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Properties

Label 18.12.4964669177...9487.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,3^{24}\cdot 7^{12}\cdot 127$
Root discriminant $20.72$
Ramified primes $3, 7, 127$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T459

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -18, -24, 75, 111, -139, -261, 198, 210, -132, 3, -110, 90, 27, -60, 33, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 33*x^16 - 60*x^15 + 27*x^14 + 90*x^13 - 110*x^12 + 3*x^11 - 132*x^10 + 210*x^9 + 198*x^8 - 261*x^7 - 139*x^6 + 111*x^5 + 75*x^4 - 24*x^3 - 18*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^18 - 9*x^17 + 33*x^16 - 60*x^15 + 27*x^14 + 90*x^13 - 110*x^12 + 3*x^11 - 132*x^10 + 210*x^9 + 198*x^8 - 261*x^7 - 139*x^6 + 111*x^5 + 75*x^4 - 24*x^3 - 18*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 33 x^{16} - 60 x^{15} + 27 x^{14} + 90 x^{13} - 110 x^{12} + 3 x^{11} - 132 x^{10} + 210 x^{9} + 198 x^{8} - 261 x^{7} - 139 x^{6} + 111 x^{5} + 75 x^{4} - 24 x^{3} - 18 x^{2} + 3 x + 1 \)

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:  \(-496466917716811140719487=-\,3^{24}\cdot 7^{12}\cdot 127\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.72$
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:  $3, 7, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{22376016466147} a^{17} + \frac{601700064010}{22376016466147} a^{16} + \frac{11035661110413}{22376016466147} a^{15} + \frac{7864034978183}{22376016466147} a^{14} + \frac{5079901926299}{22376016466147} a^{13} - \frac{2999148882949}{22376016466147} a^{12} - \frac{7998414045656}{22376016466147} a^{11} - \frac{8923070516289}{22376016466147} a^{10} + \frac{8222201401600}{22376016466147} a^{9} + \frac{2013773667163}{22376016466147} a^{8} - \frac{4892748544370}{22376016466147} a^{7} + \frac{10848691556256}{22376016466147} a^{6} - \frac{9528144808547}{22376016466147} a^{5} - \frac{1252599935505}{22376016466147} a^{4} + \frac{308534327184}{22376016466147} a^{3} - \frac{1831205958387}{22376016466147} a^{2} - \frac{2321700301002}{22376016466147} a + \frac{5871123624143}{22376016466147}$

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:  \( 244676.881686 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T459:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 96 conjugacy class representatives for t18n459 are not computed
Character table for t18n459 is not computed

Intermediate fields

3.3.3969.2, \(\Q(\zeta_{9})^+\), 3.3.3969.1, \(\Q(\zeta_{7})^+\), 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
3Data not computed
7Data not computed
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$