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Properties

Label 18.0.33000790449...3503.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 109^{17}$
Root discriminant $436.43$
Ramified primes $3, 109$
Class number $1780239708$ (GRH)
Class group $[3, 593413236]$ (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![381578493952, 398639222784, 665349107904, -380206748136, 61464190068, -18049683870, 5184150199, -356748498, 250790688, -23442412, 4236939, -2601612, 3052, 1962, 8829, -218, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 218*x^15 + 8829*x^14 + 1962*x^13 + 3052*x^12 - 2601612*x^11 + 4236939*x^10 - 23442412*x^9 + 250790688*x^8 - 356748498*x^7 + 5184150199*x^6 - 18049683870*x^5 + 61464190068*x^4 - 380206748136*x^3 + 665349107904*x^2 + 398639222784*x + 381578493952)
 
gp: K = bnfinit(x^18 - 218*x^15 + 8829*x^14 + 1962*x^13 + 3052*x^12 - 2601612*x^11 + 4236939*x^10 - 23442412*x^9 + 250790688*x^8 - 356748498*x^7 + 5184150199*x^6 - 18049683870*x^5 + 61464190068*x^4 - 380206748136*x^3 + 665349107904*x^2 + 398639222784*x + 381578493952, 1)
 

Normalized defining polynomial

\( x^{18} - 218 x^{15} + 8829 x^{14} + 1962 x^{13} + 3052 x^{12} - 2601612 x^{11} + 4236939 x^{10} - 23442412 x^{9} + 250790688 x^{8} - 356748498 x^{7} + 5184150199 x^{6} - 18049683870 x^{5} + 61464190068 x^{4} - 380206748136 x^{3} + 665349107904 x^{2} + 398639222784 x + 381578493952 \)

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:  \(-330007904499468387933830810627328948148827793503=-\,3^{27}\cdot 109^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $436.43$
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:  $3, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(981=3^{2}\cdot 109\)
Dirichlet character group:    $\lbrace$$\chi_{981}(256,·)$, $\chi_{981}(1,·)$, $\chi_{981}(965,·)$, $\chi_{981}(16,·)$, $\chi_{981}(980,·)$, $\chi_{981}(725,·)$, $\chi_{981}(790,·)$, $\chi_{981}(154,·)$, $\chi_{981}(797,·)$, $\chi_{981}(479,·)$, $\chi_{981}(868,·)$, $\chi_{981}(809,·)$, $\chi_{981}(172,·)$, $\chi_{981}(113,·)$, $\chi_{981}(502,·)$, $\chi_{981}(184,·)$, $\chi_{981}(827,·)$, $\chi_{981}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{7} - \frac{1}{16} a^{5} + \frac{5}{32} a^{3} - \frac{1}{8} a$, $\frac{1}{128} a^{8} + \frac{1}{64} a^{6} + \frac{1}{128} a^{4} - \frac{1}{32} a^{2}$, $\frac{1}{128} a^{9} - \frac{1}{64} a^{7} - \frac{7}{128} a^{5} - \frac{1}{16} a^{3} + \frac{1}{8} a$, $\frac{1}{256} a^{10} - \frac{1}{256} a^{9} - \frac{1}{128} a^{7} - \frac{3}{256} a^{6} - \frac{1}{256} a^{5} - \frac{3}{128} a^{4} + \frac{1}{64} a^{3} + \frac{1}{32} a^{2}$, $\frac{1}{2048} a^{11} + \frac{1}{1024} a^{10} - \frac{3}{2048} a^{9} - \frac{1}{1024} a^{8} + \frac{7}{2048} a^{7} + \frac{27}{1024} a^{6} - \frac{105}{2048} a^{5} - \frac{7}{1024} a^{4} - \frac{119}{512} a^{3} + \frac{59}{256} a^{2} + \frac{1}{32} a - \frac{1}{2}$, $\frac{1}{188416} a^{12} - \frac{1}{8192} a^{11} - \frac{245}{188416} a^{10} + \frac{697}{188416} a^{9} - \frac{311}{188416} a^{8} - \frac{1881}{188416} a^{7} + \frac{1713}{188416} a^{6} - \frac{10205}{188416} a^{5} + \frac{1737}{94208} a^{4} + \frac{3061}{47104} a^{3} + \frac{1853}{23552} a^{2} + \frac{1459}{2944} a + \frac{73}{184}$, $\frac{1}{753664} a^{13} - \frac{1}{376832} a^{12} + \frac{1}{94208} a^{11} - \frac{1}{23552} a^{10} + \frac{907}{376832} a^{9} + \frac{105}{188416} a^{8} + \frac{2881}{188416} a^{7} + \frac{829}{94208} a^{6} + \frac{43089}{753664} a^{5} + \frac{21991}{376832} a^{4} - \frac{22541}{188416} a^{3} - \frac{14183}{94208} a^{2} - \frac{5361}{11776} a - \frac{307}{736}$, $\frac{1}{1507328} a^{14} - \frac{1}{1507328} a^{13} - \frac{1}{753664} a^{12} + \frac{5}{47104} a^{11} - \frac{1073}{753664} a^{10} - \frac{1671}{753664} a^{9} - \frac{101}{47104} a^{8} + \frac{2413}{376832} a^{7} + \frac{18353}{1507328} a^{6} - \frac{66809}{1507328} a^{5} - \frac{28155}{753664} a^{4} - \frac{34179}{376832} a^{3} + \frac{25257}{188416} a^{2} - \frac{8697}{23552} a + \frac{581}{1472}$, $\frac{1}{259260416} a^{15} + \frac{19}{259260416} a^{14} + \frac{1}{129630208} a^{13} + \frac{25}{16203776} a^{12} + \frac{163}{129630208} a^{11} + \frac{160785}{129630208} a^{10} + \frac{11495}{4050944} a^{9} - \frac{216681}{64815104} a^{8} + \frac{647481}{259260416} a^{7} + \frac{4216995}{259260416} a^{6} - \frac{7058157}{129630208} a^{5} + \frac{3642551}{64815104} a^{4} - \frac{3617453}{32407552} a^{3} - \frac{730461}{8101888} a^{2} - \frac{112481}{1012736} a + \frac{211}{1472}$, $\frac{1}{2074083328} a^{16} + \frac{1}{1037041664} a^{15} + \frac{195}{2074083328} a^{14} - \frac{75}{1037041664} a^{13} + \frac{1063}{1037041664} a^{12} - \frac{113977}{518520832} a^{11} + \frac{331627}{1037041664} a^{10} + \frac{1923233}{518520832} a^{9} + \frac{3882557}{2074083328} a^{8} + \frac{6050117}{1037041664} a^{7} - \frac{27174729}{2074083328} a^{6} + \frac{52597993}{1037041664} a^{5} - \frac{22293863}{518520832} a^{4} + \frac{61650131}{259260416} a^{3} - \frac{214051}{1409024} a^{2} - \frac{91415}{253184} a + \frac{9}{2944}$, $\frac{1}{24367592705502871803317893579095329944894322115289808896} a^{17} + \frac{4848171701058646194025826866073935573958502265}{24367592705502871803317893579095329944894322115289808896} a^{16} - \frac{30064813828428148118085362777828195910114575567}{24367592705502871803317893579095329944894322115289808896} a^{15} - \frac{4689916167232038579681490000914241833795117442993}{24367592705502871803317893579095329944894322115289808896} a^{14} + \frac{2849790655450121285407699804745707371918949603637}{6091898176375717950829473394773832486223580528822452224} a^{13} + \frac{10612541154357869620806309690917590374798649244143}{12183796352751435901658946789547664972447161057644904448} a^{12} + \frac{1072484562882849331780034950121550888412965641453}{12183796352751435901658946789547664972447161057644904448} a^{11} + \frac{4399464765236249109227316847985821346651438500054463}{12183796352751435901658946789547664972447161057644904448} a^{10} + \frac{40857077598307387881591215972185099710936308535934745}{24367592705502871803317893579095329944894322115289808896} a^{9} - \frac{42370383773343384309813272296085132538626900352442907}{24367592705502871803317893579095329944894322115289808896} a^{8} + \frac{174395700945843779830677911325132948824836492023094109}{24367592705502871803317893579095329944894322115289808896} a^{7} - \frac{436317320156769909797490392572454994362357061093223133}{24367592705502871803317893579095329944894322115289808896} a^{6} + \frac{374145470038859295352466091979654550889920658139012257}{12183796352751435901658946789547664972447161057644904448} a^{5} - \frac{156302008668614491088734246552139168507689889150207707}{6091898176375717950829473394773832486223580528822452224} a^{4} + \frac{629586111449315453391574541035639116002706429966419021}{3045949088187858975414736697386916243111790264411226112} a^{3} + \frac{58659925788409638407289524048165817601485928348030629}{380743636023482371926842087173364530388973783051403264} a^{2} + \frac{2099559370042025953105373312771006608643416832230591}{5949119312866912061356907612083820787327715360178176} a - \frac{9348031273629416764446178528175308219232666053661}{34587902981784372449749462860952446437951833489408}$

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

Class group and class number

$C_{3}\times C_{593413236}$, which has order $1780239708$ (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:  \( 3866670056388.9624 \) (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

\(\Q(\sqrt{-327}) \), 3.3.11881.1, 6.0.415428467823.1, 9.9.10589294828624773798161.3

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{18}$ R $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
3Data not computed
109Data not computed