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Properties

Label 16.16.6124178019...5808.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 17^{15}\cdot 67^{4}\cdot 63443^{4}$
Root discriminant $1293.27$
Ramified primes $2, 17, 67, 63443$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5549872123325448369911507057, 0, -24953450052120111870485864, 0, 39890402242602894406078, 0, -31155934760944574792, 0, 13361907896488851, 0, -3278343720046, 0, 456624845, 0, -33473, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 33473*x^14 + 456624845*x^12 - 3278343720046*x^10 + 13361907896488851*x^8 - 31155934760944574792*x^6 + 39890402242602894406078*x^4 - 24953450052120111870485864*x^2 + 5549872123325448369911507057)
 
gp: K = bnfinit(x^16 - 33473*x^14 + 456624845*x^12 - 3278343720046*x^10 + 13361907896488851*x^8 - 31155934760944574792*x^6 + 39890402242602894406078*x^4 - 24953450052120111870485864*x^2 + 5549872123325448369911507057, 1)
 

Normalized defining polynomial

\( x^{16} - 33473 x^{14} + 456624845 x^{12} - 3278343720046 x^{10} + 13361907896488851 x^{8} - 31155934760944574792 x^{6} + 39890402242602894406078 x^{4} - 24953450052120111870485864 x^{2} + 5549872123325448369911507057 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(61241780195042689588946649619585484552657695735808=2^{16}\cdot 17^{15}\cdot 67^{4}\cdot 63443^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1293.27$
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:  $2, 17, 67, 63443$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4250681} a^{10} - \frac{33473}{4250681} a^{8} + \frac{1801978}{4250681} a^{6} - \frac{1748115}{4250681} a^{4} + \frac{13778}{4250681} a^{2}$, $\frac{1}{4250681} a^{11} - \frac{33473}{4250681} a^{9} + \frac{1801978}{4250681} a^{7} - \frac{1748115}{4250681} a^{5} + \frac{13778}{4250681} a^{3}$, $\frac{1}{18068288963761} a^{12} - \frac{33473}{18068288963761} a^{10} + \frac{456624845}{18068288963761} a^{8} - \frac{3278343720046}{18068288963761} a^{6} - \frac{8625936694289}{18068288963761} a^{4} - \frac{1685449}{4250681} a^{2}$, $\frac{1}{18068288963761} a^{13} - \frac{33473}{18068288963761} a^{11} + \frac{456624845}{18068288963761} a^{9} - \frac{3278343720046}{18068288963761} a^{7} - \frac{8625936694289}{18068288963761} a^{5} - \frac{1685449}{4250681} a^{3}$, $\frac{1}{104050097542623116564405343109943642383876124317951950656556497862184159} a^{14} - \frac{22802233764470181417224917371661128879714656206308540192}{1010195121773039966644712069028579052270641983669436414141325221962953} a^{12} + \frac{6131389448032500634820305050338989289137298816498454624074289595}{104050097542623116564405343109943642383876124317951950656556497862184159} a^{10} - \frac{46754766836114104211523629300935108305872240641538006735234795370588251}{104050097542623116564405343109943642383876124317951950656556497862184159} a^{8} + \frac{48846422284071546261378746290884919831642114210971343347652679239039794}{104050097542623116564405343109943642383876124317951950656556497862184159} a^{6} - \frac{12150856661432106243858268594324082552303586306630420247479992276}{24478453580172945597283198412194103105802605351460613171526279639} a^{4} + \frac{423995407588946710367885338013973972361331182064904810093}{5758713387377915585122289443078439220868986722706458840719} a^{2} + \frac{427129302437038734474866185788766908374114467108720}{1354774302606550711550052672284379660781175233499399}$, $\frac{1}{104050097542623116564405343109943642383876124317951950656556497862184159} a^{15} - \frac{22802233764470181417224917371661128879714656206308540192}{1010195121773039966644712069028579052270641983669436414141325221962953} a^{13} + \frac{6131389448032500634820305050338989289137298816498454624074289595}{104050097542623116564405343109943642383876124317951950656556497862184159} a^{11} - \frac{46754766836114104211523629300935108305872240641538006735234795370588251}{104050097542623116564405343109943642383876124317951950656556497862184159} a^{9} + \frac{48846422284071546261378746290884919831642114210971343347652679239039794}{104050097542623116564405343109943642383876124317951950656556497862184159} a^{7} - \frac{12150856661432106243858268594324082552303586306630420247479992276}{24478453580172945597283198412194103105802605351460613171526279639} a^{5} + \frac{423995407588946710367885338013973972361331182064904810093}{5758713387377915585122289443078439220868986722706458840719} a^{3} + \frac{427129302437038734474866185788766908374114467108720}{1354774302606550711550052672284379660781175233499399} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $15$
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:  \( 10922170490900000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 R $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
17Data not computed
67Data not computed
63443Data not computed