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Properties

Label 18.0.46526487881...6816.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{27}\cdot 7^{9}\cdot 37^{14}$
Root discriminant $574.54$
Ramified primes $2, 3, 7, 37$
Class number $1395216576$ (GRH)
Class group $[3, 6, 6, 12, 1076556]$ (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![360860185926073, 210401935775676, 81392406986196, 16826397242406, 2468568430875, 142282696212, 23205936379, 6463629012, 2070194571, 177967040, 6831186, -2117214, 279460, 57276, 7551, -518, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 33*x^16 - 518*x^15 + 7551*x^14 + 57276*x^13 + 279460*x^12 - 2117214*x^11 + 6831186*x^10 + 177967040*x^9 + 2070194571*x^8 + 6463629012*x^7 + 23205936379*x^6 + 142282696212*x^5 + 2468568430875*x^4 + 16826397242406*x^3 + 81392406986196*x^2 + 210401935775676*x + 360860185926073)
 
gp: K = bnfinit(x^18 - 33*x^16 - 518*x^15 + 7551*x^14 + 57276*x^13 + 279460*x^12 - 2117214*x^11 + 6831186*x^10 + 177967040*x^9 + 2070194571*x^8 + 6463629012*x^7 + 23205936379*x^6 + 142282696212*x^5 + 2468568430875*x^4 + 16826397242406*x^3 + 81392406986196*x^2 + 210401935775676*x + 360860185926073, 1)
 

Normalized defining polynomial

\( x^{18} - 33 x^{16} - 518 x^{15} + 7551 x^{14} + 57276 x^{13} + 279460 x^{12} - 2117214 x^{11} + 6831186 x^{10} + 177967040 x^{9} + 2070194571 x^{8} + 6463629012 x^{7} + 23205936379 x^{6} + 142282696212 x^{5} + 2468568430875 x^{4} + 16826397242406 x^{3} + 81392406986196 x^{2} + 210401935775676 x + 360860185926073 \)

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:  \(-46526487881757546143902493465169189274448943906816=-\,2^{24}\cdot 3^{27}\cdot 7^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $574.54$
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:  $2, 3, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{222} a^{12} + \frac{5}{74} a^{10} - \frac{15}{74} a^{8} - \frac{1}{2} a^{7} - \frac{38}{111} a^{6} - \frac{1}{2} a^{5} + \frac{4}{37} a^{4} - \frac{1}{3} a^{3} - \frac{29}{74} a^{2} - \frac{1}{2} a + \frac{5}{111}$, $\frac{1}{222} a^{13} + \frac{5}{74} a^{11} - \frac{4}{111} a^{9} + \frac{35}{222} a^{7} - \frac{1}{6} a^{6} + \frac{4}{37} a^{5} + \frac{1}{6} a^{4} + \frac{4}{37} a^{3} + \frac{5}{111} a + \frac{1}{6}$, $\frac{1}{222} a^{14} - \frac{11}{222} a^{10} + \frac{22}{111} a^{8} + \frac{1}{3} a^{7} + \frac{9}{37} a^{6} - \frac{1}{3} a^{5} + \frac{18}{37} a^{4} - \frac{17}{222} a^{2} - \frac{1}{3} a + \frac{12}{37}$, $\frac{1}{666} a^{15} + \frac{1}{666} a^{14} + \frac{1}{666} a^{13} + \frac{1}{666} a^{12} - \frac{11}{222} a^{11} + \frac{41}{666} a^{10} - \frac{1}{666} a^{9} - \frac{1}{666} a^{8} - \frac{53}{111} a^{7} + \frac{7}{37} a^{6} + \frac{29}{333} a^{5} + \frac{140}{333} a^{4} - \frac{67}{666} a^{3} + \frac{229}{666} a^{2} + \frac{89}{222} a - \frac{29}{666}$, $\frac{1}{666} a^{16} - \frac{1}{666} a^{12} - \frac{1}{18} a^{11} + \frac{1}{74} a^{10} + \frac{307}{666} a^{8} - \frac{1}{6} a^{7} - \frac{245}{666} a^{6} - \frac{1}{6} a^{5} - \frac{221}{666} a^{4} + \frac{5}{18} a^{3} - \frac{140}{333} a^{2} - \frac{1}{9} a - \frac{307}{666}$, $\frac{1}{3892923890216311501031887144122493326344329379653046277958602697914591126} a^{17} + \frac{39582258398037556783992362637906454345753947018352065579539653179631}{3892923890216311501031887144122493326344329379653046277958602697914591126} a^{16} + \frac{925740361629855920878512868200313759202472233516924826448180693719820}{1946461945108155750515943572061246663172164689826523138979301348957295563} a^{15} - \frac{1018989406824477966449190901997209316279433257294919802402121743616723}{1946461945108155750515943572061246663172164689826523138979301348957295563} a^{14} + \frac{936999131177831130015813588340326594622234810937716141575292228279610}{1946461945108155750515943572061246663172164689826523138979301348957295563} a^{13} - \frac{1220342595607532174565940197408138715196124668377023135600182914735723}{648820648369385250171981190687082221057388229942174379659767116319098521} a^{12} - \frac{189116681758850116332733728128829353399100861561366937458175255458153745}{3892923890216311501031887144122493326344329379653046277958602697914591126} a^{11} - \frac{77656440504270362518198474886582071828239434695145542927660597629133325}{3892923890216311501031887144122493326344329379653046277958602697914591126} a^{10} + \frac{175773596983922683289443116273090086685756570057911697921067155635501525}{3892923890216311501031887144122493326344329379653046277958602697914591126} a^{9} - \frac{724505688706303890106891152403134670919113085965573836650832553384538978}{1946461945108155750515943572061246663172164689826523138979301348957295563} a^{8} - \frac{1209264708359586742838312283947422101869864472833121336258019191000755183}{3892923890216311501031887144122493326344329379653046277958602697914591126} a^{7} - \frac{425024829277482593333235721426087065596229112555320301251550032894598841}{3892923890216311501031887144122493326344329379653046277958602697914591126} a^{6} - \frac{176555842845200530775945368382870684969052063000770240872267446748322023}{648820648369385250171981190687082221057388229942174379659767116319098521} a^{5} + \frac{985620375266100191472270567858070937263165766649208408744443876199149749}{3892923890216311501031887144122493326344329379653046277958602697914591126} a^{4} + \frac{844574634471215142905149784230449595385862063431539318119576873709961909}{3892923890216311501031887144122493326344329379653046277958602697914591126} a^{3} - \frac{455649341105046496447243434118644176013691029210947648353834823180472056}{1946461945108155750515943572061246663172164689826523138979301348957295563} a^{2} - \frac{137112273159875329383698599923616648499315053709971490933659730684138765}{432547098912923500114654127124721480704925486628116253106511410879399014} a - \frac{99110130743443620945342906746016015473488888028672366030288925017307265}{1946461945108155750515943572061246663172164689826523138979301348957295563}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}\times C_{12}\times C_{1076556}$, which has order $1395216576$ (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:  \( 118546543.87559307 \) (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

\(\Q(\sqrt{-21}) \), 3.3.148.1, 3.3.110889.1, 6.0.3245647104.8, 6.0.809789763859776.2, 9.9.3228844269788073792.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$