Properties

Label 18.0.12213256064...4031.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{33}\cdot 13^{3}$
Root discriminant $11.49$
Ramified primes $3, 13$
Class number $1$
Class group Trivial
Galois Group $C_3\times S_3\wr C_2$ (as 18T93)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 3, -3, -15, 12, 81, 12, -150, -56, 141, 60, -78, -30, 24, 9, -3, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 3*x^16 + 9*x^15 + 24*x^14 - 30*x^13 - 78*x^12 + 60*x^11 + 141*x^10 - 56*x^9 - 150*x^8 + 12*x^7 + 81*x^6 + 12*x^5 - 15*x^4 - 3*x^3 + 3*x^2 + 3*x + 1)
gp: K = bnfinit(x^18 - 3*x^17 - 3*x^16 + 9*x^15 + 24*x^14 - 30*x^13 - 78*x^12 + 60*x^11 + 141*x^10 - 56*x^9 - 150*x^8 + 12*x^7 + 81*x^6 + 12*x^5 - 15*x^4 - 3*x^3 + 3*x^2 + 3*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 3 x^{17} \) \(\mathstrut -\mathstrut 3 x^{16} \) \(\mathstrut +\mathstrut 9 x^{15} \) \(\mathstrut +\mathstrut 24 x^{14} \) \(\mathstrut -\mathstrut 30 x^{13} \) \(\mathstrut -\mathstrut 78 x^{12} \) \(\mathstrut +\mathstrut 60 x^{11} \) \(\mathstrut +\mathstrut 141 x^{10} \) \(\mathstrut -\mathstrut 56 x^{9} \) \(\mathstrut -\mathstrut 150 x^{8} \) \(\mathstrut +\mathstrut 12 x^{7} \) \(\mathstrut +\mathstrut 81 x^{6} \) \(\mathstrut +\mathstrut 12 x^{5} \) \(\mathstrut -\mathstrut 15 x^{4} \) \(\mathstrut -\mathstrut 3 x^{3} \) \(\mathstrut +\mathstrut 3 x^{2} \) \(\mathstrut +\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-12213256064722484031=-\,3^{33}\cdot 13^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.49$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{9} - \frac{1}{3} a^{6} + \frac{2}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{1}{3} a^{7} + \frac{2}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{9} - \frac{1}{9} a^{6} + \frac{1}{3} a^{3} - \frac{2}{9}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{10} - \frac{1}{9} a^{7} + \frac{1}{3} a^{4} - \frac{2}{9} a$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{4}{9} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{4}{9}$

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:  \( -\frac{11}{9} a^{17} + \frac{41}{9} a^{16} + \frac{4}{9} a^{15} - \frac{104}{9} a^{14} - 21 a^{13} + \frac{467}{9} a^{12} + \frac{532}{9} a^{11} - \frac{1052}{9} a^{10} - \frac{266}{3} a^{9} + \frac{1187}{9} a^{8} + \frac{769}{9} a^{7} - \frac{649}{9} a^{6} - \frac{358}{9} a^{5} + \frac{26}{3} a^{4} + \frac{46}{9} a^{3} + \frac{23}{9} a^{2} - \frac{25}{9} a - \frac{5}{3} \) (order $18$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{5}{9} a^{17} - \frac{19}{9} a^{16} + 5 a^{14} + \frac{86}{9} a^{13} - \frac{73}{3} a^{12} - \frac{221}{9} a^{11} + \frac{470}{9} a^{10} + 39 a^{9} - \frac{545}{9} a^{8} - \frac{347}{9} a^{7} + 34 a^{6} + \frac{62}{3} a^{5} - \frac{47}{9} a^{4} - \frac{11}{3} a^{3} - \frac{19}{9} a^{2} + \frac{10}{9} a + 1 \),  \( \frac{4}{9} a^{17} - \frac{11}{9} a^{16} - \frac{16}{9} a^{15} + \frac{40}{9} a^{14} + \frac{94}{9} a^{13} - \frac{106}{9} a^{12} - \frac{341}{9} a^{11} + \frac{265}{9} a^{10} + \frac{569}{9} a^{9} - \frac{295}{9} a^{8} - \frac{559}{9} a^{7} + \frac{154}{9} a^{6} + \frac{263}{9} a^{5} - \frac{25}{9} a^{4} - \frac{26}{9} a^{3} - \frac{4}{9} a^{2} + \frac{11}{9} a + \frac{10}{9} \),  \( \frac{23}{9} a^{17} - \frac{85}{9} a^{16} - a^{15} + \frac{211}{9} a^{14} + \frac{403}{9} a^{13} - \frac{968}{9} a^{12} - \frac{1099}{9} a^{11} + \frac{2135}{9} a^{10} + \frac{1711}{9} a^{9} - \frac{2489}{9} a^{8} - \frac{1652}{9} a^{7} + 164 a^{6} + \frac{803}{9} a^{5} - \frac{343}{9} a^{4} - \frac{130}{9} a^{3} + \frac{43}{9} a^{2} + \frac{58}{9} a + \frac{29}{9} \),  \( \frac{17}{9} a^{17} - 7 a^{16} - \frac{2}{3} a^{15} + \frac{52}{3} a^{14} + \frac{296}{9} a^{13} - \frac{718}{9} a^{12} - \frac{806}{9} a^{11} + \frac{1591}{9} a^{10} + \frac{1246}{9} a^{9} - \frac{1853}{9} a^{8} - \frac{410}{3} a^{7} + 125 a^{6} + \frac{208}{3} a^{5} - \frac{263}{9} a^{4} - \frac{131}{9} a^{3} + \frac{8}{9} a^{2} + \frac{47}{9} a + \frac{32}{9} \),  \( \frac{1}{9} a^{16} - \frac{2}{3} a^{15} + a^{14} + \frac{2}{3} a^{13} + \frac{1}{9} a^{12} - \frac{79}{9} a^{11} + \frac{55}{9} a^{10} + \frac{154}{9} a^{9} - \frac{38}{3} a^{8} - \frac{187}{9} a^{7} + \frac{41}{3} a^{6} + \frac{35}{3} a^{5} - \frac{17}{3} a^{4} - \frac{10}{9} a^{3} + \frac{10}{9} a^{2} - \frac{10}{9} a + \frac{2}{9} \),  \( \frac{20}{9} a^{17} - \frac{76}{9} a^{16} - \frac{2}{9} a^{15} + \frac{188}{9} a^{14} + 38 a^{13} - \frac{890}{9} a^{12} - \frac{922}{9} a^{11} + \frac{1972}{9} a^{10} + \frac{1468}{9} a^{9} - \frac{2330}{9} a^{8} - \frac{1517}{9} a^{7} + \frac{1421}{9} a^{6} + \frac{850}{9} a^{5} - \frac{112}{3} a^{4} - \frac{211}{9} a^{3} + \frac{25}{9} a^{2} + \frac{74}{9} a + \frac{32}{9} \),  \( \frac{7}{9} a^{17} - \frac{25}{9} a^{16} - a^{15} + \frac{76}{9} a^{14} + 14 a^{13} - \frac{299}{9} a^{12} - \frac{415}{9} a^{11} + \frac{737}{9} a^{10} + \frac{221}{3} a^{9} - \frac{940}{9} a^{8} - \frac{644}{9} a^{7} + \frac{214}{3} a^{6} + \frac{302}{9} a^{5} - 23 a^{4} - \frac{10}{9} a^{3} + \frac{22}{9} a^{2} - \frac{8}{9} a + 1 \),  \( \frac{10}{9} a^{17} - \frac{14}{3} a^{16} + \frac{17}{9} a^{15} + \frac{28}{3} a^{14} + \frac{131}{9} a^{13} - \frac{482}{9} a^{12} - \frac{236}{9} a^{11} + 116 a^{10} + \frac{182}{9} a^{9} - \frac{1141}{9} a^{8} - \frac{17}{3} a^{7} + \frac{622}{9} a^{6} - \frac{34}{3} a^{5} - \frac{92}{9} a^{4} + \frac{92}{9} a^{3} + \frac{2}{9} a^{2} + \frac{4}{3} a - \frac{5}{9} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 1025.11738531 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3\times S_3\wr C_2$ (as 18T93):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 216
The 27 conjugacy class representatives for $C_3\times S_3\wr C_2$
Character table for $C_3\times S_3\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 6.0.255879.1

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
3Data not computed
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$