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Properties

Label 18.0.15544228349...0784.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{6}\cdot 3^{18}\cdot 7^{12}\cdot 673^{2}$
Root discriminant $28.52$
Ramified primes $2, 3, 7, 673$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8191, -40815, 87003, -120978, 140685, -137937, 112022, -80064, 52227, -29916, 15258, -7215, 3033, -1113, 384, -114, 27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 27*x^16 - 114*x^15 + 384*x^14 - 1113*x^13 + 3033*x^12 - 7215*x^11 + 15258*x^10 - 29916*x^9 + 52227*x^8 - 80064*x^7 + 112022*x^6 - 137937*x^5 + 140685*x^4 - 120978*x^3 + 87003*x^2 - 40815*x + 8191)
 
gp: K = bnfinit(x^18 - 6*x^17 + 27*x^16 - 114*x^15 + 384*x^14 - 1113*x^13 + 3033*x^12 - 7215*x^11 + 15258*x^10 - 29916*x^9 + 52227*x^8 - 80064*x^7 + 112022*x^6 - 137937*x^5 + 140685*x^4 - 120978*x^3 + 87003*x^2 - 40815*x + 8191, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 27 x^{16} - 114 x^{15} + 384 x^{14} - 1113 x^{13} + 3033 x^{12} - 7215 x^{11} + 15258 x^{10} - 29916 x^{9} + 52227 x^{8} - 80064 x^{7} + 112022 x^{6} - 137937 x^{5} + 140685 x^{4} - 120978 x^{3} + 87003 x^{2} - 40815 x + 8191 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-155442283494504211376990784=-\,2^{6}\cdot 3^{18}\cdot 7^{12}\cdot 673^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.52$
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:  $2, 3, 7, 673$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{901765267249967805718065682015199} a^{17} - \frac{131006075447567577706724303623592}{901765267249967805718065682015199} a^{16} - \frac{57491777847920957202189751064104}{901765267249967805718065682015199} a^{15} + \frac{172880597118665443295467754922045}{901765267249967805718065682015199} a^{14} + \frac{360821445982785977184314722938139}{901765267249967805718065682015199} a^{13} - \frac{409666281823452214453113312489645}{901765267249967805718065682015199} a^{12} - \frac{164949136178090477611972139314788}{901765267249967805718065682015199} a^{11} - \frac{18716937220210246948650512956764}{901765267249967805718065682015199} a^{10} + \frac{92265774512171226596022891992265}{901765267249967805718065682015199} a^{9} + \frac{161955409080173737377207647675235}{901765267249967805718065682015199} a^{8} - \frac{228615961622628150279463804343864}{901765267249967805718065682015199} a^{7} - \frac{149624443888416798470526559524470}{901765267249967805718065682015199} a^{6} - \frac{135082050668424940547255010857782}{901765267249967805718065682015199} a^{5} - \frac{181375304587901193794412641740914}{901765267249967805718065682015199} a^{4} - \frac{340080002286487368647581332402985}{901765267249967805718065682015199} a^{3} - \frac{169868746035824864822359959090657}{901765267249967805718065682015199} a^{2} + \frac{248327042837611326422962616218172}{901765267249967805718065682015199} a - \frac{412712759692232887348513014873736}{901765267249967805718065682015199}$

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:  $8$
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:  \( 333782.575232 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.3.2315685267.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
7Data not computed
673Data not computed