac{35580464869883497479635372737815390232303044588785962}{217027582540823009482125894205964428839114887435275343} a^{14} - \frac{31812956631665748912316472078537796409033996114436591}{217027582540823009482125894205964428839114887435275343} a^{13} + \frac{56976617227727430172923163393984909959050678494051859}{434055165081646018964251788411928857678229774870550686} a^{12} - \frac{44946738783956868960758401789679186075272769424320499}{217027582540823009482125894205964428839114887435275343} a^{11} - \frac{15776296616064118699976738866928614911595864699249869}{217027582540823009482125894205964428839114887435275343} a^{10} + \frac{57514473071681216611591143873115347924242143943771385}{434055165081646018964251788411928857678229774870550686} a^{9} + \frac{51088964758476741369451431099947768528649322835184308}{217027582540823009482125894205964428839114887435275343} a^{8} - \frac{72775388762620113824641442008556004950414465087097106}{217027582540823009482125894205964428839114887435275343} a^{7} - \frac{102471116860733470535448361903209249556459676666363723}{217027582540823009482125894205964428839114887435275343} a^{6} + \frac{29063484903835805972124470171151585737093968696491925}{434055165081646018964251788411928857678229774870550686} a^{5} - \frac{61233018352127233049697427211266022893668840120464785}{217027582540823009482125894205964428839114887435275343} a^{4} + \frac{16185418899120935605806966065126667736607368661251675}{868110330163292037928503576823857715356459549741101372} a^{3} + \frac{373622981323338623518437947023803618417710821748044441}{868110330163292037928503576823857715356459549741101372} a^{2} + \frac{14129069964822202956088418386914907460575860025680807}{217027582540823009482125894205964428839114887435275343} a + \frac{67674688112408424667309996962623652707202343391789799}{217027582540823009482125894205964428839114887435275343}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{13338}$, which has order $240084$ (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

\(\Q(\sqrt{-31}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, 6.0.195458751.1, 6.0.71528191.1, 6.0.469296461151.6, 6.0.469296461151.7, 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
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}$
$31$31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$