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Properties

Label 18.0.38276746860...0992.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{30}\cdot 13^{14}\cdot 23^{9}$
Root discriminant $440.04$
Ramified primes $2, 3, 13, 23$
Class number $38283338772$ (GRH)
Class group $[3, 3, 3, 6, 236316906]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11595603968, 1876512768, 3840924672, 855399936, 667973280, 146257296, 73994292, 13613802, 4287204, 481455, 258483, -3414, -5418, 1572, 1512, 12, -42, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 42*x^16 + 12*x^15 + 1512*x^14 + 1572*x^13 - 5418*x^12 - 3414*x^11 + 258483*x^10 + 481455*x^9 + 4287204*x^8 + 13613802*x^7 + 73994292*x^6 + 146257296*x^5 + 667973280*x^4 + 855399936*x^3 + 3840924672*x^2 + 1876512768*x + 11595603968)
 
gp: K = bnfinit(x^18 - 3*x^17 - 42*x^16 + 12*x^15 + 1512*x^14 + 1572*x^13 - 5418*x^12 - 3414*x^11 + 258483*x^10 + 481455*x^9 + 4287204*x^8 + 13613802*x^7 + 73994292*x^6 + 146257296*x^5 + 667973280*x^4 + 855399936*x^3 + 3840924672*x^2 + 1876512768*x + 11595603968, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 42 x^{16} + 12 x^{15} + 1512 x^{14} + 1572 x^{13} - 5418 x^{12} - 3414 x^{11} + 258483 x^{10} + 481455 x^{9} + 4287204 x^{8} + 13613802 x^{7} + 73994292 x^{6} + 146257296 x^{5} + 667973280 x^{4} + 855399936 x^{3} + 3840924672 x^{2} + 1876512768 x + 11595603968 \)

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:  \(-382767468601979969221900924428079566404384980992=-\,2^{18}\cdot 3^{30}\cdot 13^{14}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $440.04$
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:  $2, 3, 13, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{10} + \frac{1}{32} a^{9} - \frac{1}{16} a^{7} - \frac{1}{32} a^{6} + \frac{7}{32} a^{5} - \frac{1}{8} a^{4} + \frac{5}{16} a^{3} - \frac{3}{8} a^{2}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{32} a^{7} - \frac{3}{32} a^{5} + \frac{3}{16} a^{4} + \frac{1}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{10} - \frac{1}{32} a^{9} + \frac{1}{64} a^{8} - \frac{1}{8} a^{7} + \frac{5}{64} a^{6} - \frac{1}{32} a^{5} - \frac{3}{32} a^{4} - \frac{5}{16} a^{3} - \frac{1}{2} a$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{12} - \frac{1}{128} a^{11} - \frac{1}{128} a^{10} - \frac{5}{128} a^{9} - \frac{1}{128} a^{8} - \frac{3}{128} a^{7} + \frac{9}{128} a^{6} + \frac{1}{32} a^{5} + \frac{5}{64} a^{4} - \frac{7}{32} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{512} a^{14} + \frac{1}{512} a^{13} + \frac{3}{512} a^{12} + \frac{5}{512} a^{11} - \frac{5}{512} a^{10} - \frac{23}{512} a^{9} + \frac{1}{512} a^{8} - \frac{21}{512} a^{7} - \frac{5}{128} a^{6} + \frac{3}{256} a^{5} - \frac{19}{128} a^{4} - \frac{3}{16} a^{3} - \frac{5}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{142336} a^{15} + \frac{69}{71168} a^{14} - \frac{41}{17792} a^{13} - \frac{85}{35584} a^{12} - \frac{3}{256} a^{11} - \frac{21}{17792} a^{10} + \frac{3731}{71168} a^{9} - \frac{13}{556} a^{8} + \frac{15211}{142336} a^{7} - \frac{5637}{71168} a^{6} + \frac{13173}{71168} a^{5} - \frac{5929}{35584} a^{4} - \frac{1595}{8896} a^{3} - \frac{2215}{4448} a^{2} + \frac{11}{278} a - \frac{117}{278}$, $\frac{1}{6919522304} a^{16} + \frac{3367}{1729880576} a^{15} - \frac{1995641}{3459761152} a^{14} - \frac{7363901}{3459761152} a^{13} - \frac{5390959}{3459761152} a^{12} + \frac{24613897}{3459761152} a^{11} + \frac{24230501}{1729880576} a^{10} - \frac{33280597}{3459761152} a^{9} - \frac{320181131}{6919522304} a^{8} - \frac{373840163}{3459761152} a^{7} - \frac{36398189}{3459761152} a^{6} - \frac{275630473}{1729880576} a^{5} - \frac{40765759}{216235072} a^{4} - \frac{22824441}{216235072} a^{3} + \frac{8774457}{54058768} a^{2} - \frac{2539179}{6757346} a + \frac{294742}{3378673}$, $\frac{1}{14931613688928035877592040623487827613795024371712} a^{17} + \frac{669401012943878820408676230530697382481}{14931613688928035877592040623487827613795024371712} a^{16} + \frac{24959349589859939482809980436632669390209589}{7465806844464017938796020311743913806897512185856} a^{15} - \frac{2395194814672969270208901380292828302398785}{26855420303827402657539641409150769089559396352} a^{14} - \frac{3704983711267723749740939623484028188263196817}{1866451711116004484699005077935978451724378046464} a^{13} + \frac{21642571026824254799666818933036101048126114401}{3732903422232008969398010155871956903448756092928} a^{12} + \frac{4249226574798087232511643676809469129788997907}{7465806844464017938796020311743913806897512185856} a^{11} + \frac{57339576769499330963498442831565840632781805969}{7465806844464017938796020311743913806897512185856} a^{10} - \frac{178540157021257024006836726934098212647646540581}{14931613688928035877592040623487827613795024371712} a^{9} - \frac{2809201425471520295552318099503554785960817013}{14931613688928035877592040623487827613795024371712} a^{8} + \frac{3796265656739109107945576030045835755075707181}{116653231944750280293687817370998653232773627904} a^{7} - \frac{168357454079526983234056366165511314010497923467}{7465806844464017938796020311743913806897512185856} a^{6} + \frac{907315783080058843706536075395824624150912376895}{3732903422232008969398010155871956903448756092928} a^{5} - \frac{53584797155547356337936741006243286465559147519}{233306463889500560587375634741997306465547255808} a^{4} - \frac{127175925988517232524384450194845200925435346851}{466612927779001121174751269483994612931094511616} a^{3} + \frac{50288834985902601437655838543237456190553708253}{116653231944750280293687817370998653232773627904} a^{2} + \frac{4460031807692844674414265318523391043444426383}{29163307986187570073421954342749663308193406976} a + \frac{511212525349747275109025768158627914794990693}{7290826996546892518355488585687415827048351744}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{6}\times C_{236316906}$, which has order $38283338772$ (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:  \( -1 \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3533133948.6916637 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.3.13689.1, 3.3.2808.1, 6.0.2279958568407.4, 6.0.95935140288.2, 9.9.460990789028310528.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$3$3.9.15.29$x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 12$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
3.9.15.29$x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 12$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.5.4$x^{6} + 26$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.4$x^{6} + 26$$6$$1$$5$$C_6$$[\ ]_{6}$
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$