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Properties

Label 18.12.5189863033...6207.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{14}\cdot 83^{5}\cdot 181^{5}$
Root discriminant $65.69$
Ramified primes $7, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3031, -49210, -222992, -171844, 616323, 716395, -453143, -636218, -22645, 144577, 85568, 15250, -3341, -1277, -188, -15, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^16 - 15*x^15 - 188*x^14 - 1277*x^13 - 3341*x^12 + 15250*x^11 + 85568*x^10 + 144577*x^9 - 22645*x^8 - 636218*x^7 - 453143*x^6 + 716395*x^5 + 616323*x^4 - 171844*x^3 - 222992*x^2 - 49210*x - 3031)
 
gp: K = bnfinit(x^18 - 7*x^16 - 15*x^15 - 188*x^14 - 1277*x^13 - 3341*x^12 + 15250*x^11 + 85568*x^10 + 144577*x^9 - 22645*x^8 - 636218*x^7 - 453143*x^6 + 716395*x^5 + 616323*x^4 - 171844*x^3 - 222992*x^2 - 49210*x - 3031, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{16} - 15 x^{15} - 188 x^{14} - 1277 x^{13} - 3341 x^{12} + 15250 x^{11} + 85568 x^{10} + 144577 x^{9} - 22645 x^{8} - 636218 x^{7} - 453143 x^{6} + 716395 x^{5} + 616323 x^{4} - 171844 x^{3} - 222992 x^{2} - 49210 x - 3031 \)

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:  \(-518986303303243273250774395826207=-\,7^{14}\cdot 83^{5}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.69$
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:  $7, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{4}{13} a^{13} - \frac{4}{13} a^{12} + \frac{1}{13} a^{11} - \frac{1}{13} a^{10} - \frac{6}{13} a^{9} - \frac{4}{13} a^{8} - \frac{5}{13} a^{7} - \frac{5}{13} a^{6} - \frac{6}{13} a^{5} - \frac{4}{13} a^{3} - \frac{4}{13} a^{2} - \frac{4}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{15} + \frac{6}{13} a^{13} + \frac{4}{13} a^{12} - \frac{5}{13} a^{11} - \frac{2}{13} a^{10} - \frac{6}{13} a^{9} - \frac{2}{13} a^{8} + \frac{2}{13} a^{7} + \frac{1}{13} a^{6} - \frac{2}{13} a^{5} - \frac{4}{13} a^{4} - \frac{1}{13} a^{3} - \frac{1}{13} a^{2} - \frac{5}{13} a + \frac{6}{13}$, $\frac{1}{13} a^{16} + \frac{6}{13} a^{13} + \frac{6}{13} a^{12} + \frac{5}{13} a^{11} - \frac{5}{13} a^{9} + \frac{5}{13} a^{7} + \frac{2}{13} a^{6} + \frac{6}{13} a^{5} - \frac{1}{13} a^{4} - \frac{3}{13} a^{3} + \frac{6}{13} a^{2} + \frac{4}{13} a - \frac{4}{13}$, $\frac{1}{183548506573190062581664007260735744599582973816397} a^{17} + \frac{5508055883180753590022509774569950835943136265851}{183548506573190062581664007260735744599582973816397} a^{16} + \frac{2239091054442219883809859959012686767206489875206}{183548506573190062581664007260735744599582973816397} a^{15} + \frac{513656638514146700135060494874846291251923651281}{14119115890245389429358769789287364969198690293569} a^{14} - \frac{43366207358667537608853349759955162536757278387879}{183548506573190062581664007260735744599582973816397} a^{13} - \frac{49048083779745640324460156589251891427504445391863}{183548506573190062581664007260735744599582973816397} a^{12} + \frac{23220972388383894588085815455957018114264871802356}{183548506573190062581664007260735744599582973816397} a^{11} + \frac{43149010476012809228770386009705893174351641659261}{183548506573190062581664007260735744599582973816397} a^{10} + \frac{12057352590741661730248333716987803183986407861937}{183548506573190062581664007260735744599582973816397} a^{9} - \frac{46169577321072367542108782279933069357682901394171}{183548506573190062581664007260735744599582973816397} a^{8} - \frac{1835684917721619751637112818243909143822789317561}{183548506573190062581664007260735744599582973816397} a^{7} - \frac{8664162567401910745179728677912894575091640473043}{183548506573190062581664007260735744599582973816397} a^{6} - \frac{56923944958636775836732238278278774954914740449226}{183548506573190062581664007260735744599582973816397} a^{5} + \frac{72679442667598658478071740995747214919436685831508}{183548506573190062581664007260735744599582973816397} a^{4} - \frac{21103133531418564386393650563472193766269943416403}{183548506573190062581664007260735744599582973816397} a^{3} + \frac{23508885863711078888339745649958754142338911631113}{183548506573190062581664007260735744599582973816397} a^{2} + \frac{62685047963003207790384533125823485561820751407746}{183548506573190062581664007260735744599582973816397} a - \frac{3904050755004052861204096811527796823660469433784}{14119115890245389429358769789287364969198690293569}$

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:  \( 16013216431.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 192 conjugacy class representatives for t18n839 are not computed
Character table for t18n839 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
83Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.4.3.4$x^{4} + 1448$$4$$1$$3$$C_4$$[\ ]_{4}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$