Properties

Label 18.0.19990046271...2128.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}$
Root discriminant $10.39$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois Group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -3, 14, -12, -12, 114, -270, 480, -636, 702, -630, 479, -300, 159, -68, 24, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 24*x^16 - 68*x^15 + 159*x^14 - 300*x^13 + 479*x^12 - 630*x^11 + 702*x^10 - 636*x^9 + 480*x^8 - 270*x^7 + 114*x^6 - 12*x^5 - 12*x^4 + 14*x^3 - 3*x^2 + 1)
gp: K = bnfinit(x^18 - 6*x^17 + 24*x^16 - 68*x^15 + 159*x^14 - 300*x^13 + 479*x^12 - 630*x^11 + 702*x^10 - 636*x^9 + 480*x^8 - 270*x^7 + 114*x^6 - 12*x^5 - 12*x^4 + 14*x^3 - 3*x^2 + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 6 x^{17} \) \(\mathstrut +\mathstrut 24 x^{16} \) \(\mathstrut -\mathstrut 68 x^{15} \) \(\mathstrut +\mathstrut 159 x^{14} \) \(\mathstrut -\mathstrut 300 x^{13} \) \(\mathstrut +\mathstrut 479 x^{12} \) \(\mathstrut -\mathstrut 630 x^{11} \) \(\mathstrut +\mathstrut 702 x^{10} \) \(\mathstrut -\mathstrut 636 x^{9} \) \(\mathstrut +\mathstrut 480 x^{8} \) \(\mathstrut -\mathstrut 270 x^{7} \) \(\mathstrut +\mathstrut 114 x^{6} \) \(\mathstrut -\mathstrut 12 x^{5} \) \(\mathstrut -\mathstrut 12 x^{4} \) \(\mathstrut +\mathstrut 14 x^{3} \) \(\mathstrut -\mathstrut 3 x^{2} \) \(\mathstrut +\mathstrut 1 \)

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:  \(-1999004627104432128=-\,2^{18}\cdot 3^{27}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.39$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10} a^{15} - \frac{3}{10} a^{14} + \frac{1}{10} a^{13} - \frac{2}{5} a^{11} + \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{3}{10} a^{7} - \frac{3}{10} a^{6} - \frac{3}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{10} a^{3} + \frac{2}{5} a + \frac{3}{10}$, $\frac{1}{10} a^{16} + \frac{1}{5} a^{14} + \frac{3}{10} a^{13} - \frac{2}{5} a^{12} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} + \frac{1}{5} a^{9} + \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} - \frac{3}{10} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2} a - \frac{1}{10}$, $\frac{1}{7570} a^{17} - \frac{28}{757} a^{16} + \frac{287}{7570} a^{15} - \frac{746}{3785} a^{14} + \frac{1701}{7570} a^{13} + \frac{2209}{7570} a^{12} - \frac{3729}{7570} a^{11} + \frac{1437}{7570} a^{10} + \frac{1361}{7570} a^{9} + \frac{204}{3785} a^{8} - \frac{1547}{7570} a^{7} - \frac{334}{757} a^{6} - \frac{1862}{3785} a^{5} - \frac{2343}{7570} a^{4} - \frac{3751}{7570} a^{3} - \frac{195}{1514} a^{2} - \frac{831}{7570} a + \frac{573}{1514}$

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

Class group and class number

Trivial Abelian group, 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:  $8$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( \frac{11131}{3785} a^{17} - \frac{12437}{757} a^{16} + \frac{465669}{7570} a^{15} - \frac{1220299}{7570} a^{14} + \frac{2648237}{7570} a^{13} - \frac{2269981}{3785} a^{12} + \frac{3238771}{3785} a^{11} - \frac{7271311}{7570} a^{10} + \frac{3324951}{3785} a^{9} - \frac{2210992}{3785} a^{8} + \frac{2203271}{7570} a^{7} - \frac{64853}{1514} a^{6} - \frac{152233}{7570} a^{5} + \frac{142547}{3785} a^{4} + \frac{11263}{7570} a^{3} - \frac{2497}{757} a^{2} + \frac{19604}{3785} a + \frac{1433}{1514} \) (order $18$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{691}{1514} a^{17} - \frac{601}{757} a^{16} - \frac{3449}{3785} a^{15} + \frac{119159}{7570} a^{14} - \frac{217834}{3785} a^{13} + \frac{239519}{1514} a^{12} - \frac{2409843}{7570} a^{11} + \frac{1986961}{3785} a^{10} - \frac{5079699}{7570} a^{9} + \frac{2641983}{3785} a^{8} - \frac{2037698}{3785} a^{7} + \frac{2394399}{7570} a^{6} - \frac{680991}{7570} a^{5} - \frac{52697}{7570} a^{4} + \frac{147304}{3785} a^{3} - \frac{21191}{1514} a^{2} + \frac{16093}{7570} a + \frac{10998}{3785} \),  \( \frac{2861}{1514} a^{17} - \frac{37528}{3785} a^{16} + \frac{270573}{7570} a^{15} - \frac{337723}{3785} a^{14} + \frac{282169}{1514} a^{13} - \frac{2282061}{7570} a^{12} + \frac{3054669}{7570} a^{11} - \frac{3080283}{7570} a^{10} + \frac{2448717}{7570} a^{9} - \frac{577606}{3785} a^{8} + \frac{227403}{7570} a^{7} + \frac{218829}{3785} a^{6} - \frac{168162}{3785} a^{5} + \frac{266781}{7570} a^{4} - \frac{9463}{1514} a^{3} + \frac{17761}{7570} a^{2} - \frac{5583}{7570} a - \frac{31}{1514} \),  \( \frac{3254}{3785} a^{17} - \frac{45561}{7570} a^{16} + \frac{188013}{7570} a^{15} - \frac{549479}{7570} a^{14} + \frac{638021}{3785} a^{13} - \frac{1202502}{3785} a^{12} + \frac{3724771}{7570} a^{11} - \frac{2349719}{3785} a^{10} + \frac{2393121}{3785} a^{9} - \frac{3836011}{7570} a^{8} + \frac{462269}{1514} a^{7} - \frac{924519}{7570} a^{6} + \frac{55421}{3785} a^{5} + \frac{119613}{7570} a^{4} - \frac{49091}{3785} a^{3} + \frac{5993}{3785} a^{2} + \frac{729}{1514} a - \frac{1259}{3785} \),  \( \frac{1719}{757} a^{17} - \frac{56872}{3785} a^{16} + \frac{458903}{7570} a^{15} - \frac{1318257}{7570} a^{14} + \frac{3042691}{7570} a^{13} - \frac{2850067}{3785} a^{12} + \frac{4412421}{3785} a^{11} - \frac{11182041}{7570} a^{10} + \frac{5784868}{3785} a^{9} - \frac{951936}{757} a^{8} + \frac{6008789}{7570} a^{7} - \frac{2579791}{7570} a^{6} + \frac{422551}{7570} a^{5} + \frac{147244}{3785} a^{4} - \frac{288361}{7570} a^{3} + \frac{30902}{3785} a^{2} + \frac{5906}{3785} a - \frac{9397}{7570} \),  \( \frac{4461}{3785} a^{17} - \frac{36366}{3785} a^{16} + \frac{317623}{7570} a^{15} - \frac{196473}{1514} a^{14} + \frac{473425}{1514} a^{13} - \frac{2331837}{3785} a^{12} + \frac{3757754}{3785} a^{11} - \frac{9964029}{7570} a^{10} + \frac{1066215}{757} a^{9} - \frac{4557637}{3785} a^{8} + \frac{5850143}{7570} a^{7} - \frac{2584573}{7570} a^{6} + \frac{67819}{1514} a^{5} + \frac{180699}{3785} a^{4} - \frac{67869}{1514} a^{3} + \frac{32041}{3785} a^{2} + \frac{16592}{3785} a - \frac{19771}{7570} \),  \( \frac{6246}{3785} a^{17} - \frac{15981}{1514} a^{16} + \frac{321777}{7570} a^{15} - \frac{914423}{7570} a^{14} + \frac{209982}{757} a^{13} - \frac{1940616}{3785} a^{12} + \frac{1188961}{1514} a^{11} - \frac{3680024}{3785} a^{10} + \frac{3702183}{3785} a^{9} - \frac{1147641}{1514} a^{8} + \frac{3326587}{7570} a^{7} - \frac{1077679}{7570} a^{6} - \frac{20961}{3785} a^{5} + \frac{360981}{7570} a^{4} - \frac{16119}{757} a^{3} + \frac{3828}{757} a^{2} + \frac{5731}{1514} a + \frac{67}{3785} \),  \( \frac{9711}{7570} a^{17} - \frac{29491}{3785} a^{16} + \frac{229911}{7570} a^{15} - \frac{317099}{3785} a^{14} + \frac{1426859}{7570} a^{13} - \frac{2580123}{7570} a^{12} + \frac{3876887}{7570} a^{11} - \frac{4715941}{7570} a^{10} + \frac{4711019}{7570} a^{9} - \frac{1845591}{3785} a^{8} + \frac{455815}{1514} a^{7} - \frac{477069}{3785} a^{6} + \frac{125482}{3785} a^{5} + \frac{11631}{7570} a^{4} - \frac{9719}{7570} a^{3} - \frac{16323}{7570} a^{2} + \frac{2681}{1514} a - \frac{3791}{7570} \),  \( \frac{2337}{1514} a^{17} - \frac{52279}{7570} a^{16} + \frac{85962}{3785} a^{15} - \frac{73491}{1514} a^{14} + \frac{668067}{7570} a^{13} - \frac{797899}{7570} a^{12} + \frac{321124}{3785} a^{11} + \frac{39581}{1514} a^{10} - \frac{1157981}{7570} a^{9} + \frac{2117223}{7570} a^{8} - \frac{1009227}{3785} a^{7} + \frac{1587421}{7570} a^{6} - \frac{264726}{3785} a^{5} + \frac{62306}{3785} a^{4} + \frac{207283}{7570} a^{3} - \frac{74211}{7570} a^{2} + \frac{17313}{3785} a + \frac{16611}{7570} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 269.804731118 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 3.1.648.1, \(\Q(\zeta_{9})\), 6.0.1259712.1, 9.3.272097792.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
3Data not computed