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Properties

Label 18.0.14952227306...4304.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 7^{15}\cdot 31^{15}$
Root discriminant $250.38$
Ramified primes $2, 7, 31$
Class number $76585824$ (GRH)
Class group $[2, 18, 126, 16884]$ (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![21429355544576, 0, 10714677772288, 0, 2009002082304, 0, 192490790912, 0, 10535881216, 0, 343561344, 0, 6681864, 0, 74648, 0, 434, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 434*x^16 + 74648*x^14 + 6681864*x^12 + 343561344*x^10 + 10535881216*x^8 + 192490790912*x^6 + 2009002082304*x^4 + 10714677772288*x^2 + 21429355544576)
 
gp: K = bnfinit(x^18 + 434*x^16 + 74648*x^14 + 6681864*x^12 + 343561344*x^10 + 10535881216*x^8 + 192490790912*x^6 + 2009002082304*x^4 + 10714677772288*x^2 + 21429355544576, 1)
 

Normalized defining polynomial

\( x^{18} + 434 x^{16} + 74648 x^{14} + 6681864 x^{12} + 343561344 x^{10} + 10535881216 x^{8} + 192490790912 x^{6} + 2009002082304 x^{4} + 10714677772288 x^{2} + 21429355544576 \)

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:  \(-14952227306849866974860634662963099734114304=-\,2^{27}\cdot 7^{15}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $250.38$
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:  $2, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1736=2^{3}\cdot 7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1736}(1,·)$, $\chi_{1736}(25,·)$, $\chi_{1736}(243,·)$, $\chi_{1736}(249,·)$, $\chi_{1736}(619,·)$, $\chi_{1736}(625,·)$, $\chi_{1736}(843,·)$, $\chi_{1736}(867,·)$, $\chi_{1736}(1017,·)$, $\chi_{1736}(1091,·)$, $\chi_{1736}(1121,·)$, $\chi_{1736}(1235,·)$, $\chi_{1736}(1241,·)$, $\chi_{1736}(1363,·)$, $\chi_{1736}(1369,·)$, $\chi_{1736}(1483,·)$, $\chi_{1736}(1513,·)$, $\chi_{1736}(1587,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{1736} a^{6}$, $\frac{1}{3472} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a$, $\frac{1}{13888} a^{8} + \frac{1}{6944} a^{6} - \frac{1}{8} a^{4} + \frac{1}{8} a^{2}$, $\frac{1}{27776} a^{9} + \frac{1}{13888} a^{7} - \frac{1}{16} a^{5} + \frac{1}{16} a^{3}$, $\frac{1}{111104} a^{10} + \frac{1}{55552} a^{8} - \frac{1}{13888} a^{6} - \frac{7}{64} a^{4}$, $\frac{1}{222208} a^{11} + \frac{1}{111104} a^{9} - \frac{1}{27776} a^{7} + \frac{9}{128} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{4629037056} a^{12} + \frac{29}{10665984} a^{10} + \frac{11}{888832} a^{8} - \frac{53}{888832} a^{6} + \frac{25}{1536} a^{4} + \frac{5}{32} a^{2} + \frac{1}{24}$, $\frac{1}{9258074112} a^{13} + \frac{29}{21331968} a^{11} + \frac{11}{1777664} a^{9} - \frac{53}{1777664} a^{7} - \frac{359}{3072} a^{5} - \frac{11}{64} a^{3} - \frac{23}{48} a$, $\frac{1}{37032296448} a^{14} + \frac{1}{18516148224} a^{12} - \frac{1}{21331968} a^{10} + \frac{21}{1015808} a^{8} - \frac{101}{666624} a^{6} + \frac{41}{1536} a^{4} + \frac{19}{192} a^{2} + \frac{11}{24}$, $\frac{1}{74064592896} a^{15} + \frac{1}{37032296448} a^{13} - \frac{1}{42663936} a^{11} + \frac{21}{2031616} a^{9} - \frac{101}{1333248} a^{7} + \frac{41}{3072} a^{5} + \frac{19}{384} a^{3} - \frac{13}{48} a$, $\frac{1}{833374799265792} a^{16} - \frac{5}{1431915462656} a^{14} + \frac{10495}{104171849908224} a^{12} - \frac{26027}{22859743232} a^{10} - \frac{76943}{3750426624} a^{8} + \frac{1784071}{7500853248} a^{6} - \frac{403043}{4320768} a^{4} + \frac{84283}{540096} a^{2} - \frac{3653}{11252}$, $\frac{1}{1666749598531584} a^{17} - \frac{5}{2863830925312} a^{15} + \frac{10495}{208343699816448} a^{13} - \frac{26027}{45719486464} a^{11} - \frac{76943}{7500853248} a^{9} + \frac{1784071}{15001706496} a^{7} - \frac{403043}{8641536} a^{5} - \frac{185765}{1080192} a^{3} + \frac{7599}{22504} a$

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

Class group and class number

$C_{2}\times C_{18}\times C_{126}\times C_{16884}$, which has order $76585824$ (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:  \( 2999047.7597124763 \) (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

\(\Q(\sqrt{-434}) \), 3.3.961.1, \(\Q(\zeta_{7})^+\), 3.3.47089.1, 3.3.47089.2, 6.0.5027736982016.2, 6.0.256357036544.2, Deg 6, 6.0.246359112118784.1, 9.9.104413920565969.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
7Data not computed
31Data not computed