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Properties

Label 18.0.69197667779...3184.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 1129^{8}$
Root discriminant $45.48$
Ramified primes $2, 1129$
Class number $178$ (GRH)
Class group $[178]$ (GRH)
Galois group $D_{18}$ (as 18T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 3756, 0, 13258, 0, 19532, 0, 15603, 0, 7345, 0, 2070, 0, 338, 0, 29, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 29*x^16 + 338*x^14 + 2070*x^12 + 7345*x^10 + 15603*x^8 + 19532*x^6 + 13258*x^4 + 3756*x^2 + 1)
 
gp: K = bnfinit(x^18 + 29*x^16 + 338*x^14 + 2070*x^12 + 7345*x^10 + 15603*x^8 + 19532*x^6 + 13258*x^4 + 3756*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 29 x^{16} + 338 x^{14} + 2070 x^{12} + 7345 x^{10} + 15603 x^{8} + 19532 x^{6} + 13258 x^{4} + 3756 x^{2} + 1 \)

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:  \(-691976677792049302709773533184=-\,2^{18}\cdot 1129^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.48$
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:  $2, 1129$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} + \frac{5}{19} a^{12} + \frac{7}{19} a^{10} - \frac{8}{19} a^{8} - \frac{1}{19} a^{6} + \frac{6}{19} a^{4} - \frac{9}{19} a^{2} - \frac{9}{19}$, $\frac{1}{19} a^{15} + \frac{5}{19} a^{13} + \frac{7}{19} a^{11} - \frac{8}{19} a^{9} - \frac{1}{19} a^{7} + \frac{6}{19} a^{5} - \frac{9}{19} a^{3} - \frac{9}{19} a$, $\frac{1}{60401} a^{16} + \frac{529}{60401} a^{14} + \frac{7339}{60401} a^{12} + \frac{28436}{60401} a^{10} - \frac{1162}{3553} a^{8} - \frac{9676}{60401} a^{6} + \frac{1553}{3179} a^{4} - \frac{6055}{60401} a^{2} + \frac{18559}{60401}$, $\frac{1}{60401} a^{17} + \frac{529}{60401} a^{15} + \frac{7339}{60401} a^{13} + \frac{28436}{60401} a^{11} - \frac{1162}{3553} a^{9} - \frac{9676}{60401} a^{7} + \frac{1553}{3179} a^{5} - \frac{6055}{60401} a^{3} + \frac{18559}{60401} a$

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

Class group and class number

$C_{178}$, which has order $178$ (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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{4585}{60401} a^{17} + \frac{123869}{60401} a^{15} + \frac{1302990}{60401} a^{13} + \frac{6874910}{60401} a^{11} + \frac{1155894}{3553} a^{9} + \frac{30055930}{60401} a^{7} + \frac{20972755}{60401} a^{5} + \frac{1408329}{60401} a^{3} - \frac{3699634}{60401} a \) (order $4$)
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:  \( 290542.983381 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{18}$ (as 18T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $D_{18}$
Character table for $D_{18}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.3.1129.1, 6.0.81577024.1, 9.9.1624709678881.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 18 sibling: 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
1129Data not computed