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Properties

Label 18.0.24016469656...2736.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{27}\cdot 31^{15}$
Root discriminant $257.06$
Ramified primes $2, 3, 31$
Class number $29474928$ (GRH)
Class group $[2, 6, 342, 7182]$ (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![1686858891264, 0, 1265144168448, 0, 342643212288, 0, 45983342592, 0, 3427494912, 0, 148347648, 0, 3732648, 0, 52452, 0, 372, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 372*x^16 + 52452*x^14 + 3732648*x^12 + 148347648*x^10 + 3427494912*x^8 + 45983342592*x^6 + 342643212288*x^4 + 1265144168448*x^2 + 1686858891264)
 
gp: K = bnfinit(x^18 + 372*x^16 + 52452*x^14 + 3732648*x^12 + 148347648*x^10 + 3427494912*x^8 + 45983342592*x^6 + 342643212288*x^4 + 1265144168448*x^2 + 1686858891264, 1)
 

Normalized defining polynomial

\( x^{18} + 372 x^{16} + 52452 x^{14} + 3732648 x^{12} + 148347648 x^{10} + 3427494912 x^{8} + 45983342592 x^{6} + 342643212288 x^{4} + 1265144168448 x^{2} + 1686858891264 \)

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:  \(-24016469656532628829808489513181922235252736=-\,2^{27}\cdot 3^{27}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $257.06$
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:  $2, 3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2232=2^{3}\cdot 3^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{2232}(1,·)$, $\chi_{2232}(1091,·)$, $\chi_{2232}(2113,·)$, $\chi_{2232}(1489,·)$, $\chi_{2232}(1235,·)$, $\chi_{2232}(25,·)$, $\chi_{2232}(1369,·)$, $\chi_{2232}(1859,·)$, $\chi_{2232}(347,·)$, $\chi_{2232}(491,·)$, $\chi_{2232}(1115,·)$, $\chi_{2232}(1513,·)$, $\chi_{2232}(1835,·)$, $\chi_{2232}(625,·)$, $\chi_{2232}(371,·)$, $\chi_{2232}(745,·)$, $\chi_{2232}(1979,·)$, $\chi_{2232}(769,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{744} a^{6}$, $\frac{1}{1488} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{5952} a^{8} - \frac{1}{1488} a^{6} + \frac{1}{16} a^{4} + \frac{1}{8} a^{2}$, $\frac{1}{11904} a^{9} - \frac{1}{2976} a^{7} - \frac{3}{32} a^{5} - \frac{3}{16} a^{3} - \frac{1}{2} a$, $\frac{1}{47616} a^{10} - \frac{1}{11904} a^{8} + \frac{1}{11904} a^{6} - \frac{3}{64} a^{4} + \frac{1}{8} a^{2}$, $\frac{1}{95232} a^{11} - \frac{1}{23808} a^{9} + \frac{1}{23808} a^{7} + \frac{13}{128} a^{5} - \frac{3}{16} a^{3}$, $\frac{1}{283410432} a^{12} - \frac{1}{253952} a^{10} + \frac{15}{253952} a^{8} - \frac{5}{126976} a^{6} + \frac{19}{256} a^{4} + \frac{9}{64} a^{2} + \frac{1}{8}$, $\frac{1}{566820864} a^{13} - \frac{1}{507904} a^{11} + \frac{15}{507904} a^{9} - \frac{5}{253952} a^{7} + \frac{19}{512} a^{5} - \frac{23}{128} a^{3} + \frac{1}{16} a$, $\frac{1}{11336417280} a^{14} + \frac{1}{2834104320} a^{12} - \frac{81}{10158080} a^{10} - \frac{309}{5079040} a^{8} + \frac{77}{952320} a^{6} - \frac{303}{2560} a^{4} + \frac{69}{320} a^{2} + \frac{3}{10}$, $\frac{1}{22672834560} a^{15} + \frac{1}{5668208640} a^{13} - \frac{81}{20316160} a^{11} - \frac{309}{10158080} a^{9} + \frac{77}{1904640} a^{7} + \frac{337}{5120} a^{5} - \frac{91}{640} a^{3} + \frac{3}{20} a$, $\frac{1}{49426779340800} a^{16} - \frac{7}{1372966092800} a^{14} + \frac{353}{494267793408} a^{12} - \frac{196661}{22144614400} a^{10} + \frac{235349}{4152115200} a^{8} + \frac{105571}{346009600} a^{6} + \frac{5077}{55808} a^{4} - \frac{1301}{10900} a^{2} + \frac{1389}{5450}$, $\frac{1}{98853558681600} a^{17} - \frac{7}{2745932185600} a^{15} + \frac{353}{988535586816} a^{13} - \frac{196661}{44289228800} a^{11} + \frac{235349}{8304230400} a^{9} + \frac{105571}{692019200} a^{7} - \frac{8875}{111616} a^{5} - \frac{1301}{21800} a^{3} - \frac{4061}{10900} a$

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

Class group and class number

$C_{2}\times C_{6}\times C_{342}\times C_{7182}$, which has order $29474928$ (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:  \( 4617140.625511786 \) (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

\(\Q(\sqrt{-186}) \), \(\Q(\zeta_{9})^+\), 3.3.77841.1, 3.3.961.1, 3.3.77841.2, 6.0.300224641536.8, 6.0.288515880516096.2, 6.0.395769383424.3, 6.0.288515880516096.1, 9.9.471655843734321.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
31Data not computed