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Properties

Label 18.6.18147624991...0000.2
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 5^{9}\cdot 197^{8}$
Root discriminant $37.15$
Ramified primes $2, 5, 197$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![25, 25, -225, 35, -174, -418, 701, 350, -375, -74, 288, -410, 253, 25, -26, -24, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 24*x^15 - 26*x^14 + 25*x^13 + 253*x^12 - 410*x^11 + 288*x^10 - 74*x^9 - 375*x^8 + 350*x^7 + 701*x^6 - 418*x^5 - 174*x^4 + 35*x^3 - 225*x^2 + 25*x + 25)
 
gp: K = bnfinit(x^18 - 2*x^17 - 24*x^15 - 26*x^14 + 25*x^13 + 253*x^12 - 410*x^11 + 288*x^10 - 74*x^9 - 375*x^8 + 350*x^7 + 701*x^6 - 418*x^5 - 174*x^4 + 35*x^3 - 225*x^2 + 25*x + 25, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 24 x^{15} - 26 x^{14} + 25 x^{13} + 253 x^{12} - 410 x^{11} + 288 x^{10} - 74 x^{9} - 375 x^{8} + 350 x^{7} + 701 x^{6} - 418 x^{5} - 174 x^{4} + 35 x^{3} - 225 x^{2} + 25 x + 25 \)

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:  \(18147624991591898888000000000=2^{12}\cdot 5^{9}\cdot 197^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.15$
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:  $2, 5, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{15} a^{15} + \frac{1}{15} a^{12} + \frac{2}{5} a^{11} + \frac{7}{15} a^{10} + \frac{7}{15} a^{9} - \frac{2}{5} a^{8} + \frac{1}{15} a^{7} - \frac{2}{15} a^{6} - \frac{4}{15} a^{5} + \frac{7}{15} a^{4} + \frac{1}{3} a^{3} + \frac{2}{15} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{75} a^{16} + \frac{2}{75} a^{15} - \frac{2}{25} a^{14} + \frac{4}{75} a^{13} - \frac{1}{75} a^{12} - \frac{7}{15} a^{11} + \frac{9}{25} a^{10} + \frac{23}{75} a^{9} + \frac{7}{75} a^{8} - \frac{6}{25} a^{7} - \frac{7}{15} a^{6} + \frac{32}{75} a^{5} - \frac{11}{75} a^{4} - \frac{1}{5} a^{2} + \frac{1}{3}$, $\frac{1}{2145952095913100592986025} a^{17} + \frac{9503768230171518903166}{2145952095913100592986025} a^{16} + \frac{17039696615855194907099}{715317365304366864328675} a^{15} - \frac{1137296832919692040501}{429190419182620118597205} a^{14} - \frac{39640443272593330297159}{429190419182620118597205} a^{13} + \frac{69546429609906446249476}{2145952095913100592986025} a^{12} - \frac{558165184794440661724963}{2145952095913100592986025} a^{11} + \frac{169674683856429748060276}{2145952095913100592986025} a^{10} + \frac{224582151544411529555704}{2145952095913100592986025} a^{9} - \frac{71692584654928641515609}{429190419182620118597205} a^{8} - \frac{213344717024096646524054}{715317365304366864328675} a^{7} + \frac{280795647658593993400889}{715317365304366864328675} a^{6} - \frac{207248248020138155497271}{715317365304366864328675} a^{5} - \frac{753109315581703660409}{69224261158487115902775} a^{4} + \frac{18240183877322561384342}{429190419182620118597205} a^{3} + \frac{161244467662288195003208}{429190419182620118597205} a^{2} + \frac{82038381895448358551}{2768970446339484636111} a + \frac{11463381639353007001584}{28612694612174674573147}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 12157959.8424 \) (assuming GRH)
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

3.3.788.1, 3.3.985.1, 6.2.4851125.1, 9.9.12049107848000.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$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}$
$197$197.2.0.1$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$$[\ ]_{2}^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$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}$