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Properties

Label 18.14.1484875323...7936.1
Degree $18$
Signature $[14, 2]$
Discriminant $2^{18}\cdot 3^{27}\cdot 73\cdot 3189897037^{2}$
Root discriminant $150.04$
Ramified primes $2, 3, 73, 3189897037$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T857

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4128916321, -29023484352, 50028443868, -34368736612, 5498508363, 5466517950, -2854359357, 64877160, 276982620, -55245248, -9250809, 3906366, -45249, -114246, 9183, 1418, -174, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 174*x^16 + 1418*x^15 + 9183*x^14 - 114246*x^13 - 45249*x^12 + 3906366*x^11 - 9250809*x^10 - 55245248*x^9 + 276982620*x^8 + 64877160*x^7 - 2854359357*x^6 + 5466517950*x^5 + 5498508363*x^4 - 34368736612*x^3 + 50028443868*x^2 - 29023484352*x + 4128916321)
 
gp: K = bnfinit(x^18 - 6*x^17 - 174*x^16 + 1418*x^15 + 9183*x^14 - 114246*x^13 - 45249*x^12 + 3906366*x^11 - 9250809*x^10 - 55245248*x^9 + 276982620*x^8 + 64877160*x^7 - 2854359357*x^6 + 5466517950*x^5 + 5498508363*x^4 - 34368736612*x^3 + 50028443868*x^2 - 29023484352*x + 4128916321, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 174 x^{16} + 1418 x^{15} + 9183 x^{14} - 114246 x^{13} - 45249 x^{12} + 3906366 x^{11} - 9250809 x^{10} - 55245248 x^{9} + 276982620 x^{8} + 64877160 x^{7} - 2854359357 x^{6} + 5466517950 x^{5} + 5498508363 x^{4} - 34368736612 x^{3} + 50028443868 x^{2} - 29023484352 x + 4128916321 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1484875323272941639219759100572157607936=2^{18}\cdot 3^{27}\cdot 73\cdot 3189897037^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $150.04$
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:  $2, 3, 73, 3189897037$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{9} a^{12} + \frac{1}{3} a^{11} - \frac{4}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} + \frac{1}{3} a^{5} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{4}{9} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{2}{9} a^{7} - \frac{1}{3} a^{6} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{14} - \frac{4}{9} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{2}{9} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{14} - \frac{1}{18} a^{13} + \frac{2}{9} a^{11} - \frac{5}{18} a^{10} - \frac{4}{9} a^{9} - \frac{4}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{4}{9} a^{4} + \frac{1}{6} a^{3} + \frac{5}{18} a^{2} - \frac{2}{9} a + \frac{7}{18}$, $\frac{1}{18} a^{16} - \frac{1}{18} a^{13} - \frac{1}{6} a^{11} - \frac{1}{18} a^{10} - \frac{1}{3} a^{9} + \frac{1}{9} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{7}{18} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{2964869179965995750123605545495807823588164674254248606} a^{17} - \frac{63814508856622960980228247435220816037773277029498227}{2964869179965995750123605545495807823588164674254248606} a^{16} - \frac{7192465542035681855620865772043183854595114180576249}{988289726655331916707868515165269274529388224751416202} a^{15} - \frac{67152977449221644736710680327821266765283629444849446}{1482434589982997875061802772747903911794082337127124303} a^{14} - \frac{51539912636583828838227901516727829516011233479249005}{1482434589982997875061802772747903911794082337127124303} a^{13} + \frac{50188072717947190496020814248771823308653776071342405}{2964869179965995750123605545495807823588164674254248606} a^{12} - \frac{172831409088752230117188927902229252388256045352652807}{494144863327665958353934257582634637264694112375708101} a^{11} - \frac{44969591521178465797192911690640492240216574308622}{182813489947342196949291253267715367097556090409067} a^{10} + \frac{693723757367091953283564180834758437063798370522642014}{1482434589982997875061802772747903911794082337127124303} a^{9} + \frac{164863250590655908883273366554935125073951607405352735}{494144863327665958353934257582634637264694112375708101} a^{8} + \frac{56384218600762719697215651299798931648638116056891848}{164714954442555319451311419194211545754898037458569367} a^{7} - \frac{717181523828700920826336790473505375654271184009188423}{1482434589982997875061802772747903911794082337127124303} a^{6} + \frac{362765074694220724531766986023414650549005655719942745}{2964869179965995750123605545495807823588164674254248606} a^{5} - \frac{734296255280753447520543786365743431773749220115053285}{2964869179965995750123605545495807823588164674254248606} a^{4} - \frac{16407301791850154254104167468435917730095617728698757}{1482434589982997875061802772747903911794082337127124303} a^{3} + \frac{901660108316097757072707366345001190815461838918591763}{2964869179965995750123605545495807823588164674254248606} a^{2} - \frac{512005569822563115818770074031342674729793362676351072}{1482434589982997875061802772747903911794082337127124303} a - \frac{6080574818444334446294227586678630574270255364885072}{40065799729270212839508183047240646264704928030462819}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 43097163361600 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T857:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 159 conjugacy class representatives for t18n857 are not computed
Character table for t18n857 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{36})^+\)

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

Sibling fields

Degree 18 sibling: 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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{7}$ $18$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
73Data not computed
3189897037Data not computed