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)

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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)

Normalized defining 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

$C_3\times C_3:S_3$ (as 18T23):

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

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.1714608.2, 6.0.107163.1, 6.0.21168.1, 6.0.34992.1

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ 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$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$