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Properties

Label 18.6.53914013089...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 5^{8}\cdot 7^{8}\cdot 197^{6}$
Root discriminant $44.85$
Ramified primes $2, 5, 7, 197$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3:S_3:S_4$ (as 18T154)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-125, -1705, 8980, -24556, 42545, -50637, 42574, -21097, -160, 7689, -3104, 613, -640, 29, 147, -8, -8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 8*x^16 - 8*x^15 + 147*x^14 + 29*x^13 - 640*x^12 + 613*x^11 - 3104*x^10 + 7689*x^9 - 160*x^8 - 21097*x^7 + 42574*x^6 - 50637*x^5 + 42545*x^4 - 24556*x^3 + 8980*x^2 - 1705*x - 125)
 
gp: K = bnfinit(x^18 - 3*x^17 - 8*x^16 - 8*x^15 + 147*x^14 + 29*x^13 - 640*x^12 + 613*x^11 - 3104*x^10 + 7689*x^9 - 160*x^8 - 21097*x^7 + 42574*x^6 - 50637*x^5 + 42545*x^4 - 24556*x^3 + 8980*x^2 - 1705*x - 125, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 8 x^{16} - 8 x^{15} + 147 x^{14} + 29 x^{13} - 640 x^{12} + 613 x^{11} - 3104 x^{10} + 7689 x^{9} - 160 x^{8} - 21097 x^{7} + 42574 x^{6} - 50637 x^{5} + 42545 x^{4} - 24556 x^{3} + 8980 x^{2} - 1705 x - 125 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(539140130893112269326400000000=2^{12}\cdot 5^{8}\cdot 7^{8}\cdot 197^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.85$
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:  $2, 5, 7, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{394} a^{16} - \frac{77}{394} a^{15} + \frac{92}{197} a^{14} - \frac{13}{394} a^{13} + \frac{191}{394} a^{12} - \frac{51}{394} a^{11} + \frac{119}{394} a^{10} - \frac{35}{394} a^{9} + \frac{85}{394} a^{8} - \frac{133}{394} a^{7} + \frac{91}{394} a^{6} - \frac{5}{394} a^{5} - \frac{77}{394} a^{4} - \frac{73}{394} a^{3} + \frac{47}{197} a^{2} + \frac{33}{197} a + \frac{111}{394}$, $\frac{1}{891483154769091919381672416271930} a^{17} - \frac{455237809234899528505189876319}{445741577384545959690836208135965} a^{16} + \frac{122186537259225402191705909277537}{891483154769091919381672416271930} a^{15} - \frac{169707690011525984738490329488283}{891483154769091919381672416271930} a^{14} - \frac{167402873286041203995919313714219}{445741577384545959690836208135965} a^{13} - \frac{99948332920684533863285352796288}{445741577384545959690836208135965} a^{12} + \frac{3603001165926067344889019253791}{89148315476909191938167241627193} a^{11} - \frac{68975566806551763991084675436246}{445741577384545959690836208135965} a^{10} + \frac{113849476492893204954867399405493}{445741577384545959690836208135965} a^{9} - \frac{140628907307182339525910881485638}{445741577384545959690836208135965} a^{8} - \frac{41064236756821054247527755144400}{89148315476909191938167241627193} a^{7} - \frac{52518228193532218324719006666031}{445741577384545959690836208135965} a^{6} + \frac{41980254203216138159961329805812}{445741577384545959690836208135965} a^{5} - \frac{209501028390889571654356770797836}{445741577384545959690836208135965} a^{4} + \frac{72908883047908087435345645924657}{178296630953818383876334483254386} a^{3} + \frac{35724308213143985494506588107597}{445741577384545959690836208135965} a^{2} - \frac{1555673713338915395115064983177}{178296630953818383876334483254386} a + \frac{15926749709629456144797412306067}{178296630953818383876334483254386}$

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:  $11$
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:  \( 238902706.397 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T154):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.985.1, 6.2.970225.1, 9.9.734261622920000.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$197$$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
197.2.1.2$x^{2} + 394$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.2$x^{2} + 394$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$