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Properties

Label 18.0.29888055988...0208.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 17^{8}$
Root discriminant $20.15$
Ramified primes $2, 3, 17$
Class number $1$
Class group Trivial
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -384, -768, 1504, 480, -2184, 644, 1344, -768, -660, 546, 132, -141, -9, 0, 11, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 11*x^15 - 9*x^13 - 141*x^12 + 132*x^11 + 546*x^10 - 660*x^9 - 768*x^8 + 1344*x^7 + 644*x^6 - 2184*x^5 + 480*x^4 + 1504*x^3 - 768*x^2 - 384*x + 256)
 
gp: K = bnfinit(x^18 - 3*x^17 + 11*x^15 - 9*x^13 - 141*x^12 + 132*x^11 + 546*x^10 - 660*x^9 - 768*x^8 + 1344*x^7 + 644*x^6 - 2184*x^5 + 480*x^4 + 1504*x^3 - 768*x^2 - 384*x + 256, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 11 x^{15} - 9 x^{13} - 141 x^{12} + 132 x^{11} + 546 x^{10} - 660 x^{9} - 768 x^{8} + 1344 x^{7} + 644 x^{6} - 2184 x^{5} + 480 x^{4} + 1504 x^{3} - 768 x^{2} - 384 x + 256 \)

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:  \(-298880559887628015710208=-\,2^{12}\cdot 3^{21}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.15$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{3}{8} a^{8} - \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{8} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{3}{16} a^{9} - \frac{7}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{16} a^{13} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} - \frac{5}{32} a^{10} + \frac{1}{32} a^{9} - \frac{1}{16} a^{8} - \frac{5}{16} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{1088} a^{16} + \frac{3}{1088} a^{15} - \frac{15}{544} a^{14} - \frac{41}{1088} a^{13} + \frac{53}{544} a^{12} - \frac{125}{1088} a^{11} + \frac{5}{1088} a^{10} - \frac{47}{544} a^{9} + \frac{143}{544} a^{8} - \frac{13}{34} a^{7} - \frac{2}{17} a^{6} + \frac{9}{68} a^{5} - \frac{127}{272} a^{4} - \frac{11}{68} a^{3} + \frac{13}{34} a^{2} + \frac{15}{34} a - \frac{7}{17}$, $\frac{1}{57246441087116416} a^{17} - \frac{23178915969143}{57246441087116416} a^{16} - \frac{94435437230279}{7155805135889552} a^{15} + \frac{516198049146591}{57246441087116416} a^{14} + \frac{356776702043451}{14311610271779104} a^{13} - \frac{3432938820325109}{57246441087116416} a^{12} - \frac{5221597342281}{1396254660661376} a^{11} - \frac{270925137252949}{14311610271779104} a^{10} + \frac{5472664004737407}{28623220543558208} a^{9} - \frac{2333542274390157}{14311610271779104} a^{8} + \frac{43529377002051}{7155805135889552} a^{7} + \frac{410638072021663}{894475641986194} a^{6} + \frac{3720790976695777}{14311610271779104} a^{5} - \frac{3114549342376675}{7155805135889552} a^{4} - \frac{1220473302528737}{3577902567944776} a^{3} + \frac{467978712869867}{1788951283972388} a^{2} - \frac{22183758907023}{447237820993097} a - \frac{174691884413845}{447237820993097}$

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:  \( -\frac{33747226629}{140861612608} a^{17} + \frac{35868966671}{70430806304} a^{16} + \frac{62716597629}{140861612608} a^{15} - \frac{315409515073}{140861612608} a^{14} - \frac{277586722103}{140861612608} a^{13} + \frac{59464192267}{140861612608} a^{12} + \frac{58754710115}{1717824544} a^{11} - \frac{233871670305}{140861612608} a^{10} - \frac{4656164306373}{35215403152} a^{9} + \frac{2929024477819}{70430806304} a^{8} + \frac{3877624370801}{17607701576} a^{7} - \frac{278040785589}{2200962697} a^{6} - \frac{9343212247085}{35215403152} a^{5} + \frac{10102093902617}{35215403152} a^{4} + \frac{1215313124323}{8803850788} a^{3} - \frac{1029433880067}{4401925394} a^{2} - \frac{57854647844}{2200962697} a + \frac{148560363351}{2200962697} \) (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:  \( 88502.84742509261 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.105212405952.1

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

Sibling fields

Degree 9 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.1$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$$[\ ]_{3}^{2}$
3Data not computed
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{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}$