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Properties

Label 18.0.61512776395...4944.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 31^{15}$
Root discriminant $34.98$
Ramified primes $2, 31$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $S_3\times A_4$ (as 18T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1984, 0, -5952, 0, -5952, 0, 14152, 0, 18160, 0, 8710, 0, 2245, 0, 337, 0, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 28*x^16 + 337*x^14 + 2245*x^12 + 8710*x^10 + 18160*x^8 + 14152*x^6 - 5952*x^4 - 5952*x^2 + 1984)
 
gp: K = bnfinit(x^18 + 28*x^16 + 337*x^14 + 2245*x^12 + 8710*x^10 + 18160*x^8 + 14152*x^6 - 5952*x^4 - 5952*x^2 + 1984, 1)
 

Normalized defining polynomial

\( x^{18} + 28 x^{16} + 337 x^{14} + 2245 x^{12} + 8710 x^{10} + 18160 x^{8} + 14152 x^{6} - 5952 x^{4} - 5952 x^{2} + 1984 \)

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:  \(-6151277639590133348421074944=-\,2^{18}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.98$
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:  $2, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{8} + \frac{1}{8} a^{6} + \frac{1}{4} a^{4}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{192} a^{14} + \frac{1}{24} a^{12} - \frac{19}{192} a^{10} + \frac{1}{192} a^{8} + \frac{5}{32} a^{6} + \frac{7}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{5}{12}$, $\frac{1}{192} a^{15} + \frac{1}{24} a^{13} - \frac{19}{192} a^{11} + \frac{1}{192} a^{9} + \frac{5}{32} a^{7} - \frac{1}{16} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{5}{12} a$, $\frac{1}{187190751744} a^{16} - \frac{62733097}{31198458624} a^{14} - \frac{2357774243}{187190751744} a^{12} + \frac{5774842985}{62396917248} a^{10} - \frac{63154051}{5849710992} a^{8} + \frac{23193087}{1299935776} a^{6} - \frac{10546681603}{23398843968} a^{4} - \frac{376620073}{11699421984} a^{2} - \frac{1767152713}{5849710992}$, $\frac{1}{187190751744} a^{17} - \frac{62733097}{31198458624} a^{15} - \frac{2357774243}{187190751744} a^{13} + \frac{5774842985}{62396917248} a^{11} - \frac{63154051}{5849710992} a^{9} + \frac{23193087}{1299935776} a^{7} + \frac{1152740381}{23398843968} a^{5} + \frac{5473090919}{11699421984} a^{3} - \frac{1}{2} a^{2} - \frac{1767152713}{5849710992} a$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 390018.833589 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times A_4$ (as 18T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $S_3\times A_4$
Character table for $S_3\times A_4$

Intermediate fields

3.3.961.1, 3.1.31.1, 6.0.1832265664.1, 9.3.27512614111.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
$31$31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$