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Properties

Label 16.0.52703064995...1609.3
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 67^{6}$
Root discriminant $40.51$
Ramified primes $17, 67$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T330)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 6656, 1664, -6368, 2864, 1208, 1740, -1766, 15, -464, 401, 81, 175, 16, 18, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 18*x^14 + 16*x^13 + 175*x^12 + 81*x^11 + 401*x^10 - 464*x^9 + 15*x^8 - 1766*x^7 + 1740*x^6 + 1208*x^5 + 2864*x^4 - 6368*x^3 + 1664*x^2 + 6656*x + 4096)
 
gp: K = bnfinit(x^16 - x^15 + 18*x^14 + 16*x^13 + 175*x^12 + 81*x^11 + 401*x^10 - 464*x^9 + 15*x^8 - 1766*x^7 + 1740*x^6 + 1208*x^5 + 2864*x^4 - 6368*x^3 + 1664*x^2 + 6656*x + 4096, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 18 x^{14} + 16 x^{13} + 175 x^{12} + 81 x^{11} + 401 x^{10} - 464 x^{9} + 15 x^{8} - 1766 x^{7} + 1740 x^{6} + 1208 x^{5} + 2864 x^{4} - 6368 x^{3} + 1664 x^{2} + 6656 x + 4096 \)

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:  \(52703064995487500398531609=17^{12}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.51$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{3}{8} a^{7} - \frac{3}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{11} - \frac{1}{8} a^{9} + \frac{7}{32} a^{8} + \frac{15}{32} a^{7} + \frac{3}{32} a^{6} + \frac{5}{16} a^{5} + \frac{7}{32} a^{4} - \frac{1}{4} a$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{1}{16} a^{10} - \frac{1}{64} a^{9} + \frac{1}{64} a^{8} - \frac{27}{64} a^{7} - \frac{7}{16} a^{6} - \frac{29}{64} a^{5} + \frac{9}{32} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8576} a^{14} - \frac{63}{8576} a^{13} - \frac{13}{2144} a^{12} + \frac{25}{1072} a^{11} + \frac{1031}{8576} a^{10} + \frac{703}{8576} a^{9} + \frac{1911}{8576} a^{8} - \frac{993}{4288} a^{7} + \frac{57}{128} a^{6} + \frac{531}{1072} a^{5} - \frac{419}{2144} a^{4} - \frac{413}{1072} a^{3} - \frac{1}{268} a^{2} - \frac{57}{268} a - \frac{19}{67}$, $\frac{1}{426552752323785657130496} a^{15} + \frac{15725417094358354715}{426552752323785657130496} a^{14} - \frac{13842654159322973175}{11225072429573306766592} a^{13} + \frac{409924278250864327637}{53319094040473207141312} a^{12} + \frac{5664440862768296206255}{426552752323785657130496} a^{11} + \frac{49282614242025735424245}{426552752323785657130496} a^{10} - \frac{48922751761629294447859}{426552752323785657130496} a^{9} + \frac{11659081336374882692571}{106638188080946414282624} a^{8} - \frac{63435938167516750159057}{426552752323785657130496} a^{7} + \frac{64877075596455671679263}{213276376161892828565248} a^{6} - \frac{24988362128260841952143}{106638188080946414282624} a^{5} - \frac{23969254767814299155431}{53319094040473207141312} a^{4} + \frac{431979722944635120533}{26659547020236603570656} a^{3} + \frac{1620480096989990845859}{13329773510118301785328} a^{2} + \frac{478555723622487539213}{1666221688764787723166} a - \frac{248730861481335813607}{833110844382393861583}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 30710846.4203 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T330):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.19363.1, 8.0.6373738073.1, 8.2.7259687665147.2, 8.2.427040450891.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$