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Properties

Label 18.0.20311991333...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{31}\cdot 3^{18}\cdot 5^{12}$
Root discriminant $28.94$
Ramified primes $2, 3, 5$
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![32, 0, -48, 0, 336, 0, -284, 0, 360, 0, 204, 0, 166, 0, 96, 0, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 + 96*x^14 + 166*x^12 + 204*x^10 + 360*x^8 - 284*x^6 + 336*x^4 - 48*x^2 + 32)
 
gp: K = bnfinit(x^18 + 18*x^16 + 96*x^14 + 166*x^12 + 204*x^10 + 360*x^8 - 284*x^6 + 336*x^4 - 48*x^2 + 32, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} + 96 x^{14} + 166 x^{12} + 204 x^{10} + 360 x^{8} - 284 x^{6} + 336 x^{4} - 48 x^{2} + 32 \)

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:  \(-203119913336832000000000000=-\,2^{31}\cdot 3^{18}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.94$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{15045336176} a^{16} + \frac{57377441}{3761334044} a^{14} + \frac{352948573}{3761334044} a^{12} - \frac{325645509}{7522668088} a^{10} - \frac{126100954}{940333511} a^{8} + \frac{17112185}{72333347} a^{6} + \frac{822085349}{3761334044} a^{4} + \frac{442705441}{1880667022} a^{2} - \frac{179616748}{940333511}$, $\frac{1}{15045336176} a^{17} + \frac{57377441}{3761334044} a^{15} + \frac{352948573}{3761334044} a^{13} - \frac{325645509}{7522668088} a^{11} - \frac{126100954}{940333511} a^{9} + \frac{17112185}{72333347} a^{7} + \frac{822085349}{3761334044} a^{5} + \frac{442705441}{1880667022} a^{3} - \frac{179616748}{940333511} a$

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:  \( -1 \) (order $2$)
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:  \( 147550.2494992511 \) (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

\(\Q(\sqrt{-2}) \), 3.1.675.1, 6.0.233280000.1, 9.3.4920750000.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.12.22.29$x^{12} - 2 x^{6} + 4$$6$$2$$22$$C_6\times S_3$$[3]_{3}^{6}$
$3$3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
5Data not computed