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Properties

Label 18.0.39535158872...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{6}\cdot 11^{12}$
Root discriminant $23.25$
Ramified primes $2, 3, 5, 11$
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![169, -195, 561, -73, 470, 736, -155, 1104, 13, 378, 254, -234, 243, -195, 116, -50, 20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 20*x^16 - 50*x^15 + 116*x^14 - 195*x^13 + 243*x^12 - 234*x^11 + 254*x^10 + 378*x^9 + 13*x^8 + 1104*x^7 - 155*x^6 + 736*x^5 + 470*x^4 - 73*x^3 + 561*x^2 - 195*x + 169)
 
gp: K = bnfinit(x^18 - 4*x^17 + 20*x^16 - 50*x^15 + 116*x^14 - 195*x^13 + 243*x^12 - 234*x^11 + 254*x^10 + 378*x^9 + 13*x^8 + 1104*x^7 - 155*x^6 + 736*x^5 + 470*x^4 - 73*x^3 + 561*x^2 - 195*x + 169, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 20 x^{16} - 50 x^{15} + 116 x^{14} - 195 x^{13} + 243 x^{12} - 234 x^{11} + 254 x^{10} + 378 x^{9} + 13 x^{8} + 1104 x^{7} - 155 x^{6} + 736 x^{5} + 470 x^{4} - 73 x^{3} + 561 x^{2} - 195 x + 169 \)

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:  \(-3953515887295964352000000=-\,2^{12}\cdot 3^{9}\cdot 5^{6}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.25$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{3} a^{9} - \frac{1}{9} a^{8} - \frac{2}{9} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} - \frac{1}{3} a^{10} - \frac{1}{9} a^{9} - \frac{2}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} + \frac{2}{9} a^{2} - \frac{1}{9} a$, $\frac{1}{351} a^{15} + \frac{4}{117} a^{14} + \frac{2}{39} a^{13} + \frac{1}{117} a^{12} - \frac{1}{27} a^{11} - \frac{10}{351} a^{10} + \frac{70}{351} a^{9} - \frac{8}{351} a^{8} + \frac{131}{351} a^{7} - \frac{121}{351} a^{6} + \frac{67}{351} a^{5} - \frac{116}{351} a^{4} + \frac{19}{351} a^{3} - \frac{103}{351} a^{2} + \frac{76}{351} a - \frac{2}{27}$, $\frac{1}{351} a^{16} - \frac{1}{39} a^{14} - \frac{2}{39} a^{13} - \frac{49}{351} a^{12} - \frac{127}{351} a^{11} - \frac{44}{351} a^{10} - \frac{29}{351} a^{9} - \frac{85}{351} a^{8} - \frac{94}{351} a^{7} - \frac{2}{351} a^{6} + \frac{16}{351} a^{5} - \frac{71}{351} a^{4} + \frac{98}{351} a^{3} - \frac{92}{351} a^{2} + \frac{154}{351} a$, $\frac{1}{65786034482454482235} a^{17} - \frac{1082102185838425}{4385735632163632149} a^{16} + \frac{180637006448038}{1012092838191607419} a^{15} + \frac{19310824521869453}{4385735632163632149} a^{14} + \frac{1789000199280164171}{65786034482454482235} a^{13} + \frac{9731161743394789844}{65786034482454482235} a^{12} - \frac{20777076701764291486}{65786034482454482235} a^{11} - \frac{19305450653936274538}{65786034482454482235} a^{10} - \frac{68760625548648716}{5060464190958037095} a^{9} - \frac{29766792291156504674}{65786034482454482235} a^{8} - \frac{14586341020625731648}{65786034482454482235} a^{7} + \frac{27614559516224278037}{65786034482454482235} a^{6} + \frac{498598246281285886}{21928678160818160745} a^{5} - \frac{29503510090710534692}{65786034482454482235} a^{4} + \frac{478135927338074021}{2436519795646462305} a^{3} + \frac{3350342790337724959}{13157206896490896447} a^{2} + \frac{1682589663816585092}{5060464190958037095} a - \frac{97079316219041239}{389266476227541315}$

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:  \( -\frac{154267751312}{23262544720935} a^{17} + \frac{917436041}{57438382027} a^{16} - \frac{33127787051}{357885303399} a^{15} + \frac{203054590415}{1550836314729} a^{14} - \frac{6753985485622}{23262544720935} a^{13} + \frac{4847567351357}{23262544720935} a^{12} + \frac{3123406107062}{23262544720935} a^{11} - \frac{10358175069589}{23262544720935} a^{10} + \frac{156706696342}{1789426516995} a^{9} - \frac{106270745838347}{23262544720935} a^{8} - \frac{105958541497684}{23262544720935} a^{7} - \frac{196630797503584}{23262544720935} a^{6} - \frac{71725541767937}{7754181573645} a^{5} - \frac{117431755163666}{23262544720935} a^{4} - \frac{55883691195643}{7754181573645} a^{3} - \frac{16459544775923}{4652508944187} a^{2} - \frac{3262946753134}{1789426516995} a - \frac{159045634627}{137648193615} \) (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:  \( 966489.2545981706 \)
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

\(\Q(\sqrt{-3}) \), 3.1.1452.1 x3, 3.1.1815.1, 6.0.9882675.1, 6.0.6324912.1, 9.1.1147971528000.1 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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$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}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$