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Properties

Label 18.0.11833920810...7183.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{18}\cdot 7^{9}\cdot 31^{14}$
Root discriminant $114.71$
Ramified primes $3, 7, 31$
Class number $280756$ (GRH)
Class group $[2, 140378]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7562752, -5924352, 8997792, -4412288, 3322872, -997932, 764284, -295545, 232287, -34024, -22956, 12183, 6441, -3807, -165, 286, -12, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 12*x^16 + 286*x^15 - 165*x^14 - 3807*x^13 + 6441*x^12 + 12183*x^11 - 22956*x^10 - 34024*x^9 + 232287*x^8 - 295545*x^7 + 764284*x^6 - 997932*x^5 + 3322872*x^4 - 4412288*x^3 + 8997792*x^2 - 5924352*x + 7562752)
 
gp: K = bnfinit(x^18 - 9*x^17 - 12*x^16 + 286*x^15 - 165*x^14 - 3807*x^13 + 6441*x^12 + 12183*x^11 - 22956*x^10 - 34024*x^9 + 232287*x^8 - 295545*x^7 + 764284*x^6 - 997932*x^5 + 3322872*x^4 - 4412288*x^3 + 8997792*x^2 - 5924352*x + 7562752, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 12 x^{16} + 286 x^{15} - 165 x^{14} - 3807 x^{13} + 6441 x^{12} + 12183 x^{11} - 22956 x^{10} - 34024 x^{9} + 232287 x^{8} - 295545 x^{7} + 764284 x^{6} - 997932 x^{5} + 3322872 x^{4} - 4412288 x^{3} + 8997792 x^{2} - 5924352 x + 7562752 \)

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:  \(-11833920810399528639829186240681437183=-\,3^{18}\cdot 7^{9}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $114.71$
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:  $3, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{16} a^{5} + \frac{3}{16} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{7}{32} a^{6} - \frac{1}{16} a^{5} - \frac{1}{32} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{3}{32} a^{9} - \frac{1}{8} a^{8} - \frac{11}{64} a^{7} + \frac{1}{32} a^{6} - \frac{7}{64} a^{5} - \frac{5}{32} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{13} - \frac{1}{128} a^{12} + \frac{1}{128} a^{11} - \frac{3}{64} a^{10} - \frac{1}{64} a^{9} - \frac{3}{128} a^{8} - \frac{19}{128} a^{7} + \frac{23}{128} a^{6} + \frac{29}{128} a^{5} - \frac{11}{64} a^{4} + \frac{7}{16} a^{3} + \frac{7}{16} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{256} a^{15} - \frac{1}{256} a^{14} - \frac{1}{256} a^{13} + \frac{1}{256} a^{12} - \frac{3}{128} a^{11} - \frac{1}{128} a^{10} + \frac{29}{256} a^{9} - \frac{19}{256} a^{8} - \frac{41}{256} a^{7} + \frac{29}{256} a^{6} + \frac{21}{128} a^{5} - \frac{1}{32} a^{4} + \frac{3}{32} a^{3} + \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{1971968} a^{16} - \frac{253}{1971968} a^{15} - \frac{5161}{1971968} a^{14} - \frac{14023}{1971968} a^{13} - \frac{14043}{985984} a^{12} - \frac{25755}{985984} a^{11} + \frac{15989}{1971968} a^{10} + \frac{67017}{1971968} a^{9} - \frac{136217}{1971968} a^{8} - \frac{31779}{1971968} a^{7} - \frac{240227}{985984} a^{6} + \frac{50877}{492992} a^{5} + \frac{2123}{246496} a^{4} + \frac{6795}{123248} a^{3} + \frac{28077}{61624} a^{2} + \frac{550}{7703} a - \frac{327}{7703}$, $\frac{1}{7524507544072771527015218454199823438723072} a^{17} + \frac{6076818011761629572124051672984167}{117570430376137055109612788346872241230048} a^{16} + \frac{3365554999305022835777802801330392311949}{1881126886018192881753804613549955859680768} a^{15} - \frac{239467397604455338872385674513687432207}{3762253772036385763507609227099911719361536} a^{14} - \frac{55503711369301922146894039615439836421619}{7524507544072771527015218454199823438723072} a^{13} + \frac{30450569913717443628351703216228875783995}{3762253772036385763507609227099911719361536} a^{12} - \frac{163457146441350291125040327749867632794545}{7524507544072771527015218454199823438723072} a^{11} - \frac{29479011481859178327955913466105038594761}{3762253772036385763507609227099911719361536} a^{10} - \frac{183101864321507966379817971005318070585095}{3762253772036385763507609227099911719361536} a^{9} - \frac{323997767735479622574497375701708039390475}{3762253772036385763507609227099911719361536} a^{8} - \frac{1131579650178603523668704742849477330199751}{7524507544072771527015218454199823438723072} a^{7} + \frac{89229854586161738584436999350612423707953}{940563443009096440876902306774977929840384} a^{6} + \frac{352616442769368304850569573250629298750849}{1881126886018192881753804613549955859680768} a^{5} - \frac{27624146211718605634423436871555330315807}{940563443009096440876902306774977929840384} a^{4} + \frac{114749818302409078685927154359506191601835}{235140860752274110219225576693744482460096} a^{3} + \frac{84071485314003414767227043630184341430663}{235140860752274110219225576693744482460096} a^{2} - \frac{6171763310745385951197396785408182436721}{29392607594034263777403197086718060307512} a + \frac{6645324036100409742430713986234684479531}{14696303797017131888701598543359030153756}$

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

Class group and class number

$C_{2}\times C_{140378}$, which has order $280756$ (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:  \( 67295442.84873295 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.3.837.1, 3.3.961.1, 6.0.316767703.1, 6.0.240295167.4, 9.9.541530783546813.1

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 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$31$31.6.4.1$x^{6} + 1085 x^{3} + 1660608$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.12.10.1$x^{12} + 69161 x^{6} + 2869530624$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$