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Properties

Label 18.12.4159994509...9611.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,3^{27}\cdot 73^{4}\cdot 577^{3}$
Root discriminant $38.90$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1009, 318, -11655, 14308, 16926, -42879, 16988, 22929, -25929, 5574, 6198, -5115, 1164, 459, -390, 101, 0, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 101*x^15 - 390*x^14 + 459*x^13 + 1164*x^12 - 5115*x^11 + 6198*x^10 + 5574*x^9 - 25929*x^8 + 22929*x^7 + 16988*x^6 - 42879*x^5 + 16926*x^4 + 14308*x^3 - 11655*x^2 + 318*x + 1009)
 
gp: K = bnfinit(x^18 - 6*x^17 + 101*x^15 - 390*x^14 + 459*x^13 + 1164*x^12 - 5115*x^11 + 6198*x^10 + 5574*x^9 - 25929*x^8 + 22929*x^7 + 16988*x^6 - 42879*x^5 + 16926*x^4 + 14308*x^3 - 11655*x^2 + 318*x + 1009, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 101 x^{15} - 390 x^{14} + 459 x^{13} + 1164 x^{12} - 5115 x^{11} + 6198 x^{10} + 5574 x^{9} - 25929 x^{8} + 22929 x^{7} + 16988 x^{6} - 42879 x^{5} + 16926 x^{4} + 14308 x^{3} - 11655 x^{2} + 318 x + 1009 \)

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:  \(-41599945090131789253856059611=-\,3^{27}\cdot 73^{4}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.90$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} - \frac{1}{19} a^{14} + \frac{1}{19} a^{13} + \frac{4}{19} a^{12} - \frac{2}{19} a^{11} - \frac{3}{19} a^{10} - \frac{7}{19} a^{9} + \frac{2}{19} a^{8} - \frac{6}{19} a^{7} - \frac{4}{19} a^{6} + \frac{5}{19} a^{5} + \frac{3}{19} a^{4} - \frac{6}{19} a^{3} + \frac{6}{19} a^{2} + \frac{7}{19} a + \frac{2}{19}$, $\frac{1}{19} a^{16} + \frac{5}{19} a^{13} + \frac{2}{19} a^{12} - \frac{5}{19} a^{11} + \frac{9}{19} a^{10} - \frac{5}{19} a^{9} - \frac{4}{19} a^{8} + \frac{9}{19} a^{7} + \frac{1}{19} a^{6} + \frac{8}{19} a^{5} - \frac{3}{19} a^{4} - \frac{6}{19} a^{2} + \frac{9}{19} a + \frac{2}{19}$, $\frac{1}{111632319252865681379793847} a^{17} - \frac{333730058797560834049738}{111632319252865681379793847} a^{16} + \frac{2158680945407961356155362}{111632319252865681379793847} a^{15} - \frac{2192275026641700373848382}{111632319252865681379793847} a^{14} - \frac{11968837358469351170327794}{111632319252865681379793847} a^{13} + \frac{22950263749788749391650861}{111632319252865681379793847} a^{12} + \frac{41601372550086952921624641}{111632319252865681379793847} a^{11} + \frac{25395234405857242550774437}{111632319252865681379793847} a^{10} - \frac{22521462407798338476682890}{111632319252865681379793847} a^{9} - \frac{33719096784173601137250075}{111632319252865681379793847} a^{8} - \frac{34025723346535229324773071}{111632319252865681379793847} a^{7} - \frac{18200039317831679434475281}{111632319252865681379793847} a^{6} + \frac{49941233305243220715218733}{111632319252865681379793847} a^{5} - \frac{677035764878858187745949}{5875385223835035862094413} a^{4} + \frac{42534187850265331824028876}{111632319252865681379793847} a^{3} - \frac{18831556811371181911515583}{111632319252865681379793847} a^{2} + \frac{38572585549017062244073197}{111632319252865681379793847} a + \frac{24552124001435022470039492}{111632319252865681379793847}$

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:  \( 62896797.1415 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

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 t18n767 are not computed
Character table for t18n767 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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
73Data not computed
577Data not computed