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Properties

Label 16.0.94438653106...6913.1
Degree $16$
Signature $[0, 8]$
Discriminant $43^{8}\cdot 97^{7}$
Root discriminant $48.52$
Ramified primes $43, 97$
Class number $1$ (GRH)
Class group Trivial (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![2304, -3648, 2192, 328, -280, 1522, -1081, 19, 183, 108, 303, -47, 17, -32, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^14 - 32*x^13 + 17*x^12 - 47*x^11 + 303*x^10 + 108*x^9 + 183*x^8 + 19*x^7 - 1081*x^6 + 1522*x^5 - 280*x^4 + 328*x^3 + 2192*x^2 - 3648*x + 2304)
 
gp: K = bnfinit(x^16 - x^15 + x^14 - 32*x^13 + 17*x^12 - 47*x^11 + 303*x^10 + 108*x^9 + 183*x^8 + 19*x^7 - 1081*x^6 + 1522*x^5 - 280*x^4 + 328*x^3 + 2192*x^2 - 3648*x + 2304, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + x^{14} - 32 x^{13} + 17 x^{12} - 47 x^{11} + 303 x^{10} + 108 x^{9} + 183 x^{8} + 19 x^{7} - 1081 x^{6} + 1522 x^{5} - 280 x^{4} + 328 x^{3} + 2192 x^{2} - 3648 x + 2304 \)

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:  \(944386531066764936008646913=43^{8}\cdot 97^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.52$
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:  $43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{6} + \frac{1}{6} a^{5} + \frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{4} a^{2} + \frac{1}{6} a$, $\frac{1}{12} a^{7} - \frac{1}{4} a^{5} + \frac{1}{6} a^{4} + \frac{1}{12} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{12} a^{2}$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{1}{24} a^{6} - \frac{1}{6} a^{5} - \frac{5}{24} a^{4} + \frac{1}{8} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{48} a^{10} - \frac{1}{48} a^{9} + \frac{1}{48} a^{7} - \frac{1}{16} a^{5} + \frac{11}{48} a^{4} - \frac{1}{24} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{9} + \frac{1}{48} a^{8} + \frac{1}{48} a^{7} + \frac{1}{48} a^{6} - \frac{1}{6} a^{5} - \frac{11}{48} a^{4} + \frac{1}{8} a^{3} - \frac{5}{12} a^{2} - \frac{1}{3} a$, $\frac{1}{288} a^{12} - \frac{1}{144} a^{11} + \frac{5}{288} a^{8} - \frac{1}{24} a^{6} + \frac{17}{144} a^{5} - \frac{11}{288} a^{4} + \frac{1}{12} a^{3} - \frac{7}{36} a^{2} + \frac{1}{12} a$, $\frac{1}{576} a^{13} + \frac{1}{288} a^{11} - \frac{1}{96} a^{10} + \frac{5}{576} a^{9} + \frac{1}{36} a^{8} - \frac{1}{48} a^{7} - \frac{1}{72} a^{6} - \frac{7}{192} a^{5} - \frac{5}{288} a^{4} + \frac{11}{72} a^{3} - \frac{29}{72} a^{2} - \frac{5}{12} a$, $\frac{1}{1728} a^{14} + \frac{1}{1728} a^{13} - \frac{1}{864} a^{12} - \frac{1}{216} a^{11} - \frac{1}{1728} a^{10} + \frac{1}{192} a^{9} - \frac{13}{432} a^{8} + \frac{1}{108} a^{7} - \frac{65}{1728} a^{6} + \frac{169}{1728} a^{5} + \frac{91}{864} a^{4} + \frac{7}{36} a^{3} - \frac{31}{216} a^{2} + \frac{13}{36} a - \frac{1}{3}$, $\frac{1}{69234048} a^{15} - \frac{1561}{23078016} a^{14} + \frac{3235}{23078016} a^{13} + \frac{12689}{11539008} a^{12} - \frac{112367}{69234048} a^{11} + \frac{676723}{69234048} a^{10} - \frac{1399813}{69234048} a^{9} - \frac{115985}{34617024} a^{8} - \frac{756367}{23078016} a^{7} + \frac{223799}{23078016} a^{6} - \frac{636713}{5325696} a^{5} - \frac{1019689}{8654256} a^{4} - \frac{435859}{2163564} a^{3} + \frac{2645371}{8654256} a^{2} + \frac{39059}{360594} a + \frac{14522}{60099}$

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:  $7$
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:  \( 120368330.892 \) (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

\(\Q(\sqrt{-43}) \), 4.0.179353.1, 8.0.3120247365073.2

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$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}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$