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Properties

Label 18.0.78155389553...3712.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{31}\cdot 13^{6}$
Root discriminant $31.19$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 192, -288, -576, 1632, 1476, -1515, -1752, -324, -416, -102, 828, 774, 132, -72, -24, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 - 24*x^15 - 72*x^14 + 132*x^13 + 774*x^12 + 828*x^11 - 102*x^10 - 416*x^9 - 324*x^8 - 1752*x^7 - 1515*x^6 + 1476*x^5 + 1632*x^4 - 576*x^3 - 288*x^2 + 192*x + 64)
 
gp: K = bnfinit(x^18 + 6*x^16 - 24*x^15 - 72*x^14 + 132*x^13 + 774*x^12 + 828*x^11 - 102*x^10 - 416*x^9 - 324*x^8 - 1752*x^7 - 1515*x^6 + 1476*x^5 + 1632*x^4 - 576*x^3 - 288*x^2 + 192*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} + 6 x^{16} - 24 x^{15} - 72 x^{14} + 132 x^{13} + 774 x^{12} + 828 x^{11} - 102 x^{10} - 416 x^{9} - 324 x^{8} - 1752 x^{7} - 1515 x^{6} + 1476 x^{5} + 1632 x^{4} - 576 x^{3} - 288 x^{2} + 192 x + 64 \)

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:  \(-781553895537094671760883712=-\,2^{18}\cdot 3^{31}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.19$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{4} + \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{12} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{12} + \frac{1}{24} a^{10} + \frac{1}{24} a^{9} - \frac{1}{8} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{5}{24} a^{4} - \frac{5}{24} a^{3} - \frac{1}{4} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{72} a^{14} + \frac{1}{72} a^{13} - \frac{1}{36} a^{12} + \frac{5}{72} a^{11} - \frac{1}{72} a^{10} + \frac{1}{36} a^{9} - \frac{5}{24} a^{8} + \frac{1}{24} a^{7} - \frac{1}{3} a^{6} - \frac{19}{72} a^{5} + \frac{17}{72} a^{4} - \frac{2}{9} a^{3} + \frac{7}{18} a^{2} + \frac{2}{9} a - \frac{4}{9}$, $\frac{1}{72} a^{15} - \frac{1}{36} a^{12} - \frac{1}{12} a^{10} + \frac{1}{18} a^{9} - \frac{11}{36} a^{6} - \frac{1}{4} a^{4} - \frac{1}{72} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{2}{9}$, $\frac{1}{1008} a^{16} - \frac{1}{252} a^{15} + \frac{1}{252} a^{14} - \frac{5}{504} a^{13} + \frac{1}{28} a^{12} - \frac{5}{72} a^{11} + \frac{1}{63} a^{9} - \frac{2}{21} a^{8} + \frac{85}{504} a^{7} + \frac{97}{252} a^{6} - \frac{137}{504} a^{5} - \frac{389}{1008} a^{4} - \frac{5}{21} a^{3} + \frac{16}{63} a^{2} + \frac{10}{21} a + \frac{8}{63}$, $\frac{1}{5490591568331720256} a^{17} + \frac{177614821203493}{915098594721953376} a^{16} + \frac{1653810059932729}{305032864907317792} a^{15} + \frac{888802893648087}{152516432453658896} a^{14} - \frac{602941283376823}{343161973020732516} a^{13} - \frac{31393564398819241}{1372647892082930064} a^{12} + \frac{20286371872963069}{392185112023694304} a^{11} + \frac{3172304502426161}{171580986510366258} a^{10} - \frac{132439143691792375}{2745295784165860128} a^{9} - \frac{88767005107265095}{1372647892082930064} a^{8} + \frac{1905452236779979}{21788061779094128} a^{7} + \frac{43351065609648493}{228774648680488344} a^{6} + \frac{128137665943020767}{1830197189443906752} a^{5} + \frac{687453483141595781}{2745295784165860128} a^{4} + \frac{23289523716412477}{196092556011847152} a^{3} - \frac{137573668994748751}{686323946041465032} a^{2} - \frac{121582539004843765}{343161973020732516} a - \frac{17680656272880341}{171580986510366258}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{4801125193}{114590730048} a^{17} - \frac{170632683}{19098455008} a^{16} + \frac{15036152261}{57295365024} a^{15} - \frac{10078936511}{9549227504} a^{14} - \frac{39110310367}{14323841256} a^{13} + \frac{170104184185}{28647682512} a^{12} + \frac{1741847757259}{57295365024} a^{11} + \frac{413765020993}{14323841256} a^{10} - \frac{60908827561}{19098455008} a^{9} - \frac{122394471931}{28647682512} a^{8} - \frac{29463888015}{9549227504} a^{7} - \frac{471986451481}{7161920628} a^{6} - \frac{1726290497513}{38196910016} a^{5} + \frac{3423014335129}{57295365024} a^{4} + \frac{1013633713667}{28647682512} a^{3} - \frac{506218354741}{14323841256} a^{2} + \frac{5221075213}{7161920628} a + \frac{7512133399}{1193653438} \) (order $6$)
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:  \( 18999317.186917692 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.216.1, 6.0.139968.1, 6.0.3326427.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: data not computed

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
3Data not computed
$13$13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$