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Properties

Label 18.0.53632908573...6928.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 43^{12}$
Root discriminant $34.72$
Ramified primes $2, 43$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, 0, 4736, 0, 2824, 0, 3020, 0, 2454, 0, 2865, 0, 924, 0, 178, 0, 22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 22*x^16 + 178*x^14 + 924*x^12 + 2865*x^10 + 2454*x^8 + 3020*x^6 + 2824*x^4 + 4736*x^2 + 512)
 
gp: K = bnfinit(x^18 + 22*x^16 + 178*x^14 + 924*x^12 + 2865*x^10 + 2454*x^8 + 3020*x^6 + 2824*x^4 + 4736*x^2 + 512, 1)
 

Normalized defining polynomial

\( x^{18} + 22 x^{16} + 178 x^{14} + 924 x^{12} + 2865 x^{10} + 2454 x^{8} + 3020 x^{6} + 2824 x^{4} + 4736 x^{2} + 512 \)

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:  \(-5363290857327411623968636928=-\,2^{27}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.72$
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:  $2, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} + \frac{1}{16} a^{6} + \frac{1}{16} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{32} a^{9} + \frac{1}{32} a^{8} + \frac{1}{32} a^{7} - \frac{1}{32} a^{6} + \frac{1}{32} a^{5} - \frac{1}{32} a^{4} - \frac{1}{16} a^{3} + \frac{1}{16} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{96} a^{12} + \frac{1}{24} a^{8} - \frac{1}{16} a^{6} - \frac{7}{32} a^{4} - \frac{1}{2} a^{3} + \frac{11}{48} a^{2} + \frac{1}{6}$, $\frac{1}{96} a^{13} + \frac{1}{24} a^{9} - \frac{1}{16} a^{7} - \frac{7}{32} a^{5} + \frac{11}{48} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{384} a^{14} + \frac{5}{192} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} + \frac{3}{128} a^{6} - \frac{7}{48} a^{4} - \frac{1}{8} a^{3} - \frac{41}{96} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{1536} a^{15} - \frac{1}{384} a^{13} + \frac{5}{768} a^{11} + \frac{7}{192} a^{9} + \frac{43}{512} a^{7} + \frac{7}{384} a^{5} - \frac{1}{4} a^{4} + \frac{27}{128} a^{3} - \frac{1}{4} a^{2} - \frac{5}{48} a - \frac{1}{2}$, $\frac{1}{135188054016} a^{16} - \frac{23385839}{33797013504} a^{14} + \frac{113821223}{22531342336} a^{12} + \frac{208045493}{16898506752} a^{10} + \frac{3750895457}{135188054016} a^{8} - \frac{4019921855}{33797013504} a^{6} - \frac{5235091807}{33797013504} a^{4} - \frac{1}{2} a^{3} + \frac{130259497}{704104448} a^{2} - \frac{231091879}{528078336}$, $\frac{1}{135188054016} a^{17} - \frac{1382575}{33797013504} a^{15} + \frac{165437557}{67594027008} a^{13} - \frac{210016523}{16898506752} a^{11} - \frac{1}{32} a^{10} - \frac{3994253471}{135188054016} a^{9} + \frac{1}{32} a^{8} - \frac{2237657471}{33797013504} a^{7} - \frac{1}{32} a^{6} - \frac{483510133}{11265671168} a^{5} + \frac{7}{32} a^{4} + \frac{322788057}{704104448} a^{3} + \frac{5}{16} a^{2} + \frac{241978297}{528078336} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 194465438.339 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-2}) \), 3.1.14792.1 x3, 3.3.1849.1, 6.0.1750426112.1, 6.0.1750426112.2, 6.0.946688.1 x2, 9.3.3236537881088.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.946688.1
Degree 9 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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$43$43.9.6.1$x^{9} + 1290 x^{6} + 552851 x^{3} + 79507000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
43.9.6.1$x^{9} + 1290 x^{6} + 552851 x^{3} + 79507000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$