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Properties

Label 18.18.4626610379...0576.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{20}\cdot 3^{32}\cdot 47^{8}$
Root discriminant $84.30$
Ramified primes $2, 3, 47$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3:S_4$ (as 18T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-173889, 0, 593001, 0, -843588, 0, 651252, 0, -297342, 0, 82062, 0, -13444, 0, 1236, 0, -57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 57*x^16 + 1236*x^14 - 13444*x^12 + 82062*x^10 - 297342*x^8 + 651252*x^6 - 843588*x^4 + 593001*x^2 - 173889)
 
gp: K = bnfinit(x^18 - 57*x^16 + 1236*x^14 - 13444*x^12 + 82062*x^10 - 297342*x^8 + 651252*x^6 - 843588*x^4 + 593001*x^2 - 173889, 1)
 

Normalized defining polynomial

\( x^{18} - 57 x^{16} + 1236 x^{14} - 13444 x^{12} + 82062 x^{10} - 297342 x^{8} + 651252 x^{6} - 843588 x^{4} + 593001 x^{2} - 173889 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(46266103792173518288122441368600576=2^{20}\cdot 3^{32}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.30$
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:  $2, 3, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{3}{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} + \frac{1}{16} a^{2} + \frac{3}{16}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{16} a^{3} + \frac{3}{16} a$, $\frac{1}{48} a^{12} - \frac{1}{16} a^{8} - \frac{1}{12} a^{6} - \frac{3}{16} a^{4} - \frac{1}{4} a^{2} - \frac{7}{16}$, $\frac{1}{96} a^{13} - \frac{1}{96} a^{12} - \frac{1}{32} a^{9} + \frac{1}{32} a^{8} - \frac{1}{24} a^{7} + \frac{1}{24} a^{6} + \frac{5}{32} a^{5} - \frac{5}{32} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{32} a - \frac{1}{32}$, $\frac{1}{96} a^{14} - \frac{1}{96} a^{12} - \frac{1}{32} a^{10} - \frac{1}{96} a^{8} - \frac{5}{96} a^{6} - \frac{1}{32} a^{4} + \frac{13}{32} a^{2} - \frac{9}{32}$, $\frac{1}{576} a^{15} - \frac{1}{192} a^{14} - \frac{1}{192} a^{13} - \frac{1}{192} a^{12} - \frac{5}{192} a^{11} + \frac{1}{64} a^{10} - \frac{19}{576} a^{9} - \frac{5}{192} a^{8} + \frac{3}{64} a^{7} + \frac{13}{192} a^{6} - \frac{9}{64} a^{5} + \frac{15}{64} a^{4} - \frac{31}{192} a^{3} - \frac{21}{64} a^{2} - \frac{1}{64} a + \frac{3}{64}$, $\frac{1}{1291392} a^{16} - \frac{47}{13452} a^{14} + \frac{409}{107616} a^{12} - \frac{4067}{161424} a^{10} - \frac{529}{71744} a^{8} - \frac{1}{8} a^{7} - \frac{1}{152} a^{6} + \frac{1}{8} a^{5} - \frac{4375}{107616} a^{4} + \frac{1}{8} a^{3} - \frac{727}{17936} a^{2} + \frac{3}{8} a - \frac{57935}{143488}$, $\frac{1}{179503488} a^{17} + \frac{4607}{9972416} a^{15} - \frac{1}{192} a^{14} + \frac{26803}{9972416} a^{13} + \frac{1}{192} a^{12} + \frac{1224679}{89751744} a^{11} + \frac{1}{64} a^{10} - \frac{105263}{1869828} a^{9} - \frac{11}{192} a^{8} - \frac{27707}{507072} a^{7} - \frac{19}{192} a^{6} - \frac{5984801}{29917248} a^{5} + \frac{1}{64} a^{4} - \frac{1651899}{9972416} a^{3} + \frac{11}{64} a^{2} - \frac{4822185}{19944832} a + \frac{29}{64}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $17$
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:  \( 851357937003 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_4$ (as 18T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3:S_4$
Character table for $C_3:S_4$

Intermediate fields

3.3.45684.2, 3.3.564.1, 3.3.45684.1, 3.3.11421.1, 6.6.2087027856.1, 9.9.13443473060864064.2

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$3$3.3.5.1$x^{3} + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.1$x^{3} + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
47Data not computed