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Properties

Label 18.12.8071667041...8416.2
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 7^{12}\cdot 71^{2}$
Root discriminant $40.36$
Ramified primes $2, 3, 7, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T459

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![127, 558, -1791, -7179, -228, 14742, 7440, -8799, -6411, 899, 786, -309, 170, 36, -12, 55, -18, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 18*x^16 + 55*x^15 - 12*x^14 + 36*x^13 + 170*x^12 - 309*x^11 + 786*x^10 + 899*x^9 - 6411*x^8 - 8799*x^7 + 7440*x^6 + 14742*x^5 - 228*x^4 - 7179*x^3 - 1791*x^2 + 558*x + 127)
 
gp: K = bnfinit(x^18 - 3*x^17 - 18*x^16 + 55*x^15 - 12*x^14 + 36*x^13 + 170*x^12 - 309*x^11 + 786*x^10 + 899*x^9 - 6411*x^8 - 8799*x^7 + 7440*x^6 + 14742*x^5 - 228*x^4 - 7179*x^3 - 1791*x^2 + 558*x + 127, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 18 x^{16} + 55 x^{15} - 12 x^{14} + 36 x^{13} + 170 x^{12} - 309 x^{11} + 786 x^{10} + 899 x^{9} - 6411 x^{8} - 8799 x^{7} + 7440 x^{6} + 14742 x^{5} - 228 x^{4} - 7179 x^{3} - 1791 x^{2} + 558 x + 127 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-80716670418377815414668988416=-\,2^{12}\cdot 3^{24}\cdot 7^{12}\cdot 71^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.36$
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:  $2, 3, 7, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{40} a^{15} + \frac{1}{10} a^{14} + \frac{1}{40} a^{13} + \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{9} + \frac{3}{20} a^{8} + \frac{13}{40} a^{7} - \frac{19}{40} a^{6} + \frac{13}{40} a^{5} - \frac{1}{4} a^{4} - \frac{1}{5} a^{3} - \frac{17}{40} a^{2} - \frac{1}{8} a + \frac{7}{40}$, $\frac{1}{80} a^{16} - \frac{1}{80} a^{15} - \frac{9}{80} a^{14} - \frac{1}{8} a^{13} + \frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{1}{16} a^{10} - \frac{39}{80} a^{9} - \frac{37}{80} a^{8} - \frac{3}{10} a^{7} - \frac{1}{40} a^{6} + \frac{5}{16} a^{5} - \frac{1}{10} a^{4} - \frac{27}{80} a^{3} + \frac{3}{8} a^{2} - \frac{1}{10} a - \frac{5}{16}$, $\frac{1}{528807304436529247256141723360} a^{17} + \frac{69130233664261526731189444}{16525228263641538976754428855} a^{16} - \frac{2531720896460518194004864259}{264403652218264623628070861680} a^{15} + \frac{21452383878585410131426564089}{528807304436529247256141723360} a^{14} + \frac{10394968929289033237193235871}{105761460887305849451228344672} a^{13} - \frac{4331513135254283921740001471}{105761460887305849451228344672} a^{12} + \frac{11853923913170702676048779185}{105761460887305849451228344672} a^{11} - \frac{77967754211902116192151110207}{264403652218264623628070861680} a^{10} - \frac{64531344074253321130306338677}{132201826109132311814035430840} a^{9} - \frac{210761386836733372033857742757}{528807304436529247256141723360} a^{8} - \frac{45094829959461108009186577499}{264403652218264623628070861680} a^{7} + \frac{2380572207137028335088283487}{528807304436529247256141723360} a^{6} + \frac{251866183681848638620591570317}{528807304436529247256141723360} a^{5} - \frac{54860286937191076295601354439}{528807304436529247256141723360} a^{4} - \frac{190267308111013524271676819393}{528807304436529247256141723360} a^{3} + \frac{81765095688448778659220943611}{264403652218264623628070861680} a^{2} + \frac{214521305427862272233767404003}{528807304436529247256141723360} a - \frac{36655762133797112494633289829}{105761460887305849451228344672}$

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:  $14$
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:  \( 143010703.79 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T459:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 96 conjugacy class representatives for t18n459 are not computed
Character table for t18n459 is not computed

Intermediate fields

3.3.3969.2, \(\Q(\zeta_{9})^+\), 3.3.3969.1, \(\Q(\zeta_{7})^+\), 9.9.62523502209.1

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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
3Data not computed
7Data not computed
71Data not computed