# Properties

 Label 18.0.50407427849...3712.1 Degree $18$ Signature $[0, 9]$ Discriminant $-\,2^{12}\cdot 3^{21}\cdot 7^{6}$ Root discriminant $10.94$ Ramified primes $2, 3, 7$ Class number $1$ Class group Trivial Galois Group $C_3\times C_3:S_3$ (as 18T23)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -5, 0, 0, 12, 0, 0, -9, 0, 0, 2, 0, 0, -1, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^15 + 2*x^12 - 9*x^9 + 12*x^6 - 5*x^3 + 1)
gp: K = bnfinit(x^18 - x^15 + 2*x^12 - 9*x^9 + 12*x^6 - 5*x^3 + 1, 1)

## Normalizeddefining polynomial

$$x^{18}$$ $$\mathstrut -\mathstrut x^{15}$$ $$\mathstrut +\mathstrut 2 x^{12}$$ $$\mathstrut -\mathstrut 9 x^{9}$$ $$\mathstrut +\mathstrut 12 x^{6}$$ $$\mathstrut -\mathstrut 5 x^{3}$$ $$\mathstrut +\mathstrut 1$$

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: $$-5040742784941043712=-\,2^{12}\cdot 3^{21}\cdot 7^{6}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $10.94$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 3, 7$ 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 $$a$$)

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

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: $$-\frac{2}{3} a^{15} + \frac{1}{3} a^{12} - \frac{4}{3} a^{9} + \frac{16}{3} a^{6} - 6 a^{3} + 2$$ (order $6$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$\frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{2}{3} a^{5} - \frac{5}{3} a^{2}$$,  $$\frac{2}{3} a^{16} - \frac{1}{3} a^{13} + \frac{4}{3} a^{10} - \frac{16}{3} a^{7} + 6 a^{4} - 2 a$$,  $$\frac{2}{9} a^{17} - \frac{2}{9} a^{16} - \frac{1}{9} a^{15} - \frac{2}{9} a^{14} + \frac{1}{9} a^{12} + \frac{5}{9} a^{11} - \frac{4}{9} a^{10} - \frac{2}{9} a^{9} - \frac{16}{9} a^{8} + \frac{4}{3} a^{7} + \frac{10}{9} a^{6} + 3 a^{5} - \frac{11}{9} a^{4} - \frac{8}{9} a^{3} - \frac{13}{9} a^{2} - \frac{5}{9} a + \frac{7}{9}$$,  $$\frac{2}{9} a^{16} - \frac{2}{9} a^{15} + \frac{4}{9} a^{14} - \frac{4}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{3} a^{10} - \frac{4}{9} a^{9} + \frac{7}{9} a^{8} - \frac{23}{9} a^{7} + \frac{16}{9} a^{6} - \frac{28}{9} a^{5} + \frac{37}{9} a^{4} - 2 a^{3} + \frac{20}{9} a^{2} - \frac{16}{9} a$$,  $$\frac{7}{9} a^{17} + \frac{2}{9} a^{16} - \frac{1}{9} a^{15} - \frac{5}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{3} a^{12} + \frac{4}{3} a^{11} + \frac{4}{9} a^{10} - \frac{1}{9} a^{9} - \frac{61}{9} a^{8} - \frac{16}{9} a^{7} + \frac{13}{9} a^{6} + \frac{65}{9} a^{5} + 2 a^{4} - \frac{20}{9} a^{3} - \frac{14}{9} a^{2} + 1$$,  $$\frac{2}{9} a^{17} - \frac{2}{9} a^{15} - \frac{5}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{3} a^{12} + \frac{4}{9} a^{11} + \frac{2}{9} a^{10} - \frac{1}{3} a^{9} - \frac{23}{9} a^{8} + \frac{1}{3} a^{7} + \frac{19}{9} a^{6} + \frac{46}{9} a^{5} - \frac{1}{3} a^{4} - \frac{10}{3} a^{3} - \frac{20}{9} a^{2} - \frac{2}{9} a + \frac{11}{9}$$,  $$\frac{11}{9} a^{17} - \frac{5}{9} a^{16} - a^{14} + \frac{2}{9} a^{13} - \frac{1}{9} a^{12} + \frac{19}{9} a^{11} - \frac{7}{9} a^{10} - \frac{2}{9} a^{9} - \frac{32}{3} a^{8} + \frac{41}{9} a^{7} - \frac{1}{3} a^{6} + \frac{113}{9} a^{5} - \frac{31}{9} a^{4} + \frac{1}{3} a^{3} - \frac{25}{9} a^{2} - \frac{4}{9} a + \frac{2}{9}$$,  $$\frac{2}{9} a^{16} + \frac{5}{9} a^{15} - \frac{1}{3} a^{14} + \frac{1}{9} a^{13} - \frac{4}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{3} a^{10} + \frac{8}{9} a^{9} - \frac{7}{9} a^{8} - \frac{4}{3} a^{7} - 5 a^{6} + \frac{19}{9} a^{5} + \frac{2}{9} a^{4} + \frac{47}{9} a^{3} - \frac{7}{9} a^{2} + a - \frac{14}{9}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$153.95173876$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 54 The 18 conjugacy class representatives for $C_3\times C_3:S_3$ Character table for $C_3\times C_3:S_3$

## Intermediate fields

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

## Sibling fields

 Degree 18 siblings: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3} 7.3.0.1x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3} 7.9.6.1x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$