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Properties

Label 18.0.52301766015...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 5^{13}$
Root discriminant $18.29$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![184, 264, 336, 108, 132, 48, 118, -54, 234, -83, 123, -27, 46, -6, 27, -1, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - x^15 + 27*x^14 - 6*x^13 + 46*x^12 - 27*x^11 + 123*x^10 - 83*x^9 + 234*x^8 - 54*x^7 + 118*x^6 + 48*x^5 + 132*x^4 + 108*x^3 + 336*x^2 + 264*x + 184)
 
gp: K = bnfinit(x^18 + 9*x^16 - x^15 + 27*x^14 - 6*x^13 + 46*x^12 - 27*x^11 + 123*x^10 - 83*x^9 + 234*x^8 - 54*x^7 + 118*x^6 + 48*x^5 + 132*x^4 + 108*x^3 + 336*x^2 + 264*x + 184, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - x^{15} + 27 x^{14} - 6 x^{13} + 46 x^{12} - 27 x^{11} + 123 x^{10} - 83 x^{9} + 234 x^{8} - 54 x^{7} + 118 x^{6} + 48 x^{5} + 132 x^{4} + 108 x^{3} + 336 x^{2} + 264 x + 184 \)

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:  \(-52301766015000000000000=-\,2^{12}\cdot 3^{21}\cdot 5^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} + \frac{1}{8} a^{13} + \frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{3}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{27882973565285463544} a^{17} + \frac{724742868896178051}{13941486782642731772} a^{16} + \frac{1751583392624608159}{27882973565285463544} a^{15} + \frac{148986323832846773}{27882973565285463544} a^{14} - \frac{3851882324736178437}{27882973565285463544} a^{13} - \frac{2965385047695421113}{13941486782642731772} a^{12} - \frac{1164071425588785552}{3485371695660682943} a^{11} + \frac{7491001354207525741}{27882973565285463544} a^{10} - \frac{5049652355235848063}{27882973565285463544} a^{9} - \frac{2547404381653158607}{27882973565285463544} a^{8} - \frac{1282156548887863349}{13941486782642731772} a^{7} + \frac{2493561284752079311}{6970743391321365886} a^{6} - \frac{3685701019241627349}{13941486782642731772} a^{5} - \frac{1485644687458797959}{6970743391321365886} a^{4} + \frac{2693626464215244367}{6970743391321365886} a^{3} + \frac{1205270388218778100}{3485371695660682943} a^{2} - \frac{824039586447887616}{3485371695660682943} a + \frac{48979700725401577}{151537899811334041}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (order $2$)
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:  \( 4522.356871832279 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.108.1, 6.0.4374000.1, 9.3.787320000.1

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

Sibling fields

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.10.2$x^{12} + 15 x^{6} + 100$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$