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Properties

Label 18.0.50259261193...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 5^{12}\cdot 43^{9}$
Root discriminant $30.44$
Ramified primes $2, 5, 43$
Class number $9$
Class group $[3, 3]$
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8471, -14018, 32286, -27518, 6984, 100, 5855, -10366, 5130, -1000, 462, -586, 553, -200, 24, 22, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 6*x^16 + 22*x^15 + 24*x^14 - 200*x^13 + 553*x^12 - 586*x^11 + 462*x^10 - 1000*x^9 + 5130*x^8 - 10366*x^7 + 5855*x^6 + 100*x^5 + 6984*x^4 - 27518*x^3 + 32286*x^2 - 14018*x + 8471)
 
gp: K = bnfinit(x^18 - 2*x^17 - 6*x^16 + 22*x^15 + 24*x^14 - 200*x^13 + 553*x^12 - 586*x^11 + 462*x^10 - 1000*x^9 + 5130*x^8 - 10366*x^7 + 5855*x^6 + 100*x^5 + 6984*x^4 - 27518*x^3 + 32286*x^2 - 14018*x + 8471, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 6 x^{16} + 22 x^{15} + 24 x^{14} - 200 x^{13} + 553 x^{12} - 586 x^{11} + 462 x^{10} - 1000 x^{9} + 5130 x^{8} - 10366 x^{7} + 5855 x^{6} + 100 x^{5} + 6984 x^{4} - 27518 x^{3} + 32286 x^{2} - 14018 x + 8471 \)

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:  \(-502592611936843000000000000=-\,2^{12}\cdot 5^{12}\cdot 43^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.44$
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:  $2, 5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6}$, $\frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{18} a^{8} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{7}{18} a^{4} - \frac{1}{2} a^{3} + \frac{1}{18} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{9} + \frac{1}{18} a^{8} - \frac{1}{9} a^{7} - \frac{1}{6} a^{6} + \frac{1}{18} a^{5} - \frac{4}{9} a^{4} + \frac{1}{18} a^{3} + \frac{1}{6} a^{2} - \frac{1}{9} a + \frac{7}{18}$, $\frac{1}{378} a^{12} + \frac{5}{189} a^{11} - \frac{23}{378} a^{9} - \frac{1}{18} a^{8} + \frac{13}{378} a^{7} + \frac{2}{63} a^{6} + \frac{53}{189} a^{5} - \frac{17}{63} a^{4} + \frac{85}{378} a^{3} + \frac{5}{18} a^{2} + \frac{79}{378} a - \frac{85}{378}$, $\frac{1}{756} a^{13} + \frac{5}{756} a^{11} + \frac{19}{756} a^{10} + \frac{41}{756} a^{9} - \frac{29}{756} a^{8} + \frac{113}{756} a^{7} - \frac{5}{108} a^{6} - \frac{19}{108} a^{5} - \frac{155}{756} a^{4} - \frac{325}{756} a^{3} - \frac{47}{756} a^{2} + \frac{11}{54} a + \frac{283}{756}$, $\frac{1}{6804} a^{14} + \frac{1}{1701} a^{13} + \frac{1}{972} a^{12} + \frac{143}{6804} a^{11} - \frac{5}{252} a^{10} - \frac{79}{6804} a^{9} + \frac{181}{2268} a^{8} + \frac{779}{6804} a^{7} + \frac{253}{2268} a^{6} + \frac{2129}{6804} a^{5} + \frac{653}{2268} a^{4} - \frac{3109}{6804} a^{3} + \frac{800}{1701} a^{2} - \frac{305}{972} a - \frac{233}{3402}$, $\frac{1}{34020} a^{15} - \frac{1}{34020} a^{14} - \frac{1}{8505} a^{13} + \frac{1}{1890} a^{12} - \frac{193}{34020} a^{11} + \frac{389}{34020} a^{10} - \frac{2293}{34020} a^{9} + \frac{449}{34020} a^{8} - \frac{2911}{34020} a^{7} + \frac{1097}{34020} a^{6} - \frac{12619}{34020} a^{5} - \frac{5497}{34020} a^{4} - \frac{1832}{8505} a^{3} - \frac{316}{945} a^{2} - \frac{3419}{11340} a - \frac{15067}{34020}$, $\frac{1}{612360} a^{16} + \frac{1}{153090} a^{15} + \frac{41}{612360} a^{14} - \frac{1}{8505} a^{13} - \frac{113}{612360} a^{12} - \frac{19}{43740} a^{11} - \frac{2353}{102060} a^{10} + \frac{3121}{153090} a^{9} + \frac{148}{3645} a^{8} - \frac{8567}{153090} a^{7} - \frac{2203}{34020} a^{6} + \frac{149159}{306180} a^{5} + \frac{192737}{612360} a^{4} + \frac{9857}{51030} a^{3} + \frac{11029}{87480} a^{2} + \frac{18041}{76545} a + \frac{3649}{17496}$, $\frac{1}{485560967122765800} a^{17} + \frac{2388373513}{6474146228303544} a^{16} - \frac{4741090498601}{485560967122765800} a^{15} - \frac{273338690923}{97112193424553160} a^{14} - \frac{32599444926953}{69365852446109400} a^{13} + \frac{1831368972487}{2569105646152200} a^{12} + \frac{1206523401784948}{60695120890345725} a^{11} + \frac{401395420656971}{242780483561382900} a^{10} + \frac{4415411472127109}{121390241780691450} a^{9} + \frac{352287005819233}{121390241780691450} a^{8} - \frac{9137328531627503}{242780483561382900} a^{7} + \frac{2004944938991329}{17341463111527350} a^{6} - \frac{56320256186257601}{485560967122765800} a^{5} - \frac{24399533238667031}{69365852446109400} a^{4} + \frac{6748800329585609}{97112193424553160} a^{3} - \frac{16730549414078831}{161853655707588600} a^{2} - \frac{32952802030944991}{97112193424553160} a + \frac{44097991937995297}{485560967122765800}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 400345.116 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-43}) \), 3.1.4300.2 x3, 3.1.172.1 x3, 3.1.4300.1 x3, 3.1.1075.1 x3, 6.0.795070000.1, 6.0.1272112.1, 6.0.795070000.2, 6.0.49691875.1, 9.1.3418801000000.2 x9

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$