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Properties

Label 18.2.46756258693...0656.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{20}\cdot 41^{9}$
Root discriminant $34.45$
Ramified primes $2, 3, 41$
Class number $2$ (GRH)
Class group $[2]$ (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![-512, 0, 0, 11712, 0, 0, 200, 0, 0, -1493, 0, 0, 331, 0, 0, -29, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 29*x^15 + 331*x^12 - 1493*x^9 + 200*x^6 + 11712*x^3 - 512)
 
gp: K = bnfinit(x^18 - 29*x^15 + 331*x^12 - 1493*x^9 + 200*x^6 + 11712*x^3 - 512, 1)
 

Normalized defining polynomial

\( x^{18} - 29 x^{15} + 331 x^{12} - 1493 x^{9} + 200 x^{6} + 11712 x^{3} - 512 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4675625869369624999536070656=2^{12}\cdot 3^{20}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.45$
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:  $2, 3, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} - \frac{1}{2} a^{4} - \frac{1}{6} a$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{8} + \frac{1}{4} a^{5} - \frac{1}{12} a^{2}$, $\frac{1}{24} a^{12} + \frac{1}{8} a^{9} + \frac{11}{24} a^{6} + \frac{11}{24} a^{3} + \frac{1}{3}$, $\frac{1}{144} a^{13} + \frac{1}{72} a^{12} - \frac{1}{36} a^{11} + \frac{1}{48} a^{10} - \frac{5}{72} a^{9} + \frac{1}{36} a^{8} - \frac{37}{144} a^{7} - \frac{29}{72} a^{6} - \frac{1}{12} a^{5} + \frac{11}{144} a^{4} - \frac{13}{72} a^{3} + \frac{1}{36} a^{2} - \frac{5}{18} a - \frac{4}{9}$, $\frac{1}{288} a^{14} + \frac{1}{72} a^{12} + \frac{11}{288} a^{11} - \frac{1}{18} a^{10} + \frac{1}{24} a^{9} + \frac{11}{32} a^{8} + \frac{1}{18} a^{7} + \frac{35}{72} a^{6} + \frac{35}{288} a^{5} - \frac{1}{6} a^{4} + \frac{11}{72} a^{3} + \frac{1}{3} a^{2} + \frac{1}{18} a + \frac{4}{9}$, $\frac{1}{28618560} a^{15} - \frac{42413}{3179840} a^{12} - \frac{290333}{28618560} a^{9} + \frac{1852097}{9539520} a^{6} + \frac{274681}{894330} a^{3} - \frac{67631}{447165}$, $\frac{1}{57237120} a^{16} + \frac{15763}{57237120} a^{13} + \frac{1}{72} a^{12} - \frac{1}{36} a^{11} + \frac{902107}{57237120} a^{10} - \frac{5}{72} a^{9} + \frac{1}{36} a^{8} + \frac{19468091}{57237120} a^{7} - \frac{29}{72} a^{6} - \frac{1}{12} a^{5} + \frac{1645259}{7154640} a^{4} - \frac{13}{72} a^{3} + \frac{1}{36} a^{2} + \frac{43703}{298110} a - \frac{4}{9}$, $\frac{1}{114474240} a^{17} + \frac{15763}{114474240} a^{14} + \frac{1}{72} a^{12} + \frac{1360649}{38158080} a^{11} - \frac{1}{18} a^{10} + \frac{1}{24} a^{9} + \frac{5429417}{38158080} a^{8} + \frac{1}{18} a^{7} + \frac{35}{72} a^{6} - \frac{4316941}{14309280} a^{5} - \frac{1}{6} a^{4} + \frac{11}{72} a^{3} - \frac{406453}{894330} a^{2} + \frac{1}{18} a + \frac{4}{9}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 20136087.093132745 \) (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

\(\Q(\sqrt{41}) \), 3.1.108.1, 3.1.492.1, 6.2.200973636.2 x2, 6.2.803894544.3, 6.2.9924624.1, 9.1.260461832256.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$41$41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$