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Properties

Label 18.12.5320678958...1867.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,3^{27}\cdot 7^{12}\cdot 71^{2}$
Root discriminant $30.53$
Ramified primes $3, 7, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![127, 4173, 648, -13241, 4263, 17313, -12972, -9180, 12687, 93, -5415, 1614, 1007, -558, -63, 80, -3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 3*x^16 + 80*x^15 - 63*x^14 - 558*x^13 + 1007*x^12 + 1614*x^11 - 5415*x^10 + 93*x^9 + 12687*x^8 - 9180*x^7 - 12972*x^6 + 17313*x^5 + 4263*x^4 - 13241*x^3 + 648*x^2 + 4173*x + 127)
 
gp: K = bnfinit(x^18 - 6*x^17 - 3*x^16 + 80*x^15 - 63*x^14 - 558*x^13 + 1007*x^12 + 1614*x^11 - 5415*x^10 + 93*x^9 + 12687*x^8 - 9180*x^7 - 12972*x^6 + 17313*x^5 + 4263*x^4 - 13241*x^3 + 648*x^2 + 4173*x + 127, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 3 x^{16} + 80 x^{15} - 63 x^{14} - 558 x^{13} + 1007 x^{12} + 1614 x^{11} - 5415 x^{10} + 93 x^{9} + 12687 x^{8} - 9180 x^{7} - 12972 x^{6} + 17313 x^{5} + 4263 x^{4} - 13241 x^{3} + 648 x^{2} + 4173 x + 127 \)

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:  \(-532067895824267826219741867=-\,3^{27}\cdot 7^{12}\cdot 71^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.53$
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:  $3, 7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{26} a^{15} - \frac{2}{13} a^{13} + \frac{1}{26} a^{12} + \frac{6}{13} a^{11} - \frac{7}{26} a^{10} - \frac{6}{13} a^{9} - \frac{4}{13} a^{8} + \frac{2}{13} a^{7} - \frac{4}{13} a^{6} - \frac{3}{26} a^{5} + \frac{1}{13} a^{4} - \frac{11}{26} a^{3} - \frac{5}{13} a^{2} + \frac{9}{26} a + \frac{6}{13}$, $\frac{1}{26} a^{16} - \frac{2}{13} a^{14} + \frac{1}{26} a^{13} + \frac{6}{13} a^{12} - \frac{7}{26} a^{11} - \frac{6}{13} a^{10} - \frac{4}{13} a^{9} + \frac{2}{13} a^{8} - \frac{4}{13} a^{7} - \frac{3}{26} a^{6} + \frac{1}{13} a^{5} - \frac{11}{26} a^{4} - \frac{5}{13} a^{3} + \frac{9}{26} a^{2} + \frac{6}{13} a$, $\frac{1}{47374607210950708} a^{17} + \frac{654037333101669}{47374607210950708} a^{16} - \frac{113172643364414}{11843651802737677} a^{15} - \frac{157221376962809}{1822100277344258} a^{14} - \frac{2133567609522625}{47374607210950708} a^{13} - \frac{10280077941272933}{47374607210950708} a^{12} - \frac{2195354020996679}{23687303605475354} a^{11} - \frac{2569122678585328}{11843651802737677} a^{10} + \frac{19845489787195207}{47374607210950708} a^{9} - \frac{6006344751724885}{23687303605475354} a^{8} + \frac{11620620119770825}{47374607210950708} a^{7} - \frac{527240355377005}{3644200554688516} a^{6} + \frac{670114897712497}{3644200554688516} a^{5} - \frac{9241115550788035}{23687303605475354} a^{4} - \frac{15980893506804207}{47374607210950708} a^{3} + \frac{4392066301774728}{11843651802737677} a^{2} + \frac{2967711416074536}{11843651802737677} a - \frac{7398498174552345}{47374607210950708}$

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:  \( 11410494.3151 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 48 conjugacy class representatives for t18n263
Character table for t18n263 is not computed

Intermediate fields

3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$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}$