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Properties

Label 18.0.18665884703...6032.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 7^{10}\cdot 31^{10}$
Root discriminant $54.61$
Ramified primes $2, 3, 7, 31$
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![126963, -130761, 139299, -236085, 321958, -344420, 307981, -252336, 193737, -124458, 61120, -21824, 5983, -1617, 486, -114, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 16*x^16 - 114*x^15 + 486*x^14 - 1617*x^13 + 5983*x^12 - 21824*x^11 + 61120*x^10 - 124458*x^9 + 193737*x^8 - 252336*x^7 + 307981*x^6 - 344420*x^5 + 321958*x^4 - 236085*x^3 + 139299*x^2 - 130761*x + 126963)
 
gp: K = bnfinit(x^18 - 2*x^17 + 16*x^16 - 114*x^15 + 486*x^14 - 1617*x^13 + 5983*x^12 - 21824*x^11 + 61120*x^10 - 124458*x^9 + 193737*x^8 - 252336*x^7 + 307981*x^6 - 344420*x^5 + 321958*x^4 - 236085*x^3 + 139299*x^2 - 130761*x + 126963, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 16 x^{16} - 114 x^{15} + 486 x^{14} - 1617 x^{13} + 5983 x^{12} - 21824 x^{11} + 61120 x^{10} - 124458 x^{9} + 193737 x^{8} - 252336 x^{7} + 307981 x^{6} - 344420 x^{5} + 321958 x^{4} - 236085 x^{3} + 139299 x^{2} - 130761 x + 126963 \)

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:  \(-18665884703684652969692527276032=-\,2^{12}\cdot 3^{9}\cdot 7^{10}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.61$
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:  $2, 3, 7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{27} a^{12} - \frac{1}{27} a^{10} + \frac{1}{9} a^{7} - \frac{4}{27} a^{6} + \frac{1}{9} a^{5} - \frac{2}{27} a^{4} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{81} a^{13} - \frac{1}{81} a^{12} - \frac{4}{81} a^{11} + \frac{1}{81} a^{10} + \frac{1}{27} a^{9} + \frac{1}{27} a^{8} + \frac{2}{81} a^{7} + \frac{7}{81} a^{6} + \frac{25}{81} a^{5} - \frac{4}{81} a^{4} + \frac{2}{9} a^{3} - \frac{2}{27} a^{2} - \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{81} a^{14} + \frac{1}{81} a^{12} - \frac{1}{27} a^{11} - \frac{2}{81} a^{10} - \frac{1}{27} a^{9} + \frac{5}{81} a^{8} + \frac{8}{81} a^{6} + \frac{13}{27} a^{5} + \frac{2}{81} a^{4} - \frac{5}{27} a^{3} - \frac{2}{27} a^{2} + \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{243} a^{15} - \frac{1}{243} a^{13} + \frac{2}{243} a^{12} - \frac{4}{81} a^{11} + \frac{1}{243} a^{10} - \frac{10}{243} a^{9} - \frac{11}{81} a^{8} + \frac{13}{243} a^{7} - \frac{14}{243} a^{6} + \frac{29}{81} a^{5} + \frac{113}{243} a^{4} - \frac{8}{81} a^{3} + \frac{31}{81} a^{2} + \frac{10}{27} a - \frac{7}{27}$, $\frac{1}{243} a^{16} - \frac{1}{243} a^{14} - \frac{1}{243} a^{13} + \frac{13}{243} a^{11} + \frac{5}{243} a^{10} + \frac{4}{81} a^{9} + \frac{4}{243} a^{8} + \frac{7}{243} a^{7} + \frac{10}{81} a^{6} - \frac{97}{243} a^{5} - \frac{19}{81} a^{4} + \frac{4}{81} a^{3} - \frac{13}{27} a - \frac{2}{9}$, $\frac{1}{2571602451955508115404013058005339} a^{17} + \frac{4204197884143228953088347575603}{2571602451955508115404013058005339} a^{16} + \frac{63227088378140099927865971549}{285733605772834235044890339778371} a^{15} + \frac{181926122044465867680695462332}{857200817318502705134671019335113} a^{14} - \frac{583951995824976034880181733538}{285733605772834235044890339778371} a^{13} - \frac{1988972614407540873688160116114}{285733605772834235044890339778371} a^{12} - \frac{93747790477254741092333362471007}{2571602451955508115404013058005339} a^{11} + \frac{2112523047689821830412823701763}{2571602451955508115404013058005339} a^{10} + \frac{4518340105287333870278958259069}{95244535257611411681630113259457} a^{9} + \frac{85460825248346647873755069941410}{857200817318502705134671019335113} a^{8} + \frac{563195251495385464519726915265}{31748178419203803893876704419819} a^{7} + \frac{26122770473127931178942699832068}{285733605772834235044890339778371} a^{6} - \frac{546382888189958008830175990167950}{2571602451955508115404013058005339} a^{5} + \frac{1206132362693769377132696880814784}{2571602451955508115404013058005339} a^{4} + \frac{89916646068663970930424120387707}{285733605772834235044890339778371} a^{3} + \frac{222337820040408128810269531960441}{857200817318502705134671019335113} a^{2} - \frac{123810489509855262187067540325706}{285733605772834235044890339778371} a + \frac{66149769715766565538191374831762}{285733605772834235044890339778371}$

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:  \( \frac{55919525118659528}{19669355733881560089393} a^{17} + \frac{188573743120357555}{19669355733881560089393} a^{16} + \frac{299007753330834347}{6556451911293853363131} a^{15} - \frac{695969295971892493}{6556451911293853363131} a^{14} + \frac{638181969623069746}{6556451911293853363131} a^{13} - \frac{1158051162136249313}{6556451911293853363131} a^{12} + \frac{73519359275405503898}{19669355733881560089393} a^{11} - \frac{142429975104130465031}{19669355733881560089393} a^{10} - \frac{108077241736940251718}{6556451911293853363131} a^{9} + \frac{463477561192634118941}{6556451911293853363131} a^{8} - \frac{328240354406823678412}{6556451911293853363131} a^{7} - \frac{863601145902108423946}{6556451911293853363131} a^{6} + \frac{8563004073621646538261}{19669355733881560089393} a^{5} - \frac{9845432429977538068544}{19669355733881560089393} a^{4} + \frac{1580971888917259944313}{2185483970431284454377} a^{3} - \frac{4942857528174734940073}{6556451911293853363131} a^{2} + \frac{1313153521012351323796}{2185483970431284454377} a - \frac{27904668054178861310}{2185483970431284454377} \) (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:  \( 4121877351.3792276 \) (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

\(\Q(\sqrt{-3}) \), 3.1.2604.1, 6.0.1271403.2, 6.0.20342448.2

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$31$31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.5.1$x^{6} - 31$$6$$1$$5$$C_6$$[\ ]_{6}$