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Properties

Label 18.18.2638204387...8125.1
Degree $18$
Signature $[18, 0]$
Discriminant $5^{4}\cdot 13^{17}\cdot 47^{4}$
Root discriminant $37.93$
Ramified primes $5, 13, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T201

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 33, 238, 29, -2204, -1511, 6383, 3367, -7761, -3094, 4732, 1430, -1573, -347, 288, 42, -27, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 27*x^16 + 42*x^15 + 288*x^14 - 347*x^13 - 1573*x^12 + 1430*x^11 + 4732*x^10 - 3094*x^9 - 7761*x^8 + 3367*x^7 + 6383*x^6 - 1511*x^5 - 2204*x^4 + 29*x^3 + 238*x^2 + 33*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 27*x^16 + 42*x^15 + 288*x^14 - 347*x^13 - 1573*x^12 + 1430*x^11 + 4732*x^10 - 3094*x^9 - 7761*x^8 + 3367*x^7 + 6383*x^6 - 1511*x^5 - 2204*x^4 + 29*x^3 + 238*x^2 + 33*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 27 x^{16} + 42 x^{15} + 288 x^{14} - 347 x^{13} - 1573 x^{12} + 1430 x^{11} + 4732 x^{10} - 3094 x^{9} - 7761 x^{8} + 3367 x^{7} + 6383 x^{6} - 1511 x^{5} - 2204 x^{4} + 29 x^{3} + 238 x^{2} + 33 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26382043877439154041399608125=5^{4}\cdot 13^{17}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.93$
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:  $5, 13, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{18363793509725} a^{17} - \frac{1759025429706}{18363793509725} a^{16} - \frac{999125030353}{18363793509725} a^{15} - \frac{5181301763821}{18363793509725} a^{14} - \frac{1656534531103}{18363793509725} a^{13} + \frac{1253742884448}{3672758701945} a^{12} - \frac{1550139626608}{18363793509725} a^{11} - \frac{4345154562038}{18363793509725} a^{10} + \frac{6203264533709}{18363793509725} a^{9} - \frac{1011842661151}{3672758701945} a^{8} + \frac{4576194792934}{18363793509725} a^{7} + \frac{4873689974031}{18363793509725} a^{6} - \frac{8001898259316}{18363793509725} a^{5} - \frac{1835151572872}{18363793509725} a^{4} - \frac{6321545334366}{18363793509725} a^{3} - \frac{918485934882}{18363793509725} a^{2} + \frac{8911599754641}{18363793509725} a + \frac{1826315489769}{18363793509725}$

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:  $17$
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:  \( 216261759.31 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T201:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 26 conjugacy class representatives for t18n201
Character table for t18n201 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{13})^+\), 9.9.45048729067225.1

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ $18$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ $18$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13Data not computed
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$