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Properties

Label 18.0.13730383371...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $28.32$
Ramified primes $2, 3, 5, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 54, -175, 411, -705, 831, -372, -147, 603, -444, 204, 74, -105, 84, -26, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 26*x^15 + 84*x^14 - 105*x^13 + 74*x^12 + 204*x^11 - 444*x^10 + 603*x^9 - 147*x^8 - 372*x^7 + 831*x^6 - 705*x^5 + 411*x^4 - 175*x^3 + 54*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^18 + 9*x^16 - 26*x^15 + 84*x^14 - 105*x^13 + 74*x^12 + 204*x^11 - 444*x^10 + 603*x^9 - 147*x^8 - 372*x^7 + 831*x^6 - 705*x^5 + 411*x^4 - 175*x^3 + 54*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 26 x^{15} + 84 x^{14} - 105 x^{13} + 74 x^{12} + 204 x^{11} - 444 x^{10} + 603 x^{9} - 147 x^{8} - 372 x^{7} + 831 x^{6} - 705 x^{5} + 411 x^{4} - 175 x^{3} + 54 x^{2} - 9 x + 1 \)

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:  \(-137303833711203000000000000=-\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.32$
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:  $2, 3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{4}{9} a^{5} + \frac{4}{9} a^{4} + \frac{4}{9} a^{2} + \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{2}{9} a^{2} - \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{1}{9} a^{6} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{1071} a^{15} - \frac{2}{153} a^{14} - \frac{19}{1071} a^{13} + \frac{4}{153} a^{12} + \frac{2}{17} a^{11} + \frac{8}{63} a^{10} - \frac{157}{1071} a^{9} + \frac{148}{1071} a^{8} - \frac{107}{1071} a^{7} - \frac{14}{153} a^{6} - \frac{23}{1071} a^{5} + \frac{409}{1071} a^{4} - \frac{79}{1071} a^{3} + \frac{188}{1071} a^{2} + \frac{121}{357} a + \frac{424}{1071}$, $\frac{1}{1071} a^{16} + \frac{23}{1071} a^{14} + \frac{2}{51} a^{12} - \frac{4}{1071} a^{11} + \frac{9}{119} a^{10} - \frac{3}{119} a^{9} - \frac{59}{357} a^{8} - \frac{7}{153} a^{7} + \frac{11}{357} a^{6} + \frac{29}{357} a^{5} - \frac{101}{357} a^{4} + \frac{1}{7} a^{3} + \frac{377}{1071} a^{2} - \frac{206}{1071} a + \frac{5}{51}$, $\frac{1}{10681083} a^{17} + \frac{4247}{10681083} a^{16} - \frac{4139}{10681083} a^{15} - \frac{8416}{1186787} a^{14} - \frac{195056}{10681083} a^{13} - \frac{274936}{10681083} a^{12} - \frac{161003}{3560361} a^{11} + \frac{665569}{10681083} a^{10} + \frac{193250}{10681083} a^{9} + \frac{99065}{1525869} a^{8} - \frac{1230125}{10681083} a^{7} - \frac{2306}{89757} a^{6} - \frac{55732}{169541} a^{5} + \frac{980354}{3560361} a^{4} + \frac{314779}{3560361} a^{3} + \frac{4075963}{10681083} a^{2} - \frac{1705346}{3560361} a + \frac{255163}{1186787}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{902522}{3560361} a^{17} - \frac{177601}{3560361} a^{16} - \frac{8215939}{3560361} a^{15} + \frac{7266912}{1186787} a^{14} - \frac{24014203}{1186787} a^{13} + \frac{27245913}{1186787} a^{12} - \frac{54267007}{3560361} a^{11} - \frac{64096705}{1186787} a^{10} + \frac{120821291}{1186787} a^{9} - \frac{23100129}{169541} a^{8} + \frac{51381685}{3560361} a^{7} + \frac{47186122}{508623} a^{6} - \frac{98883926}{508623} a^{5} + \frac{171579262}{1186787} a^{4} - \frac{98100136}{1186787} a^{3} + \frac{114805613}{3560361} a^{2} - \frac{37599367}{3560361} a + \frac{366514}{209433} \) (order $6$)
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:  \( 732910.4094523011 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.300.1 x3, 6.0.270000.1, 9.3.2255067000000.1

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

Sibling fields

Degree 9 siblings: 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 R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5Data not computed
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$