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Properties

Label 18.8.56986226150...6707.2
Degree $18$
Signature $[8, 5]$
Discriminant $-\,3^{27}\cdot 73^{3}\cdot 577^{3}$
Root discriminant $30.65$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-53, -549, -72, 2763, 3078, -675, -2879, -1380, 930, 566, -396, -27, 82, 12, -12, -11, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 11*x^15 - 12*x^14 + 12*x^13 + 82*x^12 - 27*x^11 - 396*x^10 + 566*x^9 + 930*x^8 - 1380*x^7 - 2879*x^6 - 675*x^5 + 3078*x^4 + 2763*x^3 - 72*x^2 - 549*x - 53)
 
gp: K = bnfinit(x^18 - 6*x^17 + 15*x^16 - 11*x^15 - 12*x^14 + 12*x^13 + 82*x^12 - 27*x^11 - 396*x^10 + 566*x^9 + 930*x^8 - 1380*x^7 - 2879*x^6 - 675*x^5 + 3078*x^4 + 2763*x^3 - 72*x^2 - 549*x - 53, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 15 x^{16} - 11 x^{15} - 12 x^{14} + 12 x^{13} + 82 x^{12} - 27 x^{11} - 396 x^{10} + 566 x^{9} + 930 x^{8} - 1380 x^{7} - 2879 x^{6} - 675 x^{5} + 3078 x^{4} + 2763 x^{3} - 72 x^{2} - 549 x - 53 \)

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:  \(-569862261508654647313096707=-\,3^{27}\cdot 73^{3}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.65$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{25172553044342629521106031233} a^{17} + \frac{9833729144194309905642730985}{25172553044342629521106031233} a^{16} - \frac{8909625711716871933426039565}{25172553044342629521106031233} a^{15} + \frac{7166290248284086182343377391}{25172553044342629521106031233} a^{14} + \frac{2212310436649697669365065467}{25172553044342629521106031233} a^{13} - \frac{4311778546487525636739500722}{25172553044342629521106031233} a^{12} + \frac{522852222314849170600138702}{25172553044342629521106031233} a^{11} + \frac{12320930528582738887484691183}{25172553044342629521106031233} a^{10} + \frac{1530187816189964858693587756}{25172553044342629521106031233} a^{9} + \frac{7199059027316326322509572339}{25172553044342629521106031233} a^{8} - \frac{5006292901020427983212658795}{25172553044342629521106031233} a^{7} + \frac{8339281218908222472967212815}{25172553044342629521106031233} a^{6} - \frac{5661719452276898955061592247}{25172553044342629521106031233} a^{5} - \frac{5258147901016957186775977483}{25172553044342629521106031233} a^{4} + \frac{12358449339654085944993067746}{25172553044342629521106031233} a^{3} - \frac{1790671103784944153012236902}{25172553044342629521106031233} a^{2} + \frac{6086792567303548595450012760}{25172553044342629521106031233} a - \frac{5753829142408117053952029496}{25172553044342629521106031233}$

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:  \( 4057991.14929 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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 $18$ R $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $18$ $18$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
73Data not computed
577Data not computed