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Properties

Label 18.6.25335674079...9632.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 7^{12}\cdot 197^{9}$
Root discriminant $81.53$
Ramified primes $2, 7, 197$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_3:S_4$ (as 18T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-30249612, -23700060, 52507872, -6903624, -17698768, 7233538, 337971, -800673, 398451, -56398, -7019, 4203, -3516, 413, -97, -26, 17, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 17*x^16 - 26*x^15 - 97*x^14 + 413*x^13 - 3516*x^12 + 4203*x^11 - 7019*x^10 - 56398*x^9 + 398451*x^8 - 800673*x^7 + 337971*x^6 + 7233538*x^5 - 17698768*x^4 - 6903624*x^3 + 52507872*x^2 - 23700060*x - 30249612)
 
gp: K = bnfinit(x^18 - 5*x^17 + 17*x^16 - 26*x^15 - 97*x^14 + 413*x^13 - 3516*x^12 + 4203*x^11 - 7019*x^10 - 56398*x^9 + 398451*x^8 - 800673*x^7 + 337971*x^6 + 7233538*x^5 - 17698768*x^4 - 6903624*x^3 + 52507872*x^2 - 23700060*x - 30249612, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 17 x^{16} - 26 x^{15} - 97 x^{14} + 413 x^{13} - 3516 x^{12} + 4203 x^{11} - 7019 x^{10} - 56398 x^{9} + 398451 x^{8} - 800673 x^{7} + 337971 x^{6} + 7233538 x^{5} - 17698768 x^{4} - 6903624 x^{3} + 52507872 x^{2} - 23700060 x - 30249612 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(25335674079416182011223462348869632=2^{12}\cdot 7^{12}\cdot 197^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.53$
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:  $2, 7, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a$, $\frac{1}{612} a^{15} + \frac{19}{306} a^{14} + \frac{91}{612} a^{13} - \frac{97}{612} a^{12} - \frac{7}{306} a^{11} - \frac{89}{204} a^{10} + \frac{91}{204} a^{9} + \frac{10}{51} a^{8} + \frac{211}{612} a^{7} - \frac{73}{204} a^{6} - \frac{16}{51} a^{5} + \frac{29}{68} a^{4} + \frac{10}{51} a^{3} + \frac{58}{153} a^{2} + \frac{49}{102} a - \frac{5}{17}$, $\frac{1}{612} a^{16} - \frac{3}{68} a^{14} + \frac{5}{204} a^{13} + \frac{1}{6} a^{12} + \frac{163}{612} a^{11} + \frac{13}{68} a^{10} - \frac{13}{51} a^{9} + \frac{241}{612} a^{8} + \frac{25}{612} a^{7} + \frac{23}{51} a^{6} - \frac{31}{204} a^{5} + \frac{25}{51} a^{4} + \frac{131}{306} a^{3} + \frac{23}{306} a^{2} + \frac{2}{17} a + \frac{3}{17}$, $\frac{1}{988278457511864310376885983768422484753311765194102690927380} a^{17} - \frac{305552472356606835998680813702487120917216368919066562959}{494139228755932155188442991884211242376655882597051345463690} a^{16} - \frac{46311304243997434437553037615045339642117366460122421379}{988278457511864310376885983768422484753311765194102690927380} a^{15} + \frac{79830491264871473195806943009312569959609466345096322628301}{988278457511864310376885983768422484753311765194102690927380} a^{14} + \frac{1112466457355009957148941636244467826705579349246021352788}{49413922875593215518844299188421124237665588259705134546369} a^{13} - \frac{219145632137557503780580495303574318330539190697429081783277}{988278457511864310376885983768422484753311765194102690927380} a^{12} - \frac{6052017893973314121057817222497939587415693413948704771087}{65885230500790954025125732251228165650220784346273512728492} a^{11} - \frac{8241920136254867131832750609826507527472529693791169582163}{18301452916886376118090481180896712680616884540631531313470} a^{10} - \frac{296374268830683704202223534777704915247148938173494733685763}{988278457511864310376885983768422484753311765194102690927380} a^{9} + \frac{201809906727227831590751914476354760697320287241880069599391}{988278457511864310376885983768422484753311765194102690927380} a^{8} - \frac{2518317232386019983774463521587413558896422098302579914737}{164713076251977385062814330628070414125551960865683781821230} a^{7} - \frac{6630011852082628505748631709118426588610403247232352284669}{29947832045814070011420787386921893477373083793760687603860} a^{6} + \frac{26759503862443521021331734711588446955248096372149068531017}{164713076251977385062814330628070414125551960865683781821230} a^{5} + \frac{74529651451771804944184413192801750970461402193593847639601}{494139228755932155188442991884211242376655882597051345463690} a^{4} - \frac{28079825046007532951141389582927187770699342921394743487867}{494139228755932155188442991884211242376655882597051345463690} a^{3} + \frac{38016215848522165872042714447549198525297841675074254112119}{82356538125988692531407165314035207062775980432841890910615} a^{2} + \frac{964717436737498197342647206335929144004514003633211497133}{2495652670484505834285065615576824456447756982813390633655} a - \frac{2981664500985577255144029894088383232914757459735788591893}{9150726458443188059045240590448356340308442270315765656735}$

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:  $11$
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:  \( 14150853991.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_4$ (as 18T40):

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

3.3.38612.1, 3.3.38612.2, 3.3.9653.1, 3.3.788.1, 6.2.122325968.1, 9.9.11340523913674816.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
197Data not computed