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Properties

Label 18.0.95006081547...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 13^{6}$
Root discriminant $18.90$
Ramified primes $2, 3, 5, 13$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -8, -8, 76, -94, -48, 349, -657, 621, -134, -361, 435, -166, -93, 159, -102, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 102*x^15 + 159*x^14 - 93*x^13 - 166*x^12 + 435*x^11 - 361*x^10 - 134*x^9 + 621*x^8 - 657*x^7 + 349*x^6 - 48*x^5 - 94*x^4 + 76*x^3 - 8*x^2 - 8*x + 4)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 102*x^15 + 159*x^14 - 93*x^13 - 166*x^12 + 435*x^11 - 361*x^10 - 134*x^9 + 621*x^8 - 657*x^7 + 349*x^6 - 48*x^5 - 94*x^4 + 76*x^3 - 8*x^2 - 8*x + 4, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 102 x^{15} + 159 x^{14} - 93 x^{13} - 166 x^{12} + 435 x^{11} - 361 x^{10} - 134 x^{9} + 621 x^{8} - 657 x^{7} + 349 x^{6} - 48 x^{5} - 94 x^{4} + 76 x^{3} - 8 x^{2} - 8 x + 4 \)

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:  \(-95006081547000000000000=-\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.90$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{296} a^{16} - \frac{1}{296} a^{15} - \frac{11}{148} a^{14} - \frac{3}{296} a^{13} - \frac{31}{296} a^{12} - \frac{17}{148} a^{11} - \frac{53}{296} a^{10} - \frac{3}{8} a^{9} - \frac{17}{74} a^{8} - \frac{49}{296} a^{7} - \frac{15}{296} a^{6} - \frac{13}{37} a^{5} + \frac{45}{148} a^{4} - \frac{4}{37} a^{3} + \frac{15}{74} a^{2} + \frac{4}{37} a + \frac{7}{74}$, $\frac{1}{102131025112} a^{17} - \frac{83042435}{102131025112} a^{16} - \frac{3757565349}{51065512556} a^{15} + \frac{4757742025}{102131025112} a^{14} + \frac{17861013903}{102131025112} a^{13} - \frac{11490140239}{51065512556} a^{12} - \frac{44676973185}{102131025112} a^{11} + \frac{31774736891}{102131025112} a^{10} + \frac{4255053929}{25532756278} a^{9} - \frac{28138730213}{102131025112} a^{8} + \frac{49366124123}{102131025112} a^{7} + \frac{2464698810}{12766378139} a^{6} - \frac{25279575385}{51065512556} a^{5} - \frac{5838248783}{25532756278} a^{4} + \frac{2285701243}{25532756278} a^{3} - \frac{5425746626}{12766378139} a^{2} + \frac{4635219819}{25532756278} a + \frac{5518452639}{12766378139}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{4841767}{74602648} a^{17} - \frac{49597587}{74602648} a^{16} + \frac{118021939}{37301324} a^{15} - \frac{671335471}{74602648} a^{14} + \frac{1161001067}{74602648} a^{13} - \frac{450989303}{37301324} a^{12} - \frac{842287253}{74602648} a^{11} + \frac{3038726651}{74602648} a^{10} - \frac{746496559}{18650662} a^{9} - \frac{385384229}{74602648} a^{8} + \frac{4236984607}{74602648} a^{7} - \frac{606105452}{9325331} a^{6} + \frac{344608323}{9325331} a^{5} - \frac{207737191}{18650662} a^{4} - \frac{58503867}{18650662} a^{3} + \frac{101555603}{18650662} a^{2} + \frac{1178433}{18650662} a + \frac{2014680}{9325331} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 54339.93481731778 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

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

\(\Q(\sqrt{-3}) \), 3.1.300.1 x3, 3.3.1300.1, 6.0.45630000.1, 6.0.270000.1, 9.3.59319000000.1 x3

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$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}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$