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Properties

Label 16.0.33456573905...2593.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 43^{8}$
Root discriminant $93.39$
Ramified primes $17, 43$
Class number $48$ (GRH)
Class group $[48]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4250000, 0, -2310028, 0, 345253, 0, -20893, 0, 11322, 0, -2057, 0, 170, 0, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 17*x^14 + 170*x^12 - 2057*x^10 + 11322*x^8 - 20893*x^6 + 345253*x^4 - 2310028*x^2 + 4250000)
 
gp: K = bnfinit(x^16 - 17*x^14 + 170*x^12 - 2057*x^10 + 11322*x^8 - 20893*x^6 + 345253*x^4 - 2310028*x^2 + 4250000, 1)
 

Normalized defining polynomial

\( x^{16} - 17 x^{14} + 170 x^{12} - 2057 x^{10} + 11322 x^{8} - 20893 x^{6} + 345253 x^{4} - 2310028 x^{2} + 4250000 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33456573905268530473918973952593=17^{15}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.39$
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:  $17, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{7} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{24} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{6}$, $\frac{1}{240} a^{9} - \frac{1}{48} a^{8} + \frac{1}{40} a^{7} - \frac{1}{16} a^{6} + \frac{7}{40} a^{5} - \frac{1}{16} a^{4} - \frac{13}{80} a^{3} - \frac{3}{8} a^{2} + \frac{29}{60} a + \frac{1}{3}$, $\frac{1}{240} a^{10} + \frac{1}{240} a^{8} - \frac{1}{16} a^{7} - \frac{1}{80} a^{6} - \frac{1}{16} a^{5} + \frac{3}{20} a^{4} - \frac{1}{16} a^{3} + \frac{29}{60} a^{2} - \frac{1}{6}$, $\frac{1}{240} a^{11} - \frac{3}{80} a^{7} + \frac{9}{40} a^{5} - \frac{5}{48} a^{3} - \frac{1}{2} a^{2} + \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{960} a^{12} - \frac{1}{480} a^{10} + \frac{19}{960} a^{8} + \frac{1}{32} a^{6} - \frac{1}{4} a^{5} + \frac{113}{960} a^{4} - \frac{1}{4} a^{3} - \frac{97}{240} a^{2} + \frac{1}{4} a - \frac{5}{12}$, $\frac{1}{24000} a^{13} + \frac{23}{12000} a^{11} + \frac{43}{24000} a^{9} + \frac{117}{4000} a^{7} - \frac{4927}{24000} a^{5} - \frac{511}{6000} a^{3} - \frac{709}{1500} a$, $\frac{1}{7989661008000} a^{14} - \frac{1}{48000} a^{13} + \frac{591818453}{1141380144000} a^{12} - \frac{23}{24000} a^{11} - \frac{1622232407}{7989661008000} a^{10} - \frac{43}{48000} a^{9} - \frac{21385863589}{1141380144000} a^{8} + \frac{383}{8000} a^{7} + \frac{322177415423}{7989661008000} a^{6} - \frac{4073}{48000} a^{5} + \frac{1883490149881}{7989661008000} a^{4} - \frac{1739}{12000} a^{3} + \frac{798204104489}{1997415252000} a^{2} - \frac{41}{3000} a - \frac{422538955}{3994830504}$, $\frac{1}{7989661008000} a^{15} - \frac{662593}{285345036000} a^{13} - \frac{1}{1920} a^{12} + \frac{6700331143}{7989661008000} a^{11} + \frac{1}{960} a^{10} + \frac{76185367}{142672518000} a^{9} - \frac{19}{1920} a^{8} - \frac{401885613427}{7989661008000} a^{7} + \frac{3}{64} a^{6} - \frac{391945733947}{3994830504000} a^{5} + \frac{7}{1920} a^{4} + \frac{213925013257}{998707626000} a^{3} + \frac{7}{480} a^{2} + \frac{4852672733}{9987076260} a + \frac{11}{24}$

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

Class group and class number

$C_{48}$, which has order $48$ (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:  $7$
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:  \( 2165185142.81 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-731}) \), 4.0.9084137.1, 8.0.1402866265591073.1

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 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $16$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$