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Properties

Label 18.2.90022298341...4937.1
Degree $18$
Signature $[2, 8]$
Discriminant $23^{7}\cdot 31^{9}$
Root discriminant $18.85$
Ramified primes $23, 31$
Class number $1$
Class group Trivial
Galois group $S_3\times S_4$ (as 18T69)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -320, 576, -384, -228, 976, -1445, 1219, -568, 62, 184, -132, 81, -32, 8, 1, 1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + x^16 + x^15 + 8*x^14 - 32*x^13 + 81*x^12 - 132*x^11 + 184*x^10 + 62*x^9 - 568*x^8 + 1219*x^7 - 1445*x^6 + 976*x^5 - 228*x^4 - 384*x^3 + 576*x^2 - 320*x + 64)
 
gp: K = bnfinit(x^18 - 3*x^17 + x^16 + x^15 + 8*x^14 - 32*x^13 + 81*x^12 - 132*x^11 + 184*x^10 + 62*x^9 - 568*x^8 + 1219*x^7 - 1445*x^6 + 976*x^5 - 228*x^4 - 384*x^3 + 576*x^2 - 320*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + x^{16} + x^{15} + 8 x^{14} - 32 x^{13} + 81 x^{12} - 132 x^{11} + 184 x^{10} + 62 x^{9} - 568 x^{8} + 1219 x^{7} - 1445 x^{6} + 976 x^{5} - 228 x^{4} - 384 x^{3} + 576 x^{2} - 320 x + 64 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(90022298341717743394937=23^{7}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.85$
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:  $23, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{3}{16} a^{4} + \frac{1}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{16} - \frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{7}{32} a^{10} - \frac{1}{16} a^{9} + \frac{1}{8} a^{8} - \frac{1}{16} a^{7} + \frac{3}{8} a^{6} - \frac{5}{32} a^{5} + \frac{9}{32} a^{4} + \frac{5}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{13186047202321191232} a^{17} - \frac{118886985864805877}{13186047202321191232} a^{16} + \frac{68163539556227339}{13186047202321191232} a^{15} - \frac{693605582417340093}{13186047202321191232} a^{14} + \frac{814394559540881237}{6593023601160595616} a^{13} + \frac{237532932342108377}{3296511800580297808} a^{12} + \frac{152084593330638673}{13186047202321191232} a^{11} - \frac{392151479319417555}{6593023601160595616} a^{10} - \frac{234140632763631291}{3296511800580297808} a^{9} + \frac{319855003649811591}{6593023601160595616} a^{8} + \frac{723688274286336175}{3296511800580297808} a^{7} + \frac{6229459518598787947}{13186047202321191232} a^{6} + \frac{65412293737063973}{13186047202321191232} a^{5} - \frac{2095335006716741645}{6593023601160595616} a^{4} - \frac{309073402464783835}{1648255900290148904} a^{3} + \frac{132303590559796597}{1648255900290148904} a^{2} - \frac{47816661609904206}{206031987536268613} a - \frac{46016716135178475}{206031987536268613}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 21587.3825455 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times S_4$ (as 18T69):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 15 conjugacy class representatives for $S_3\times S_4$
Character table for $S_3\times S_4$

Intermediate fields

3.1.23.1, 3.1.31.1, 6.2.685193.1, 9.1.362467097.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$31$31.6.0.1$x^{6} - 2 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
31.12.9.1$x^{12} - 961 x^{4} + 268119$$4$$3$$9$$D_4 \times C_3$$[\ ]_{4}^{6}$