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Properties

Label 18.0.36019449641...1312.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 7^{6}\cdot 11^{14}$
Root discriminant $33.96$
Ramified primes $2, 3, 7, 11$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![93919, 19033, 36967, -22214, 9437, -24345, 13003, -5025, 8916, -6116, 2771, -819, -277, 447, -189, 21, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 + 21*x^15 - 189*x^14 + 447*x^13 - 277*x^12 - 819*x^11 + 2771*x^10 - 6116*x^9 + 8916*x^8 - 5025*x^7 + 13003*x^6 - 24345*x^5 + 9437*x^4 - 22214*x^3 + 36967*x^2 + 19033*x + 93919)
 
gp: K = bnfinit(x^18 - 6*x^17 + 15*x^16 + 21*x^15 - 189*x^14 + 447*x^13 - 277*x^12 - 819*x^11 + 2771*x^10 - 6116*x^9 + 8916*x^8 - 5025*x^7 + 13003*x^6 - 24345*x^5 + 9437*x^4 - 22214*x^3 + 36967*x^2 + 19033*x + 93919, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 15 x^{16} + 21 x^{15} - 189 x^{14} + 447 x^{13} - 277 x^{12} - 819 x^{11} + 2771 x^{10} - 6116 x^{9} + 8916 x^{8} - 5025 x^{7} + 13003 x^{6} - 24345 x^{5} + 9437 x^{4} - 22214 x^{3} + 36967 x^{2} + 19033 x + 93919 \)

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:  \(-3601944964195995655415181312=-\,2^{12}\cdot 3^{9}\cdot 7^{6}\cdot 11^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.96$
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:  $2, 3, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{11} + \frac{1}{11} a^{10} - \frac{5}{11} a^{9} + \frac{2}{11} a^{8} - \frac{4}{11} a^{7} - \frac{2}{11} a^{5} - \frac{2}{11} a^{3} + \frac{3}{11} a^{2} + \frac{2}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{13} - \frac{4}{11} a^{10} - \frac{3}{11} a^{9} - \frac{2}{11} a^{8} - \frac{4}{11} a^{7} - \frac{2}{11} a^{6} - \frac{2}{11} a^{5} - \frac{2}{11} a^{4} + \frac{1}{11} a^{3} + \frac{5}{11} a^{2} + \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{14} - \frac{4}{11} a^{11} - \frac{3}{11} a^{10} - \frac{2}{11} a^{9} - \frac{4}{11} a^{8} - \frac{2}{11} a^{7} - \frac{2}{11} a^{6} - \frac{2}{11} a^{5} + \frac{1}{11} a^{4} + \frac{5}{11} a^{3} + \frac{1}{11} a^{2} - \frac{1}{11} a$, $\frac{1}{8452235} a^{15} - \frac{1}{1690447} a^{14} + \frac{4529}{768385} a^{13} + \frac{271694}{8452235} a^{12} + \frac{541571}{8452235} a^{11} - \frac{29063}{153677} a^{10} - \frac{3229287}{8452235} a^{9} + \frac{214778}{8452235} a^{8} - \frac{3857079}{8452235} a^{7} + \frac{873474}{8452235} a^{6} + \frac{700671}{8452235} a^{5} - \frac{344294}{768385} a^{4} - \frac{1788081}{8452235} a^{3} - \frac{3952543}{8452235} a^{2} - \frac{681966}{8452235} a + \frac{1098264}{8452235}$, $\frac{1}{8452235} a^{16} + \frac{49794}{8452235} a^{14} - \frac{247596}{8452235} a^{13} + \frac{363271}{8452235} a^{12} + \frac{48112}{153677} a^{11} - \frac{1232607}{8452235} a^{10} + \frac{2509583}{8452235} a^{9} + \frac{4132276}{8452235} a^{8} - \frac{739066}{8452235} a^{7} - \frac{1847424}{8452235} a^{6} - \frac{4125804}{8452235} a^{5} - \frac{2283011}{8452235} a^{4} - \frac{2135558}{8452235} a^{3} - \frac{3540211}{8452235} a^{2} + \frac{2298744}{8452235} a - \frac{131152}{1690447}$, $\frac{1}{1046735485851171025208334695} a^{17} - \frac{43534610367253263688}{1046735485851171025208334695} a^{16} + \frac{3437351326160510211}{95157771441015547746212245} a^{15} - \frac{46061730514308279255912538}{1046735485851171025208334695} a^{14} + \frac{21687194489138196413776547}{1046735485851171025208334695} a^{13} + \frac{554861237242870338688154}{209347097170234205041666939} a^{12} + \frac{66395213229807189942299601}{209347097170234205041666939} a^{11} - \frac{155425114794227622267975551}{1046735485851171025208334695} a^{10} + \frac{393371120689617769436867938}{1046735485851171025208334695} a^{9} + \frac{160359477849722463575237857}{1046735485851171025208334695} a^{8} - \frac{45822911362529822797199984}{95157771441015547746212245} a^{7} + \frac{462789820655957067660129081}{1046735485851171025208334695} a^{6} + \frac{320022018383919957604500713}{1046735485851171025208334695} a^{5} + \frac{355746693814824392440985857}{1046735485851171025208334695} a^{4} + \frac{164010069123564282228823181}{1046735485851171025208334695} a^{3} + \frac{154715990948904566028085456}{1046735485851171025208334695} a^{2} - \frac{341201490055334311248977724}{1046735485851171025208334695} a - \frac{15623469900152188004520207}{1046735485851171025208334695}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{106653444}{5300511057865} a^{17} + \frac{508448312}{5300511057865} a^{16} - \frac{952207734}{5300511057865} a^{15} - \frac{3587155718}{5300511057865} a^{14} + \frac{16497731652}{5300511057865} a^{13} - \frac{5666231459}{1060102211573} a^{12} - \frac{1516513718}{1060102211573} a^{11} + \frac{95654295484}{5300511057865} a^{10} - \frac{217008678992}{5300511057865} a^{9} + \frac{469510631727}{5300511057865} a^{8} - \frac{428908118444}{5300511057865} a^{7} - \frac{161193888474}{5300511057865} a^{6} - \frac{465211707542}{5300511057865} a^{5} + \frac{29518691767}{5300511057865} a^{4} + \frac{1120656169666}{5300511057865} a^{3} + \frac{1009733201106}{5300511057865} a^{2} - \frac{77774608534}{5300511057865} a + \frac{2815400394673}{5300511057865} \) (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:  \( 1305071.139073367 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.44.1, 6.0.19370043.1, 6.0.52272.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
$11$11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.12.10.2$x^{12} + 143 x^{6} + 5929$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$