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Properties

Label 18.0.14350158658...6816.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{45}\cdot 17^{9}$
Root discriminant $102.03$
Ramified primes $2, 3, 17$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2904619, -6170886, 11021922, -18105654, 13758246, -4740084, 2671395, -1551690, 322596, -132088, 42570, -5040, 6729, 1044, 522, -30, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 30*x^15 + 522*x^14 + 1044*x^13 + 6729*x^12 - 5040*x^11 + 42570*x^10 - 132088*x^9 + 322596*x^8 - 1551690*x^7 + 2671395*x^6 - 4740084*x^5 + 13758246*x^4 - 18105654*x^3 + 11021922*x^2 - 6170886*x + 2904619)
 
gp: K = bnfinit(x^18 - 30*x^15 + 522*x^14 + 1044*x^13 + 6729*x^12 - 5040*x^11 + 42570*x^10 - 132088*x^9 + 322596*x^8 - 1551690*x^7 + 2671395*x^6 - 4740084*x^5 + 13758246*x^4 - 18105654*x^3 + 11021922*x^2 - 6170886*x + 2904619, 1)
 

Normalized defining polynomial

\( x^{18} - 30 x^{15} + 522 x^{14} + 1044 x^{13} + 6729 x^{12} - 5040 x^{11} + 42570 x^{10} - 132088 x^{9} + 322596 x^{8} - 1551690 x^{7} + 2671395 x^{6} - 4740084 x^{5} + 13758246 x^{4} - 18105654 x^{3} + 11021922 x^{2} - 6170886 x + 2904619 \)

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:  \(-1435015865868157212700925134821666816=-\,2^{12}\cdot 3^{45}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.03$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} + \frac{1}{6}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} + \frac{1}{6} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} + \frac{1}{6} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{3} a^{4} - \frac{1}{2} a$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{13} + \frac{1}{18} a^{12} + \frac{1}{18} a^{11} - \frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{18} a^{5} - \frac{1}{18} a^{4} + \frac{1}{18} a^{3} + \frac{2}{9} a^{2} - \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{12} + \frac{1}{18} a^{9} + \frac{7}{18} a^{6} + \frac{1}{9} a^{3} - \frac{1}{9}$, $\frac{1}{684} a^{16} - \frac{11}{684} a^{15} + \frac{5}{342} a^{14} + \frac{55}{684} a^{13} - \frac{5}{228} a^{12} - \frac{8}{171} a^{11} + \frac{1}{114} a^{10} - \frac{17}{342} a^{9} - \frac{47}{114} a^{8} - \frac{29}{171} a^{7} - \frac{139}{342} a^{6} + \frac{70}{171} a^{5} + \frac{337}{684} a^{4} + \frac{5}{12} a^{3} + \frac{40}{171} a^{2} - \frac{27}{76} a + \frac{173}{684}$, $\frac{1}{2816592895852035049506953516341051606728743198756039412} a^{17} + \frac{682569795982737250126246873421930841550477691060989}{2816592895852035049506953516341051606728743198756039412} a^{16} - \frac{881692890618829978942420516993904779222968685607806}{704148223963008762376738379085262901682185799689009853} a^{15} - \frac{31963497501860891509781177275130862045861204198901967}{2816592895852035049506953516341051606728743198756039412} a^{14} - \frac{7836473517034240600699466945709871455862704523615591}{312954766205781672167439279593450178525415910972893268} a^{13} + \frac{18173781535371247455690887856406035786065878281536678}{234716074654336254125579459695087633894061933229669951} a^{12} - \frac{15500240698866369040077988524828953046786977939442433}{469432149308672508251158919390175267788123866459339902} a^{11} + \frac{38116479201080231963130279744986143607976768236469644}{704148223963008762376738379085262901682185799689009853} a^{10} - \frac{31769762771729473569924044610352734267853936620273595}{1408296447926017524753476758170525803364371599378019706} a^{9} + \frac{253431949870624692340497971626674829177841845568260720}{704148223963008762376738379085262901682185799689009853} a^{8} - \frac{5119060853272876450787406747371533344247076891539808}{704148223963008762376738379085262901682185799689009853} a^{7} - \frac{3591602710091935381413537805196892884904392465092942}{704148223963008762376738379085262901682185799689009853} a^{6} - \frac{1194750343435124867387446977656927325427255320106806359}{2816592895852035049506953516341051606728743198756039412} a^{5} - \frac{85373579983116489685163077013325857883009573970338839}{938864298617345016502317838780350535576247732918679804} a^{4} - \frac{18559356952419216133055571386303411515258494121488541}{469432149308672508251158919390175267788123866459339902} a^{3} - \frac{430040121924889654215756175571355622962417002975769319}{938864298617345016502317838780350535576247732918679804} a^{2} - \frac{1129884026578098808585924480674706084910370669747076993}{2816592895852035049506953516341051606728743198756039412} a + \frac{71731820509931383527611615402147329312409422097647966}{704148223963008762376738379085262901682185799689009853}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 16724342128.232887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-51}) \), 3.1.243.1, 6.0.870323211.2, 9.1.2008387814976.4

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$