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Properties

Label 18.0.50900865739...3344.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{18}\cdot 23^{8}$
Root discriminant $30.46$
Ramified primes $2, 3, 23$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 27, 0, 213, 0, 824, 0, 1092, 0, 795, 0, 375, 0, 111, 0, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 + 111*x^14 + 375*x^12 + 795*x^10 + 1092*x^8 + 824*x^6 + 213*x^4 + 27*x^2 + 1)
 
gp: K = bnfinit(x^18 + 18*x^16 + 111*x^14 + 375*x^12 + 795*x^10 + 1092*x^8 + 824*x^6 + 213*x^4 + 27*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} + 111 x^{14} + 375 x^{12} + 795 x^{10} + 1092 x^{8} + 824 x^{6} + 213 x^{4} + 27 x^{2} + 1 \)

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:  \(-509008657395156683109433344=-\,2^{24}\cdot 3^{18}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.46$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{3}{8} a^{5} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{1473128} a^{16} - \frac{147803}{1473128} a^{14} - \frac{1}{8} a^{13} + \frac{14431}{368282} a^{12} - \frac{34689}{184141} a^{10} - \frac{1}{4} a^{9} - \frac{93269}{1473128} a^{8} - \frac{1}{4} a^{7} - \frac{255293}{1473128} a^{6} + \frac{3}{8} a^{5} - \frac{189163}{1473128} a^{4} + \frac{1}{4} a^{3} + \frac{269045}{1473128} a^{2} - \frac{1}{8} a + \frac{136131}{736564}$, $\frac{1}{1473128} a^{17} + \frac{18169}{736564} a^{15} - \frac{1}{8} a^{14} - \frac{126417}{1473128} a^{13} - \frac{1}{8} a^{12} + \frac{45385}{736564} a^{11} - \frac{1}{4} a^{10} - \frac{93269}{1473128} a^{9} + \frac{148565}{736564} a^{7} + \frac{1}{8} a^{6} - \frac{2511}{736564} a^{5} + \frac{1}{8} a^{4} - \frac{162915}{368282} a^{3} - \frac{3}{8} a^{2} + \frac{88121}{1473128} a - \frac{1}{8}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{92319}{736564} a^{17} + \frac{1664353}{736564} a^{15} + \frac{2573328}{184141} a^{13} + \frac{8723448}{184141} a^{11} + \frac{74325313}{736564} a^{9} + \frac{102931083}{736564} a^{7} + \frac{79405791}{736564} a^{5} + \frac{22755913}{736564} a^{3} + \frac{2064231}{368282} a \) (order $4$)
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:  \( 292555.45841582766 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.108.1, 6.0.186624.1, 9.3.352519065792.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$23$23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
23.6.4.2$x^{6} - 23 x^{3} + 3703$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
23.6.4.2$x^{6} - 23 x^{3} + 3703$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$