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Properties

Label 18.0.16906752970...8528.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 37^{7}\cdot 101^{6}$
Root discriminant $47.79$
Ramified primes $2, 37, 101$
Class number $108$ (GRH)
Class group $[6, 18]$ (GRH)
Galois group $C_2\times S_3\times S_4$ (as 18T111)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![592, 0, 4448, 0, 13168, 0, 19864, 0, 16724, 0, 8172, 0, 2329, 0, 375, 0, 31, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 31*x^16 + 375*x^14 + 2329*x^12 + 8172*x^10 + 16724*x^8 + 19864*x^6 + 13168*x^4 + 4448*x^2 + 592)
 
gp: K = bnfinit(x^18 + 31*x^16 + 375*x^14 + 2329*x^12 + 8172*x^10 + 16724*x^8 + 19864*x^6 + 13168*x^4 + 4448*x^2 + 592, 1)
 

Normalized defining polynomial

\( x^{18} + 31 x^{16} + 375 x^{14} + 2329 x^{12} + 8172 x^{10} + 16724 x^{8} + 19864 x^{6} + 13168 x^{4} + 4448 x^{2} + 592 \)

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:  \(-1690675297047726899552246038528=-\,2^{24}\cdot 37^{7}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.79$
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:  $2, 37, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{13} + \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{137936} a^{16} + \frac{1577}{137936} a^{14} + \frac{7295}{137936} a^{12} - \frac{1}{8} a^{11} + \frac{4131}{137936} a^{10} - \frac{521}{34484} a^{8} - \frac{1}{8} a^{7} - \frac{8153}{34484} a^{6} - \frac{1}{4} a^{5} - \frac{4291}{34484} a^{4} - \frac{1}{2} a^{3} - \frac{4833}{17242} a^{2} - \frac{1}{2} a - \frac{74}{233}$, $\frac{1}{137936} a^{17} + \frac{1577}{137936} a^{15} + \frac{7295}{137936} a^{13} - \frac{13111}{137936} a^{11} - \frac{521}{34484} a^{9} + \frac{9557}{68968} a^{7} + \frac{2165}{17242} a^{5} - \frac{1}{2} a^{4} + \frac{1894}{8621} a^{3} - \frac{1}{2} a^{2} + \frac{85}{466} a$

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

Class group and class number

$C_{6}\times C_{18}$, which has order $108$ (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:  \( 866680.938443 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\times S_4$ (as 18T111):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 30 conjugacy class representatives for $C_2\times S_3\times S_4$
Character table for $C_2\times S_3\times S_4$ is not computed

Intermediate fields

3.3.148.1, 3.3.404.1, 6.0.96623872.1, 9.9.3340021539392.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.6.8.4$x^{6} + 2 x^{3} + 2 x^{2} + 2$$6$$1$$8$$S_4\times C_2$$[4/3, 4/3, 2]_{3}^{2}$
2.12.16.19$x^{12} + x^{10} - 2 x^{8} - 3 x^{6} + 2 x^{4} - 3 x^{2} + 1$$6$$2$$16$$C_2\times S_4$$[4/3, 4/3, 2]_{3}^{2}$
37Data not computed
$101$101.3.0.1$x^{3} - x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$
101.3.0.1$x^{3} - x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$
101.12.6.1$x^{12} + 6181806 x^{6} - 10510100501 x^{2} + 9553681355409$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$