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Properties

Label 18.2.51305860647...5488.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{27}\cdot 3^{8}\cdot 17^{12}$
Root discriminant $30.47$
Ramified primes $2, 3, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3^2$ (as 18T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-288, 0, 2004, 0, -3854, 0, 3457, 0, -1356, 0, 140, 0, 74, 0, -12, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^16 - 12*x^14 + 74*x^12 + 140*x^10 - 1356*x^8 + 3457*x^6 - 3854*x^4 + 2004*x^2 - 288)
 
gp: K = bnfinit(x^18 - 2*x^16 - 12*x^14 + 74*x^12 + 140*x^10 - 1356*x^8 + 3457*x^6 - 3854*x^4 + 2004*x^2 - 288, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{16} - 12 x^{14} + 74 x^{12} + 140 x^{10} - 1356 x^{8} + 3457 x^{6} - 3854 x^{4} + 2004 x^{2} - 288 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(513058606471919567779135488=2^{27}\cdot 3^{8}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.47$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{8} + \frac{1}{6} a^{2}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{3}$, $\frac{1}{12} a^{10} + \frac{1}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{36} a^{11} - \frac{1}{18} a^{9} + \frac{13}{36} a^{5} - \frac{1}{2} a^{4} + \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{36} a^{12} + \frac{1}{36} a^{10} + \frac{1}{36} a^{6} - \frac{1}{2} a^{5} - \frac{17}{36} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{36} a^{13} + \frac{1}{18} a^{9} + \frac{1}{36} a^{7} - \frac{1}{3} a^{5} - \frac{1}{9} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{108} a^{14} - \frac{1}{108} a^{12} + \frac{1}{108} a^{10} + \frac{7}{108} a^{8} - \frac{1}{108} a^{6} - \frac{1}{2} a^{5} + \frac{37}{108} a^{4} + \frac{2}{9} a^{2} + \frac{1}{3}$, $\frac{1}{216} a^{15} - \frac{1}{216} a^{14} + \frac{1}{108} a^{13} + \frac{1}{216} a^{12} + \frac{1}{216} a^{11} + \frac{1}{27} a^{10} + \frac{13}{216} a^{9} - \frac{7}{216} a^{8} + \frac{1}{108} a^{7} - \frac{17}{216} a^{6} + \frac{55}{216} a^{5} - \frac{8}{27} a^{4} + \frac{11}{36} a^{3} - \frac{4}{9} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{30872232} a^{16} - \frac{140563}{30872232} a^{14} - \frac{1}{72} a^{13} + \frac{80993}{15436116} a^{12} - \frac{1}{72} a^{11} + \frac{357517}{30872232} a^{10} - \frac{1}{12} a^{9} - \frac{2100529}{30872232} a^{8} + \frac{5}{72} a^{7} + \frac{288191}{15436116} a^{6} + \frac{29}{72} a^{5} + \frac{606623}{1715124} a^{4} + \frac{1}{4} a^{3} - \frac{327121}{857562} a^{2} - \frac{1}{2} a + \frac{28674}{142927}$, $\frac{1}{30872232} a^{17} + \frac{197}{2572686} a^{15} - \frac{68287}{5145372} a^{13} - \frac{178559}{15436116} a^{11} - \frac{13471}{1715124} a^{9} + \frac{779}{5145372} a^{7} + \frac{2486521}{30872232} a^{5} + \frac{2182157}{5145372} a^{3} - \frac{85579}{285854} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 8577930.408608785 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{2}) \), 3.1.6936.1, 3.1.867.1, 6.2.384864768.1, 6.2.1331712.1 x2, 6.2.384864768.2, 9.1.1001033261568.1

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$