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Properties

Label 18.0.91730128303...5631.8
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{12}\cdot 31^{15}$
Root discriminant $276.92$
Ramified primes $3, 7, 31$
Class number $272643084$ (GRH)
Class group $[3, 3, 114, 265734]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3470262272, 6649528320, 6736075776, 4627285632, 2450615328, 717479952, -75971136, -124465920, -18115494, 9507061, 2726625, -357180, -177376, 3654, 6198, 116, -114, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 114*x^16 + 116*x^15 + 6198*x^14 + 3654*x^13 - 177376*x^12 - 357180*x^11 + 2726625*x^10 + 9507061*x^9 - 18115494*x^8 - 124465920*x^7 - 75971136*x^6 + 717479952*x^5 + 2450615328*x^4 + 4627285632*x^3 + 6736075776*x^2 + 6649528320*x + 3470262272)
 
gp: K = bnfinit(x^18 - 3*x^17 - 114*x^16 + 116*x^15 + 6198*x^14 + 3654*x^13 - 177376*x^12 - 357180*x^11 + 2726625*x^10 + 9507061*x^9 - 18115494*x^8 - 124465920*x^7 - 75971136*x^6 + 717479952*x^5 + 2450615328*x^4 + 4627285632*x^3 + 6736075776*x^2 + 6649528320*x + 3470262272, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 114 x^{16} + 116 x^{15} + 6198 x^{14} + 3654 x^{13} - 177376 x^{12} - 357180 x^{11} + 2726625 x^{10} + 9507061 x^{9} - 18115494 x^{8} - 124465920 x^{7} - 75971136 x^{6} + 717479952 x^{5} + 2450615328 x^{4} + 4627285632 x^{3} + 6736075776 x^{2} + 6649528320 x + 3470262272 \)

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:  \(-91730128303217099071904424489633811022225631=-\,3^{24}\cdot 7^{12}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $276.92$
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:  $3, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1953=3^{2}\cdot 7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1953}(1,·)$, $\chi_{1953}(781,·)$, $\chi_{1953}(718,·)$, $\chi_{1953}(466,·)$, $\chi_{1953}(340,·)$, $\chi_{1953}(88,·)$, $\chi_{1953}(25,·)$, $\chi_{1953}(1948,·)$, $\chi_{1953}(1885,·)$, $\chi_{1953}(1828,·)$, $\chi_{1953}(1576,·)$, $\chi_{1953}(1513,·)$, $\chi_{1953}(688,·)$, $\chi_{1953}(625,·)$, $\chi_{1953}(373,·)$, $\chi_{1953}(247,·)$, $\chi_{1953}(316,·)$, $\chi_{1953}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{32} a^{3}$, $\frac{1}{512} a^{10} + \frac{1}{512} a^{9} - \frac{1}{256} a^{8} - \frac{7}{256} a^{7} - \frac{7}{512} a^{6} + \frac{1}{512} a^{5} + \frac{3}{64} a^{4} - \frac{29}{128} a^{3} - \frac{1}{32} a^{2} + \frac{1}{4} a$, $\frac{1}{512} a^{11} + \frac{1}{512} a^{9} - \frac{1}{512} a^{7} + \frac{27}{512} a^{5} - \frac{1}{16} a^{4} + \frac{9}{128} a^{3} + \frac{1}{16} a^{2} - \frac{1}{8} a$, $\frac{1}{2048} a^{12} - \frac{1}{1024} a^{11} + \frac{1}{2048} a^{10} - \frac{1}{1024} a^{9} + \frac{7}{2048} a^{8} - \frac{31}{1024} a^{7} - \frac{53}{2048} a^{6} + \frac{37}{1024} a^{5} - \frac{13}{512} a^{4} - \frac{33}{256} a^{3} - \frac{13}{64} a^{2} + \frac{3}{8} a$, $\frac{1}{4096} a^{13} + \frac{1}{4096} a^{11} - \frac{1}{1024} a^{10} + \frac{3}{4096} a^{9} + \frac{1}{512} a^{8} - \frac{61}{4096} a^{7} + \frac{23}{1024} a^{6} + \frac{9}{512} a^{5} + \frac{7}{128} a^{4} - \frac{33}{256} a^{3} - \frac{13}{64} a^{2} + \frac{1}{4} a$, $\frac{1}{8192} a^{14} - \frac{1}{8192} a^{13} + \frac{1}{8192} a^{12} - \frac{5}{8192} a^{11} + \frac{7}{8192} a^{10} - \frac{27}{8192} a^{9} + \frac{27}{8192} a^{8} - \frac{39}{8192} a^{7} + \frac{43}{2048} a^{6} - \frac{17}{1024} a^{5} + \frac{23}{512} a^{4} - \frac{19}{512} a^{3} + \frac{7}{128} a^{2} - \frac{1}{16} a$, $\frac{1}{65536} a^{15} - \frac{3}{65536} a^{14} + \frac{1}{65536} a^{13} - \frac{7}{65536} a^{12} + \frac{31}{65536} a^{11} + \frac{15}{65536} a^{10} + \frac{139}{65536} a^{9} + \frac{51}{65536} a^{8} - \frac{335}{16384} a^{7} - \frac{253}{8192} a^{6} - \frac{163}{4096} a^{5} + \frac{123}{4096} a^{4} + \frac{91}{1024} a^{3} - \frac{3}{32} a^{2} - \frac{7}{16} a$, $\frac{1}{102891520} a^{16} + \frac{117}{25722880} a^{15} - \frac{259}{5144576} a^{14} - \frac{637}{12861440} a^{13} + \frac{10771}{51445760} a^{12} + \frac{7}{160768} a^{11} + \frac{15519}{25722880} a^{10} + \frac{3817}{12861440} a^{9} - \frac{195823}{102891520} a^{8} + \frac{176651}{25722880} a^{7} - \frac{32961}{12861440} a^{6} + \frac{97083}{3215360} a^{5} - \frac{203863}{6430720} a^{4} - \frac{69445}{321536} a^{3} + \frac{263}{40192} a^{2} + \frac{703}{6280} a + \frac{279}{785}$, $\frac{1}{165826007403613063106997889346600619212800} a^{17} + \frac{32547947317868132698190457905467}{16582600740361306310699788934660061921280} a^{16} - \frac{125944744820569989996063485073577901}{41456501850903265776749472336650154803200} a^{15} + \frac{83352323429802629263017437026678241}{2591031365681454111046842021040634675200} a^{14} + \frac{226299217399391923853847584859444371}{3316520148072261262139957786932012384256} a^{13} - \frac{9632266447958923114736138636201812649}{41456501850903265776749472336650154803200} a^{12} + \frac{34487971661757342772316811366136459879}{41456501850903265776749472336650154803200} a^{11} - \frac{4701425991193610632853735228948723791}{5182062731362908222093684042081269350400} a^{10} + \frac{586560474526042048088555549839761210049}{165826007403613063106997889346600619212800} a^{9} + \frac{312454306622581081495581417534007511619}{82913003701806531553498944673300309606400} a^{8} - \frac{49007979523097392131598253581057530839}{2072825092545163288837473616832507740160} a^{7} - \frac{13381491715372060498678875564511183971}{2072825092545163288837473616832507740160} a^{6} - \frac{313630577854356181788344365254448016011}{10364125462725816444187368084162538700800} a^{5} - \frac{164256686112257105713121693162931650253}{5182062731362908222093684042081269350400} a^{4} - \frac{55220958876602647747098583439850376217}{259103136568145411104684202104063467520} a^{3} + \frac{11426486761448289675013683524462169433}{161939460355090881940427626315039667200} a^{2} - \frac{2248449403687843154381004966130179187}{10121216272193180121276726644689979200} a - \frac{38126428367612359697364389350482453}{316288008506036878789897707646561850}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{114}\times C_{265734}$, which has order $272643084$ (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:  \( 219311185150.01114 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-31}) \), 3.3.3814209.2, 3.3.3814209.1, 3.3.961.1, 3.3.3969.1, 6.0.450993899166111.4, 6.0.450993899166111.1, 6.0.28629151.1, 6.0.469296461151.6, 9.9.55489838359499131329.9

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$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}$
$31$31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$