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Properties

Label 18.0.26475349193...2243.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{33}\cdot 37^{4}\cdot 71^{4}$
Root discriminant $43.11$
Ramified primes $3, 37, 71$
Class number $42$ (GRH)
Class group $[42]$ (GRH)
Galois group 18T201

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4213137, 1283616, -3303990, 2188377, -212220, -1303911, 1297998, -512811, -70938, 130548, -25020, -4662, -87, 531, 99, -51, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 12*x^16 - 51*x^15 + 99*x^14 + 531*x^13 - 87*x^12 - 4662*x^11 - 25020*x^10 + 130548*x^9 - 70938*x^8 - 512811*x^7 + 1297998*x^6 - 1303911*x^5 - 212220*x^4 + 2188377*x^3 - 3303990*x^2 + 1283616*x + 4213137)
 
gp: K = bnfinit(x^18 - 6*x^17 + 12*x^16 - 51*x^15 + 99*x^14 + 531*x^13 - 87*x^12 - 4662*x^11 - 25020*x^10 + 130548*x^9 - 70938*x^8 - 512811*x^7 + 1297998*x^6 - 1303911*x^5 - 212220*x^4 + 2188377*x^3 - 3303990*x^2 + 1283616*x + 4213137, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 12 x^{16} - 51 x^{15} + 99 x^{14} + 531 x^{13} - 87 x^{12} - 4662 x^{11} - 25020 x^{10} + 130548 x^{9} - 70938 x^{8} - 512811 x^{7} + 1297998 x^{6} - 1303911 x^{5} - 212220 x^{4} + 2188377 x^{3} - 3303990 x^{2} + 1283616 x + 4213137 \)

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:  \(-264753491934954018004342992243=-\,3^{33}\cdot 37^{4}\cdot 71^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.11$
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:  $3, 37, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} - \frac{3}{19} a^{14} + \frac{6}{19} a^{13} + \frac{1}{19} a^{12} - \frac{9}{19} a^{11} + \frac{5}{19} a^{10} + \frac{1}{19} a^{9} + \frac{4}{19} a^{8} + \frac{5}{19} a^{7} - \frac{7}{19} a^{6} + \frac{3}{19} a^{5} - \frac{7}{19} a^{4} + \frac{4}{19} a^{3} + \frac{8}{19} a^{2} - \frac{8}{19} a - \frac{4}{19}$, $\frac{1}{11951} a^{16} - \frac{79}{11951} a^{15} - \frac{3357}{11951} a^{14} - \frac{5680}{11951} a^{13} + \frac{998}{11951} a^{12} - \frac{1895}{11951} a^{11} + \frac{3269}{11951} a^{10} - \frac{4784}{11951} a^{9} + \frac{3387}{11951} a^{8} + \frac{13}{703} a^{7} + \frac{383}{11951} a^{6} - \frac{25}{703} a^{5} - \frac{15}{11951} a^{4} - \frac{2690}{11951} a^{3} - \frac{3352}{11951} a^{2} + \frac{5943}{11951} a - \frac{162}{629}$, $\frac{1}{509300638899266853915999667074317784326041144919969} a^{17} + \frac{19127334133113134087456919267113917883832268733}{509300638899266853915999667074317784326041144919969} a^{16} - \frac{5810161656613364021379142547637368406802771367147}{509300638899266853915999667074317784326041144919969} a^{15} - \frac{2329894828195390841872034746392286144010966036813}{13764882132412617673405396407413994170974084997837} a^{14} + \frac{70407632135264114869853839332096744637466020760506}{509300638899266853915999667074317784326041144919969} a^{13} + \frac{7397567475527689876344317369221842728160057506556}{509300638899266853915999667074317784326041144919969} a^{12} - \frac{228487560218540371978936041936128158477910742440081}{509300638899266853915999667074317784326041144919969} a^{11} - \frac{195074516258936067068001507079635667169201278099753}{509300638899266853915999667074317784326041144919969} a^{10} - \frac{20730343689121387216684163660128476221255245802125}{509300638899266853915999667074317784326041144919969} a^{9} - \frac{5142644328277885501511134172322417270746460005378}{13764882132412617673405396407413994170974084997837} a^{8} + \frac{3585557245052225504572910175091436036117011852386}{13764882132412617673405396407413994170974084997837} a^{7} + \frac{88179445989434056041026201298883540756814536920308}{509300638899266853915999667074317784326041144919969} a^{6} + \frac{37785833091694561103830090794971457007722071341504}{509300638899266853915999667074317784326041144919969} a^{5} + \frac{120352084488393218425200845653060493911957894246099}{509300638899266853915999667074317784326041144919969} a^{4} + \frac{98550361965848002977255480136598425976307680438091}{509300638899266853915999667074317784326041144919969} a^{3} - \frac{48153791520188838396548388021428677604810775867721}{509300638899266853915999667074317784326041144919969} a^{2} - \frac{180829047719834633631526307911109805989127375570958}{509300638899266853915999667074317784326041144919969} a + \frac{203546487740320840164811526912267387615471493301595}{509300638899266853915999667074317784326041144919969}$

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

Class group and class number

$C_{42}$, which has order $42$ (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:  \( \frac{1445875642572562332313926208183127253753073412}{509300638899266853915999667074317784326041144919969} a^{17} - \frac{4635431422123556184503740711568738028639883750}{509300638899266853915999667074317784326041144919969} a^{16} + \frac{7092745178024153083405730731284320254970940511}{509300638899266853915999667074317784326041144919969} a^{15} - \frac{66054825934729253201758217207552805983716623030}{509300638899266853915999667074317784326041144919969} a^{14} - \frac{19064129176665112313121173637824370904243235848}{509300638899266853915999667074317784326041144919969} a^{13} + \frac{30331169502187165638311003333948178283280858700}{26805296784171939679789456161806199175054797101051} a^{12} + \frac{1606778283964503106474802065127616209795428210061}{509300638899266853915999667074317784326041144919969} a^{11} - \frac{1075955934545413934809443123145364647323719148974}{509300638899266853915999667074317784326041144919969} a^{10} - \frac{37341490144126189123023282216409082645822915213085}{509300638899266853915999667074317784326041144919969} a^{9} + \frac{77904925057635683330746863485018974755853209650719}{509300638899266853915999667074317784326041144919969} a^{8} + \frac{45712932905815854579270173768213019467354254371283}{509300638899266853915999667074317784326041144919969} a^{7} - \frac{381412595150743404004800017074985226061617409152014}{509300638899266853915999667074317784326041144919969} a^{6} + \frac{755546387946926908891555328263656363401929215878091}{509300638899266853915999667074317784326041144919969} a^{5} - \frac{37756723607303196149252020643817550901465156553084}{29958861111721579642117627474959869666237714407057} a^{4} - \frac{13711970204233617030195318194505675919624397897599}{509300638899266853915999667074317784326041144919969} a^{3} + \frac{845453952649337101940082455125523226802012941870798}{509300638899266853915999667074317784326041144919969} a^{2} - \frac{1257007723053708362612642645966110952654244365447448}{509300638899266853915999667074317784326041144919969} a - \frac{907822144081140555280144824140440363154900380084289}{509300638899266853915999667074317784326041144919969} \) (order $18$)
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:  \( 5933010.89765 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T201:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 26 conjugacy class representatives for t18n201
Character table for t18n201 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 9.9.297070974648009.1

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

Sibling fields

Degree 18 siblings: 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 $18$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{10}$ $18$ $18$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ R $18$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ $18$

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
3Data not computed
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$