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Properties

Label 18.0.36542633964...0208.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 31^{8}$
Root discriminant $26.31$
Ramified primes $2, 3, 31$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
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![1, 45, 492, -869, -501, 183, -409, 1386, 2373, 61, -1344, -882, -40, 345, 174, -14, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 21*x^16 - 14*x^15 + 174*x^14 + 345*x^13 - 40*x^12 - 882*x^11 - 1344*x^10 + 61*x^9 + 2373*x^8 + 1386*x^7 - 409*x^6 + 183*x^5 - 501*x^4 - 869*x^3 + 492*x^2 + 45*x + 1)
 
gp: K = bnfinit(x^18 - 21*x^16 - 14*x^15 + 174*x^14 + 345*x^13 - 40*x^12 - 882*x^11 - 1344*x^10 + 61*x^9 + 2373*x^8 + 1386*x^7 - 409*x^6 + 183*x^5 - 501*x^4 - 869*x^3 + 492*x^2 + 45*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 21 x^{16} - 14 x^{15} + 174 x^{14} + 345 x^{13} - 40 x^{12} - 882 x^{11} - 1344 x^{10} + 61 x^{9} + 2373 x^{8} + 1386 x^{7} - 409 x^{6} + 183 x^{5} - 501 x^{4} - 869 x^{3} + 492 x^{2} + 45 x + 1 \)

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:  \(-36542633964773200992350208=-\,2^{12}\cdot 3^{21}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.31$
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:  $2, 3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{9} a^{7} + \frac{2}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{2}{9} a^{8} + \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{153} a^{15} + \frac{2}{51} a^{13} + \frac{8}{153} a^{12} - \frac{8}{17} a^{11} - \frac{7}{51} a^{10} - \frac{23}{153} a^{9} + \frac{3}{17} a^{8} + \frac{7}{51} a^{7} - \frac{22}{153} a^{6} + \frac{22}{51} a^{5} - \frac{8}{17} a^{4} + \frac{73}{153} a^{3} - \frac{1}{51} a^{2} + \frac{13}{51} a + \frac{44}{153}$, $\frac{1}{1071} a^{16} + \frac{1}{357} a^{15} + \frac{40}{1071} a^{14} - \frac{25}{1071} a^{13} - \frac{82}{1071} a^{12} + \frac{19}{119} a^{11} + \frac{220}{1071} a^{10} + \frac{20}{357} a^{9} - \frac{1}{9} a^{8} - \frac{214}{1071} a^{7} + \frac{10}{63} a^{6} + \frac{110}{357} a^{5} - \frac{449}{1071} a^{4} + \frac{89}{357} a^{3} + \frac{472}{1071} a^{2} - \frac{451}{1071} a + \frac{506}{1071}$, $\frac{1}{496533586393436928903} a^{17} - \frac{91574474731965052}{496533586393436928903} a^{16} + \frac{53761770160160918}{55170398488159658767} a^{15} + \frac{11188383073842128531}{496533586393436928903} a^{14} - \frac{6884808616180410988}{165511195464478976301} a^{13} - \frac{9298580937082505006}{70933369484776704129} a^{12} + \frac{21645730134903978511}{496533586393436928903} a^{11} - \frac{91048843869654410842}{496533586393436928903} a^{10} - \frac{4123778491035553927}{55170398488159658767} a^{9} - \frac{86794022380578516532}{496533586393436928903} a^{8} + \frac{1326180074620513852}{165511195464478976301} a^{7} + \frac{54258577510744709107}{496533586393436928903} a^{6} - \frac{237458044063937822789}{496533586393436928903} a^{5} + \frac{6659953538050188473}{496533586393436928903} a^{4} - \frac{47390388526427067530}{165511195464478976301} a^{3} - \frac{142598582837903367550}{496533586393436928903} a^{2} + \frac{62007819558377270509}{165511195464478976301} a - \frac{175069863021267989792}{496533586393436928903}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{14581671259778}{84866699310949} a^{17} - \frac{819681447173}{84866699310949} a^{16} - \frac{919079377379711}{254600097932847} a^{15} - \frac{187023331501972}{84866699310949} a^{14} + \frac{2551669904042171}{84866699310949} a^{13} + \frac{14675657262096647}{254600097932847} a^{12} - \frac{888505556155999}{84866699310949} a^{11} - \frac{12891148647065375}{84866699310949} a^{10} - \frac{56739055705281419}{254600097932847} a^{9} + \frac{2085414722832387}{84866699310949} a^{8} + \frac{34813645285428115}{84866699310949} a^{7} + \frac{55308234610692770}{254600097932847} a^{6} - \frac{7313516928054618}{84866699310949} a^{5} + \frac{2656817975876652}{84866699310949} a^{4} - \frac{23130049884648812}{254600097932847} a^{3} - \frac{12399857109759963}{84866699310949} a^{2} + \frac{7926405531756733}{84866699310949} a + \frac{1171355136931613}{254600097932847} \) (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:  \( 336399.2656843654 \) (assuming GRH)
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, 9.3.1163370485952.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$31$31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$