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Properties

Label 18.0.58165204085...4447.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,127^{17}$
Root discriminant $97.03$
Ramified prime $127$
Class number $200135$ (GRH)
Class group $[200135]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048576, -573440, -566272, 424704, 698944, -886384, 494368, -195632, 120032, -60665, 27225, -10452, 3828, -638, 110, -20, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 4*x^16 - 20*x^15 + 110*x^14 - 638*x^13 + 3828*x^12 - 10452*x^11 + 27225*x^10 - 60665*x^9 + 120032*x^8 - 195632*x^7 + 494368*x^6 - 886384*x^5 + 698944*x^4 + 424704*x^3 - 566272*x^2 - 573440*x + 1048576)
 
gp: K = bnfinit(x^18 - x^17 + 4*x^16 - 20*x^15 + 110*x^14 - 638*x^13 + 3828*x^12 - 10452*x^11 + 27225*x^10 - 60665*x^9 + 120032*x^8 - 195632*x^7 + 494368*x^6 - 886384*x^5 + 698944*x^4 + 424704*x^3 - 566272*x^2 - 573440*x + 1048576, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 4 x^{16} - 20 x^{15} + 110 x^{14} - 638 x^{13} + 3828 x^{12} - 10452 x^{11} + 27225 x^{10} - 60665 x^{9} + 120032 x^{8} - 195632 x^{7} + 494368 x^{6} - 886384 x^{5} + 698944 x^{4} + 424704 x^{3} - 566272 x^{2} - 573440 x + 1048576 \)

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:  \(-581652040856250348581103942808504447=-\,127^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.03$
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:  $127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(127\)
Dirichlet character group:    $\lbrace$$\chi_{127}(1,·)$, $\chi_{127}(68,·)$, $\chi_{127}(75,·)$, $\chi_{127}(19,·)$, $\chi_{127}(20,·)$, $\chi_{127}(22,·)$, $\chi_{127}(24,·)$, $\chi_{127}(90,·)$, $\chi_{127}(28,·)$, $\chi_{127}(99,·)$, $\chi_{127}(37,·)$, $\chi_{127}(103,·)$, $\chi_{127}(105,·)$, $\chi_{127}(107,·)$, $\chi_{127}(108,·)$, $\chi_{127}(52,·)$, $\chi_{127}(59,·)$, $\chi_{127}(126,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{32} a^{3}$, $\frac{1}{256} a^{10} - \frac{1}{256} a^{9} + \frac{3}{128} a^{7} - \frac{3}{256} a^{6} + \frac{15}{256} a^{5} - \frac{3}{128} a^{4} + \frac{3}{64} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{512} a^{11} + \frac{1}{512} a^{9} + \frac{15}{512} a^{7} + \frac{27}{512} a^{5} + \frac{5}{128} a^{3} - \frac{1}{8} a$, $\frac{1}{2048} a^{12} - \frac{1}{1024} a^{11} - \frac{3}{2048} a^{10} + \frac{1}{1024} a^{9} + \frac{7}{2048} a^{8} - \frac{11}{1024} a^{7} - \frac{41}{2048} a^{6} + \frac{7}{1024} a^{5} + \frac{25}{512} a^{4} - \frac{47}{256} a^{3} + \frac{7}{32} a^{2} - \frac{1}{16} a$, $\frac{1}{2048} a^{13} + \frac{1}{2048} a^{11} - \frac{1}{512} a^{10} + \frac{3}{2048} a^{9} - \frac{1}{256} a^{8} + \frac{3}{2048} a^{7} + \frac{15}{512} a^{6} - \frac{7}{256} a^{5} - \frac{3}{128} a^{4} + \frac{19}{128} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a$, $\frac{1}{8192} a^{14} + \frac{1}{8192} a^{13} - \frac{1}{8192} a^{12} + \frac{1}{8192} a^{11} - \frac{11}{8192} a^{10} - \frac{25}{8192} a^{9} + \frac{13}{8192} a^{8} - \frac{53}{8192} a^{7} - \frac{29}{4096} a^{6} + \frac{27}{2048} a^{5} - \frac{41}{1024} a^{4} + \frac{15}{256} a^{3} - \frac{5}{64} a^{2} + \frac{1}{16} a$, $\frac{1}{65536} a^{15} + \frac{1}{32768} a^{14} + \frac{1}{16384} a^{13} - \frac{3}{16384} a^{12} - \frac{7}{32768} a^{11} - \frac{5}{4096} a^{10} + \frac{1}{8192} a^{9} - \frac{55}{16384} a^{8} - \frac{1467}{65536} a^{7} - \frac{857}{32768} a^{6} - \frac{97}{16384} a^{5} + \frac{77}{8192} a^{4} + \frac{61}{1024} a^{3} - \frac{21}{512} a^{2} + \frac{1}{32} a$, $\frac{1}{131072} a^{16} - \frac{1}{16384} a^{14} - \frac{7}{32768} a^{13} - \frac{7}{65536} a^{12} + \frac{1}{32768} a^{11} + \frac{3}{4096} a^{10} + \frac{39}{32768} a^{9} - \frac{331}{131072} a^{8} - \frac{821}{32768} a^{7} + \frac{221}{8192} a^{6} + \frac{21}{8192} a^{5} - \frac{477}{8192} a^{4} + \frac{107}{512} a^{3} + \frac{33}{512} a^{2} - \frac{7}{32} a$, $\frac{1}{123708567156126384128} a^{17} - \frac{460082048514963}{123708567156126384128} a^{16} + \frac{45335281366443}{61854283578063192064} a^{15} + \frac{542483912833233}{15463570894515798016} a^{14} - \frac{8727872538041273}{61854283578063192064} a^{13} + \frac{9213752562820459}{61854283578063192064} a^{12} - \frac{7126007998981}{120809147613404672} a^{11} + \frac{22444299684699979}{30927141789031596032} a^{10} + \frac{463506581055048321}{123708567156126384128} a^{9} - \frac{258346518559340027}{123708567156126384128} a^{8} + \frac{1065388015005684425}{61854283578063192064} a^{7} + \frac{178935744873005689}{30927141789031596032} a^{6} - \frac{175840133893199215}{15463570894515798016} a^{5} + \frac{268683859689415}{20240276039942144} a^{4} - \frac{228430223421775433}{966473180907237376} a^{3} - \frac{56614400829781787}{241618295226809344} a^{2} - \frac{4082084364862853}{15101143451675584} a - \frac{105200267450820}{235955366432431}$

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

Class group and class number

$C_{200135}$, which has order $200135$ (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:  \( 16450307908.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-127}) \), 3.3.16129.1, 6.0.33038369407.1, 9.9.67675234241018881.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ $18$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ $18$ $18$

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
127Data not computed