Properties

Label 18.0.12827848719...6283.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 1297^{2}$
Root discriminant $11.52$
Ramified primes $3, 1297$
Class number $1$
Class group Trivial
Galois group 18T286

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 39, -106, 198, -270, 274, -186, 63, 43, -87, 90, -47, 18, 3, -9, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 9*x^15 + 3*x^14 + 18*x^13 - 47*x^12 + 90*x^11 - 87*x^10 + 43*x^9 + 63*x^8 - 186*x^7 + 274*x^6 - 270*x^5 + 198*x^4 - 106*x^3 + 39*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 - 9*x^15 + 3*x^14 + 18*x^13 - 47*x^12 + 90*x^11 - 87*x^10 + 43*x^9 + 63*x^8 - 186*x^7 + 274*x^6 - 270*x^5 + 198*x^4 - 106*x^3 + 39*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 6 x^{16} - 9 x^{15} + 3 x^{14} + 18 x^{13} - 47 x^{12} + 90 x^{11} - 87 x^{10} + 43 x^{9} + 63 x^{8} - 186 x^{7} + 274 x^{6} - 270 x^{5} + 198 x^{4} - 106 x^{3} + 39 x^{2} - 9 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-12827848719622496283=-\,3^{27}\cdot 1297^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.52$
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:  $3, 1297$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11331399151} a^{17} - \frac{5012802240}{11331399151} a^{16} - \frac{332300229}{11331399151} a^{15} - \frac{3807216236}{11331399151} a^{14} + \frac{1648854709}{11331399151} a^{13} - \frac{76066867}{11331399151} a^{12} - \frac{3914875299}{11331399151} a^{11} + \frac{3602980441}{11331399151} a^{10} + \frac{1405698519}{11331399151} a^{9} + \frac{204808519}{11331399151} a^{8} + \frac{2080157976}{11331399151} a^{7} + \frac{1759198270}{11331399151} a^{6} + \frac{4074136241}{11331399151} a^{5} - \frac{1683044718}{11331399151} a^{4} + \frac{1573467686}{11331399151} a^{3} + \frac{2111109130}{11331399151} a^{2} - \frac{2166427783}{11331399151} a + \frac{2225142671}{11331399151}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{35352706938}{11331399151} a^{17} + \frac{90854094538}{11331399151} a^{16} - \frac{172876998924}{11331399151} a^{15} + \frac{242261203095}{11331399151} a^{14} + \frac{1668131236}{11331399151} a^{13} - \frac{641319898088}{11331399151} a^{12} + \frac{1391708984120}{11331399151} a^{11} - \frac{2577536726340}{11331399151} a^{10} + \frac{1939234074618}{11331399151} a^{9} - \frac{636677436344}{11331399151} a^{8} - \frac{2576010774093}{11331399151} a^{7} + \frac{5507921744460}{11331399151} a^{6} - \frac{7298219065135}{11331399151} a^{5} + \frac{6299662986481}{11331399151} a^{4} - \frac{4104563025992}{11331399151} a^{3} + \frac{1784458218203}{11331399151} a^{2} - \frac{470870344327}{11331399151} a + \frac{52125926226}{11331399151} \) (order $18$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 784.727413524 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T286:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 34 conjugacy class representatives for t18n286
Character table for t18n286 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 9.5.689278977.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
1297Data not computed