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Properties

Label 18.0.29530722454...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 5^{9}\cdot 37^{14}$
Root discriminant $93.45$
Ramified primes $2, 5, 37$
Class number $33904$ (GRH)
Class group $[2, 2, 2, 4238]$ (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![60112841, 27726080, 53966797, 16508326, 19859011, 3906398, 4295158, 466198, 639262, 17826, 71211, -2572, 6419, -516, 421, -58, 20, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 20*x^16 - 58*x^15 + 421*x^14 - 516*x^13 + 6419*x^12 - 2572*x^11 + 71211*x^10 + 17826*x^9 + 639262*x^8 + 466198*x^7 + 4295158*x^6 + 3906398*x^5 + 19859011*x^4 + 16508326*x^3 + 53966797*x^2 + 27726080*x + 60112841)
 
gp: K = bnfinit(x^18 - 2*x^17 + 20*x^16 - 58*x^15 + 421*x^14 - 516*x^13 + 6419*x^12 - 2572*x^11 + 71211*x^10 + 17826*x^9 + 639262*x^8 + 466198*x^7 + 4295158*x^6 + 3906398*x^5 + 19859011*x^4 + 16508326*x^3 + 53966797*x^2 + 27726080*x + 60112841, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 20 x^{16} - 58 x^{15} + 421 x^{14} - 516 x^{13} + 6419 x^{12} - 2572 x^{11} + 71211 x^{10} + 17826 x^{9} + 639262 x^{8} + 466198 x^{7} + 4295158 x^{6} + 3906398 x^{5} + 19859011 x^{4} + 16508326 x^{3} + 53966797 x^{2} + 27726080 x + 60112841 \)

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:  \(-295307224547164431964209152000000000=-\,2^{24}\cdot 5^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.45$
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:  $2, 5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{7} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{75808807500227248389301534925646880307031731022} a^{17} + \frac{6190244529588368272208644559932961278065211885}{75808807500227248389301534925646880307031731022} a^{16} - \frac{553163017414020020576561072365449309327745361}{12634801250037874731550255820941146717838621837} a^{15} - \frac{14302918053900619869703522650843190307034403}{25269602500075749463100511641882293435677243674} a^{14} + \frac{563274894480557635388061644030164488518519501}{25269602500075749463100511641882293435677243674} a^{13} + \frac{1833569163476856722345196285707261427638646842}{37904403750113624194650767462823440153515865511} a^{12} + \frac{16883062475502177314171022198929798777605924093}{75808807500227248389301534925646880307031731022} a^{11} + \frac{9705545909822479995810527383260752190937555507}{75808807500227248389301534925646880307031731022} a^{10} + \frac{182500609449130336407295726137308849893146674}{37904403750113624194650767462823440153515865511} a^{9} - \frac{28540970394667463336820400599874108801695908821}{75808807500227248389301534925646880307031731022} a^{8} + \frac{3938468621442964521448714377347457032140842647}{37904403750113624194650767462823440153515865511} a^{7} - \frac{18176762021896376877633612426062349906792315400}{37904403750113624194650767462823440153515865511} a^{6} + \frac{5959716885045060038401434685873007348574292463}{12634801250037874731550255820941146717838621837} a^{5} - \frac{3808803750779934568887372197356947559697961283}{37904403750113624194650767462823440153515865511} a^{4} + \frac{14473896072425461242083335319655728918090234543}{37904403750113624194650767462823440153515865511} a^{3} - \frac{4063995251573041515837515839211541474077777797}{25269602500075749463100511641882293435677243674} a^{2} - \frac{5639028063682148387292857963857225687312016280}{12634801250037874731550255820941146717838621837} a + \frac{5628232744228508054152558130775422150867731349}{12634801250037874731550255820941146717838621837}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4238}$, which has order $33904$ (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:  \( 615797.1340659427 \) (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

\(\Q(\sqrt{-5}) \), 3.3.1369.1, 3.3.148.1, 6.0.43808000.1, 6.0.14993288000.2, 9.9.6075640136512.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$