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Properties

Label 18.18.2508135635...6656.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{8}\cdot 2351^{8}$
Root discriminant $81.48$
Ramified primes $2, 3, 2351$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3:S_4$ (as 18T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![339, -5415, -15181, 92466, 13708, -291004, 52726, 359116, -97997, -198755, 61275, 48500, -15868, -5008, 1688, 214, -73, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 73*x^16 + 214*x^15 + 1688*x^14 - 5008*x^13 - 15868*x^12 + 48500*x^11 + 61275*x^10 - 198755*x^9 - 97997*x^8 + 359116*x^7 + 52726*x^6 - 291004*x^5 + 13708*x^4 + 92466*x^3 - 15181*x^2 - 5415*x + 339)
 
gp: K = bnfinit(x^18 - 3*x^17 - 73*x^16 + 214*x^15 + 1688*x^14 - 5008*x^13 - 15868*x^12 + 48500*x^11 + 61275*x^10 - 198755*x^9 - 97997*x^8 + 359116*x^7 + 52726*x^6 - 291004*x^5 + 13708*x^4 + 92466*x^3 - 15181*x^2 - 5415*x + 339, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 73 x^{16} + 214 x^{15} + 1688 x^{14} - 5008 x^{13} - 15868 x^{12} + 48500 x^{11} + 61275 x^{10} - 198755 x^{9} - 97997 x^{8} + 359116 x^{7} + 52726 x^{6} - 291004 x^{5} + 13708 x^{4} + 92466 x^{3} - 15181 x^{2} - 5415 x + 339 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(25081356358474547401099302024646656=2^{12}\cdot 3^{8}\cdot 2351^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.48$
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:  $2, 3, 2351$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} + \frac{1}{12} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{12} a^{3} - \frac{5}{12} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{13} + \frac{1}{12} a^{12} - \frac{5}{24} a^{11} + \frac{5}{24} a^{10} - \frac{1}{12} a^{9} + \frac{1}{24} a^{8} - \frac{1}{24} a^{7} + \frac{1}{24} a^{6} + \frac{1}{12} a^{5} - \frac{7}{24} a^{4} + \frac{7}{24} a^{3} + \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{504} a^{15} + \frac{1}{56} a^{14} + \frac{1}{63} a^{13} - \frac{61}{504} a^{12} - \frac{97}{504} a^{11} - \frac{17}{126} a^{10} - \frac{83}{504} a^{9} - \frac{47}{504} a^{8} - \frac{101}{504} a^{7} + \frac{11}{126} a^{6} + \frac{41}{504} a^{5} + \frac{89}{504} a^{4} + \frac{41}{84} a^{3} - \frac{233}{504} a^{2} + \frac{83}{168} a - \frac{19}{84}$, $\frac{1}{1008} a^{16} - \frac{5}{504} a^{14} - \frac{1}{36} a^{13} - \frac{47}{504} a^{12} - \frac{13}{72} a^{11} + \frac{1}{252} a^{10} + \frac{11}{72} a^{9} - \frac{29}{144} a^{8} - \frac{101}{504} a^{7} + \frac{11}{252} a^{6} + \frac{1}{72} a^{5} - \frac{13}{84} a^{4} - \frac{205}{504} a^{3} + \frac{41}{168} a^{2} + \frac{17}{168} a + \frac{51}{112}$, $\frac{1}{227657133071878553672404764420432} a^{17} - \frac{52782457898039729671229137033}{113828566535939276836202382210216} a^{16} + \frac{3625517998177672945937824769}{4742856938997469868175099258759} a^{15} + \frac{294170526198362328190970635547}{28457141633984819209050595552554} a^{14} - \frac{316056888376027812230386296973}{18971427755989879472700397035036} a^{13} - \frac{503744348059058832836617093315}{56914283267969638418101191105108} a^{12} + \frac{3032991397255772892015459584074}{14228570816992409604525297776277} a^{11} - \frac{646049709152853980176508514157}{18971427755989879472700397035036} a^{10} + \frac{5849014701206663096497958329007}{25295237007986505963600529380048} a^{9} - \frac{9794367121135864661882213057921}{56914283267969638418101191105108} a^{8} + \frac{373103088634268434067317987651}{2189010894921909169926968888658} a^{7} + \frac{244128625535950664538056316626}{4742856938997469868175099258759} a^{6} - \frac{26357496094148098616792047127749}{113828566535939276836202382210216} a^{5} - \frac{17539149753907044624831599805911}{37942855511979758945400794070072} a^{4} - \frac{6583342743391723973148814214882}{14228570816992409604525297776277} a^{3} + \frac{11774422480498282863978049077637}{56914283267969638418101191105108} a^{2} - \frac{10692819059981097497484488159501}{25295237007986505963600529380048} a + \frac{4539020339707329391249375578835}{18971427755989879472700397035036}$

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:  $17$
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:  \( 675507999150 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_4$ (as 18T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3:S_4$
Character table for $C_3:S_4$

Intermediate fields

3.3.7053.1, 3.3.28212.1, 3.3.28212.2, 3.3.28212.3, 6.6.49744809.1, 9.9.158370945436574784.1

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
2351Data not computed