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Properties

Label 18.0.84728860944...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{25}\cdot 5^{12}$
Root discriminant $21.35$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![64, -192, 48, 528, -276, -1740, 3789, -3207, 1155, -22, 99, -621, 864, -693, 387, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 387*x^14 - 693*x^13 + 864*x^12 - 621*x^11 + 99*x^10 - 22*x^9 + 1155*x^8 - 3207*x^7 + 3789*x^6 - 1740*x^5 - 276*x^4 + 528*x^3 + 48*x^2 - 192*x + 64)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 387*x^14 - 693*x^13 + 864*x^12 - 621*x^11 + 99*x^10 - 22*x^9 + 1155*x^8 - 3207*x^7 + 3789*x^6 - 1740*x^5 - 276*x^4 + 528*x^3 + 48*x^2 - 192*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 387 x^{14} - 693 x^{13} + 864 x^{12} - 621 x^{11} + 99 x^{10} - 22 x^{9} + 1155 x^{8} - 3207 x^{7} + 3789 x^{6} - 1740 x^{5} - 276 x^{4} + 528 x^{3} + 48 x^{2} - 192 x + 64 \)

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:  \(-847288609443000000000000=-\,2^{12}\cdot 3^{25}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.35$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{12} + \frac{3}{16} a^{9} - \frac{3}{16} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{3}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{3}{16} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{16} a^{4} - \frac{3}{8} a^{3}$, $\frac{1}{64} a^{14} + \frac{1}{64} a^{13} - \frac{1}{32} a^{12} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{5}{32} a^{9} - \frac{11}{64} a^{8} + \frac{11}{64} a^{7} + \frac{1}{8} a^{6} - \frac{9}{64} a^{5} - \frac{11}{64} a^{4} + \frac{1}{4} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{64} a^{15} + \frac{1}{64} a^{13} + \frac{1}{64} a^{12} + \frac{3}{64} a^{10} - \frac{1}{64} a^{9} - \frac{5}{32} a^{8} - \frac{15}{64} a^{7} + \frac{7}{64} a^{6} - \frac{9}{32} a^{5} + \frac{7}{64} a^{4} - \frac{3}{8} a^{2} - \frac{1}{4}$, $\frac{1}{22449664} a^{16} - \frac{1}{2806208} a^{15} + \frac{133095}{22449664} a^{14} + \frac{471579}{22449664} a^{13} - \frac{38053}{5612416} a^{12} - \frac{1008307}{22449664} a^{11} - \frac{225499}{22449664} a^{10} - \frac{770327}{11224832} a^{9} - \frac{607457}{22449664} a^{8} + \frac{3562509}{22449664} a^{7} - \frac{2329721}{11224832} a^{6} - \frac{4786175}{22449664} a^{5} + \frac{3672677}{11224832} a^{4} + \frac{57837}{350776} a^{3} + \frac{183433}{701552} a^{2} + \frac{195039}{1403104} a - \frac{194609}{701552}$, $\frac{1}{23863992832} a^{17} + \frac{523}{23863992832} a^{16} + \frac{38012655}{23863992832} a^{15} + \frac{19811031}{2982999104} a^{14} - \frac{695085083}{23863992832} a^{13} + \frac{509575457}{23863992832} a^{12} + \frac{38584969}{5965998208} a^{11} - \frac{722159911}{23863992832} a^{10} - \frac{4661095035}{23863992832} a^{9} + \frac{2409058681}{11931996416} a^{8} - \frac{3903928147}{23863992832} a^{7} - \frac{5004537077}{23863992832} a^{6} + \frac{3091982693}{23863992832} a^{5} - \frac{2082407893}{11931996416} a^{4} + \frac{142985713}{372874888} a^{3} + \frac{592079317}{1491499552} a^{2} + \frac{12676283}{1491499552} a - \frac{235404543}{745749776}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{1086767}{73202432} a^{17} + \frac{18475039}{146404864} a^{16} - \frac{44473109}{73202432} a^{15} + \frac{297595855}{146404864} a^{14} - \frac{706335523}{146404864} a^{13} + \frac{599231399}{73202432} a^{12} - \frac{1382797315}{146404864} a^{11} + \frac{829502047}{146404864} a^{10} + \frac{1846651}{36601216} a^{9} + \frac{167819195}{146404864} a^{8} - \frac{2451378833}{146404864} a^{7} + \frac{1454539301}{36601216} a^{6} - \frac{5715796301}{146404864} a^{5} + \frac{852975379}{73202432} a^{4} + \frac{11971765}{2287576} a^{3} - \frac{8405983}{2287576} a^{2} - \frac{26901059}{9150304} a + \frac{10198281}{4575152} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 460641.7441064912 \) (assuming GRH)
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

\(\Q(\sqrt{-3}) \), 3.1.675.1 x3, 3.3.2700.1, 6.0.21870000.1, 6.0.1366875.1, 9.3.531441000000.1 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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
3Data not computed
$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}$