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Properties

Label 18.0.18481699108...4727.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{9}\cdot 127^{16}$
Root discriminant $196.15$
Ramified primes $7, 127$
Class number $106755327$ (GRH)
Class group $[9, 11861703]$ (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![165412864, -40402944, 24702976, 44681728, 2280064, -8927584, 6853576, 2701876, -1668350, -488799, 374289, 84268, -39184, -8242, 2558, 408, -74, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 74*x^16 + 408*x^15 + 2558*x^14 - 8242*x^13 - 39184*x^12 + 84268*x^11 + 374289*x^10 - 488799*x^9 - 1668350*x^8 + 2701876*x^7 + 6853576*x^6 - 8927584*x^5 + 2280064*x^4 + 44681728*x^3 + 24702976*x^2 - 40402944*x + 165412864)
 
gp: K = bnfinit(x^18 - 7*x^17 - 74*x^16 + 408*x^15 + 2558*x^14 - 8242*x^13 - 39184*x^12 + 84268*x^11 + 374289*x^10 - 488799*x^9 - 1668350*x^8 + 2701876*x^7 + 6853576*x^6 - 8927584*x^5 + 2280064*x^4 + 44681728*x^3 + 24702976*x^2 - 40402944*x + 165412864, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} - 74 x^{16} + 408 x^{15} + 2558 x^{14} - 8242 x^{13} - 39184 x^{12} + 84268 x^{11} + 374289 x^{10} - 488799 x^{9} - 1668350 x^{8} + 2701876 x^{7} + 6853576 x^{6} - 8927584 x^{5} + 2280064 x^{4} + 44681728 x^{3} + 24702976 x^{2} - 40402944 x + 165412864 \)

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:  \(-184816991082370630395707686096416257574727=-\,7^{9}\cdot 127^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $196.15$
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:  $7, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(889=7\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{889}(1,·)$, $\chi_{889}(195,·)$, $\chi_{889}(449,·)$, $\chi_{889}(400,·)$, $\chi_{889}(657,·)$, $\chi_{889}(146,·)$, $\chi_{889}(22,·)$, $\chi_{889}(734,·)$, $\chi_{889}(799,·)$, $\chi_{889}(545,·)$, $\chi_{889}(99,·)$, $\chi_{889}(484,·)$, $\chi_{889}(869,·)$, $\chi_{889}(230,·)$, $\chi_{889}(615,·)$, $\chi_{889}(687,·)$, $\chi_{889}(433,·)$, $\chi_{889}(636,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{32} a^{3}$, $\frac{1}{512} a^{10} - \frac{1}{512} a^{9} - \frac{1}{256} a^{8} - \frac{1}{256} a^{7} + \frac{9}{512} a^{6} + \frac{15}{512} a^{5} + \frac{1}{64} a^{4} - \frac{3}{128} a^{3} - \frac{1}{32} a^{2}$, $\frac{1}{1024} a^{11} + \frac{1}{1024} a^{9} + \frac{31}{1024} a^{7} - \frac{1}{32} a^{6} - \frac{5}{1024} a^{5} - \frac{1}{16} a^{4} - \frac{55}{256} a^{3} + \frac{3}{32} a^{2} - \frac{5}{16} a - \frac{1}{2}$, $\frac{1}{1024} a^{12} - \frac{1}{1024} a^{10} + \frac{1}{512} a^{9} + \frac{3}{1024} a^{8} - \frac{7}{256} a^{7} - \frac{23}{1024} a^{6} - \frac{15}{512} a^{5} + \frac{13}{256} a^{4} + \frac{7}{128} a^{3} - \frac{1}{32} a^{2}$, $\frac{1}{2048} a^{13} - \frac{1}{2048} a^{12} - \frac{1}{2048} a^{11} - \frac{1}{2048} a^{10} - \frac{3}{2048} a^{9} + \frac{1}{2048} a^{8} + \frac{29}{2048} a^{7} + \frac{5}{2048} a^{6} - \frac{25}{1024} a^{5} + \frac{31}{512} a^{4} - \frac{13}{256} a^{3} - \frac{1}{16} a^{2} - \frac{7}{16} a - \frac{1}{2}$, $\frac{1}{16384} a^{14} + \frac{1}{16384} a^{13} - \frac{5}{16384} a^{12} + \frac{1}{16384} a^{11} + \frac{9}{16384} a^{10} + \frac{15}{16384} a^{9} + \frac{1}{16384} a^{8} + \frac{283}{16384} a^{7} - \frac{199}{8192} a^{6} - \frac{91}{4096} a^{5} - \frac{127}{2048} a^{4} - \frac{63}{256} a^{3} + \frac{11}{128} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16384} a^{15} + \frac{1}{8192} a^{13} - \frac{1}{8192} a^{12} - \frac{1}{8192} a^{10} - \frac{19}{8192} a^{9} + \frac{17}{8192} a^{8} - \frac{449}{16384} a^{7} - \frac{219}{8192} a^{6} + \frac{249}{4096} a^{5} - \frac{29}{2048} a^{4} - \frac{3}{32} a^{3} + \frac{5}{128} a^{2} + \frac{1}{16} a$, $\frac{1}{8585216} a^{16} + \frac{3}{2146304} a^{15} + \frac{31}{4292608} a^{14} + \frac{241}{4292608} a^{13} - \frac{593}{2146304} a^{12} + \frac{1477}{4292608} a^{11} + \frac{207}{4292608} a^{10} + \frac{10791}{4292608} a^{9} - \frac{8301}{8585216} a^{8} + \frac{25409}{4292608} a^{7} + \frac{15221}{2146304} a^{6} + \frac{57223}{1073152} a^{5} - \frac{429}{16768} a^{4} - \frac{99}{512} a^{3} + \frac{1541}{8384} a^{2} + \frac{3}{2096} a + \frac{15}{262}$, $\frac{1}{5147308932188942699962417953898496} a^{17} - \frac{289213559706660330831657415}{5147308932188942699962417953898496} a^{16} - \frac{7782070714380379334827846369}{2573654466094471349981208976949248} a^{15} - \frac{8069213327170001696975199015}{643413616523617837495302244237312} a^{14} + \frac{258332583359123054476580760783}{2573654466094471349981208976949248} a^{13} + \frac{164123837878683789139006110487}{2573654466094471349981208976949248} a^{12} + \frac{2356959297582012576709153617}{80426702065452229686912780529664} a^{11} - \frac{1096046297675092170440370729493}{1286827233047235674990604488474624} a^{10} - \frac{13709490863146907964272780294383}{5147308932188942699962417953898496} a^{9} + \frac{18432851834011276415995162619457}{5147308932188942699962417953898496} a^{8} + \frac{14731143062992756891069746434165}{2573654466094471349981208976949248} a^{7} + \frac{35470795014155501889572490490017}{1286827233047235674990604488474624} a^{6} - \frac{16034008916883355058726103377331}{643413616523617837495302244237312} a^{5} - \frac{4668762193906598597389696576973}{160853404130904459373825561059328} a^{4} + \frac{1601498177428900666056031456283}{40213351032726114843456390264832} a^{3} - \frac{801035427338539178707044484361}{10053337758181528710864097566208} a^{2} - \frac{99278699019723684308677991525}{628333609886345544429006097888} a + \frac{901755898428729847463238729}{157083402471586386107251524472}$

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

Class group and class number

$C_{9}\times C_{11861703}$, which has order $106755327$ (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:  \( 16450307908.665707 \) (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

\(\Q(\sqrt{-7}) \), 3.3.16129.1, 6.0.89229611863.2, 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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
7Data not computed
$127$127.9.8.1$x^{9} - 127$$9$$1$$8$$C_9$$[\ ]_{9}$
127.9.8.1$x^{9} - 127$$9$$1$$8$$C_9$$[\ ]_{9}$