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Properties

Label 18.8.13991415038...6391.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,3^{24}\cdot 7^{12}\cdot 71^{3}$
Root discriminant $32.22$
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![-503, 1203, 2397, -9349, 7647, -1506, 3116, -1536, -3813, 2212, -75, 369, -59, -30, -39, -8, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^15 - 39*x^14 - 30*x^13 - 59*x^12 + 369*x^11 - 75*x^10 + 2212*x^9 - 3813*x^8 - 1536*x^7 + 3116*x^6 - 1506*x^5 + 7647*x^4 - 9349*x^3 + 2397*x^2 + 1203*x - 503)
 
gp: K = bnfinit(x^18 - 8*x^15 - 39*x^14 - 30*x^13 - 59*x^12 + 369*x^11 - 75*x^10 + 2212*x^9 - 3813*x^8 - 1536*x^7 + 3116*x^6 - 1506*x^5 + 7647*x^4 - 9349*x^3 + 2397*x^2 + 1203*x - 503, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{15} - 39 x^{14} - 30 x^{13} - 59 x^{12} + 369 x^{11} - 75 x^{10} + 2212 x^{9} - 3813 x^{8} - 1536 x^{7} + 3116 x^{6} - 1506 x^{5} + 7647 x^{4} - 9349 x^{3} + 2397 x^{2} + 1203 x - 503 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1399141503834185765244506391=-\,3^{24}\cdot 7^{12}\cdot 71^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.22$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{18} a^{15} - \frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{1}{9} a^{12} + \frac{1}{3} a^{11} - \frac{1}{6} a^{10} + \frac{1}{3} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{5}{18} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{18}$, $\frac{1}{18} a^{16} - \frac{1}{18} a^{13} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} + \frac{2}{9} a^{7} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} - \frac{2}{9} a + \frac{1}{6}$, $\frac{1}{172864691130004631798610882582} a^{17} - \frac{4283861268348522212434685347}{172864691130004631798610882582} a^{16} + \frac{384619818497981083191385096}{28810781855000771966435147097} a^{15} + \frac{16638582429128432891679383261}{172864691130004631798610882582} a^{14} - \frac{10458366525622953983583435965}{172864691130004631798610882582} a^{13} + \frac{3593641953628322906172806477}{57621563710001543932870294194} a^{12} + \frac{4475758199697754297723224855}{9603593951666923988811715699} a^{11} - \frac{89370134648521005168951673}{9603593951666923988811715699} a^{10} - \frac{21203643588509399533709498965}{57621563710001543932870294194} a^{9} - \frac{2174045780671888443828067480}{86432345565002315899305441291} a^{8} + \frac{5624921016955184719257886051}{86432345565002315899305441291} a^{7} - \frac{8335209944426658073381852319}{57621563710001543932870294194} a^{6} + \frac{85855207494088579510633125}{9603593951666923988811715699} a^{5} - \frac{11106343769686565666283174803}{28810781855000771966435147097} a^{4} - \frac{617555051436466120507880876}{9603593951666923988811715699} a^{3} + \frac{40129449719778400653408485903}{172864691130004631798610882582} a^{2} - \frac{48621599381440786059378647603}{172864691130004631798610882582} a - \frac{11203677797884939036752061279}{57621563710001543932870294194}$

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:  $12$
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:  \( 5606146.70867 \) (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

3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed
$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$$0$Trivial$[\ ]$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$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}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$