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Properties

Label 18.0.18402047963...4448.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{44}\cdot 3^{21}$
Root discriminant $19.61$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $S_3\wr C_2$ (as 18T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 45, -138, 309, -471, 523, -318, 162, -184, 126, 18, -17, -45, 21, 10, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 3*x^16 + 10*x^15 + 21*x^14 - 45*x^13 - 17*x^12 + 18*x^11 + 126*x^10 - 184*x^9 + 162*x^8 - 318*x^7 + 523*x^6 - 471*x^5 + 309*x^4 - 138*x^3 + 45*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 3*x^16 + 10*x^15 + 21*x^14 - 45*x^13 - 17*x^12 + 18*x^11 + 126*x^10 - 184*x^9 + 162*x^8 - 318*x^7 + 523*x^6 - 471*x^5 + 309*x^4 - 138*x^3 + 45*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 3 x^{16} + 10 x^{15} + 21 x^{14} - 45 x^{13} - 17 x^{12} + 18 x^{11} + 126 x^{10} - 184 x^{9} + 162 x^{8} - 318 x^{7} + 523 x^{6} - 471 x^{5} + 309 x^{4} - 138 x^{3} + 45 x^{2} - 9 x + 1 \)

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:  \(-184020479637478805864448=-\,2^{44}\cdot 3^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.61$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{76} a^{16} + \frac{3}{76} a^{15} + \frac{3}{38} a^{14} - \frac{1}{4} a^{13} - \frac{7}{38} a^{12} - \frac{15}{76} a^{11} - \frac{1}{4} a^{10} - \frac{9}{38} a^{9} - \frac{1}{76} a^{8} - \frac{9}{76} a^{7} + \frac{11}{38} a^{6} - \frac{29}{76} a^{5} - \frac{5}{19} a^{4} - \frac{7}{76} a^{3} - \frac{9}{76} a^{2} - \frac{17}{38} a - \frac{1}{38}$, $\frac{1}{1468477244} a^{17} + \frac{63909}{77288276} a^{16} + \frac{106551275}{1468477244} a^{15} + \frac{26300437}{734238622} a^{14} + \frac{44414203}{734238622} a^{13} - \frac{99769889}{734238622} a^{12} - \frac{156085677}{1468477244} a^{11} + \frac{71739935}{1468477244} a^{10} - \frac{167498815}{734238622} a^{9} - \frac{69122785}{1468477244} a^{8} - \frac{1075161}{2571764} a^{7} + \frac{3876049}{19322069} a^{6} - \frac{91323849}{367119311} a^{5} - \frac{24560256}{367119311} a^{4} + \frac{227740073}{1468477244} a^{3} - \frac{12637173}{1468477244} a^{2} + \frac{596592443}{1468477244} a - \frac{166570538}{367119311}$

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:  \( \frac{1953902}{1016951} a^{17} - \frac{21268381}{4067804} a^{16} - \frac{7531539}{1016951} a^{15} + \frac{35788341}{2033902} a^{14} + \frac{46870468}{1016951} a^{13} - \frac{305344553}{4067804} a^{12} - \frac{59371976}{1016951} a^{11} + \frac{49218561}{2033902} a^{10} + \frac{260358307}{1016951} a^{9} - \frac{287585332}{1016951} a^{8} + \frac{370157}{1781} a^{7} - \frac{1082759989}{2033902} a^{6} + \frac{857236124}{1016951} a^{5} - \frac{2537909865}{4067804} a^{4} + \frac{358106272}{1016951} a^{3} - \frac{258366549}{2033902} a^{2} + \frac{31942635}{1016951} a - \frac{12250665}{4067804} \) (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:  \( 110226.745818 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 18T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $S_3\wr C_2$
Character table for $S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 9.3.82556485632.2

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

Sibling fields

Degree 6 siblings: 6.0.442368.1, 6.4.35831808.1
Degree 9 sibling: 9.3.82556485632.2
Degree 12 siblings: 12.6.41085390865563648.2, 12.2.41085390865563648.27, 12.0.15850845241344.1, 12.0.18786186952704.1, 12.0.1283918464548864.8, 12.0.169075682574336.1
Degree 18 siblings: Deg 18, Deg 18

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.22.1$x^{8} + 8 x^{5} + 6 x^{4} + 16 x^{3} + 8 x^{2} + 12$$4$$2$$22$$D_4$$[3, 4]^{2}$
2.8.22.1$x^{8} + 8 x^{5} + 6 x^{4} + 16 x^{3} + 8 x^{2} + 12$$4$$2$$22$$D_4$$[3, 4]^{2}$
3Data not computed