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Properties

Label 16.0.23584074680...5057.4
Degree $16$
Signature $[0, 8]$
Discriminant $17^{13}\cdot 47^{8}$
Root discriminant $68.52$
Ramified primes $17, 47$
Class number $80$ (GRH)
Class group $[4, 20]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1666681, -1165773, 1711428, -1212534, 919138, -401814, 174382, -27038, -4164, 9420, -3974, 1062, -86, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 86*x^12 + 1062*x^11 - 3974*x^10 + 9420*x^9 - 4164*x^8 - 27038*x^7 + 174382*x^6 - 401814*x^5 + 919138*x^4 - 1212534*x^3 + 1711428*x^2 - 1165773*x + 1666681)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 86*x^12 + 1062*x^11 - 3974*x^10 + 9420*x^9 - 4164*x^8 - 27038*x^7 + 174382*x^6 - 401814*x^5 + 919138*x^4 - 1212534*x^3 + 1711428*x^2 - 1165773*x + 1666681, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 86 x^{12} + 1062 x^{11} - 3974 x^{10} + 9420 x^{9} - 4164 x^{8} - 27038 x^{7} + 174382 x^{6} - 401814 x^{5} + 919138 x^{4} - 1212534 x^{3} + 1711428 x^{2} - 1165773 x + 1666681 \)

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:  \(235840746805304140638847775057=17^{13}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.52$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{17} a^{8} - \frac{4}{17} a^{7} + \frac{7}{17} a^{6} - \frac{7}{17} a^{5} - \frac{2}{17} a^{4} - \frac{6}{17} a^{3} - \frac{7}{17} a^{2} + \frac{1}{17} a + \frac{1}{17}$, $\frac{1}{17} a^{9} + \frac{8}{17} a^{7} + \frac{4}{17} a^{6} + \frac{4}{17} a^{5} + \frac{3}{17} a^{4} + \frac{3}{17} a^{3} + \frac{7}{17} a^{2} + \frac{5}{17} a + \frac{4}{17}$, $\frac{1}{17} a^{10} + \frac{2}{17} a^{7} - \frac{1}{17} a^{6} + \frac{8}{17} a^{5} + \frac{2}{17} a^{4} + \frac{4}{17} a^{3} - \frac{7}{17} a^{2} - \frac{4}{17} a - \frac{8}{17}$, $\frac{1}{17} a^{11} + \frac{7}{17} a^{7} - \frac{6}{17} a^{6} - \frac{1}{17} a^{5} + \frac{8}{17} a^{4} + \frac{5}{17} a^{3} - \frac{7}{17} a^{2} + \frac{7}{17} a - \frac{2}{17}$, $\frac{1}{289} a^{12} - \frac{6}{289} a^{11} + \frac{8}{289} a^{10} - \frac{2}{289} a^{9} - \frac{2}{289} a^{8} + \frac{5}{289} a^{7} - \frac{78}{289} a^{6} + \frac{31}{289} a^{5} - \frac{134}{289} a^{4} - \frac{93}{289} a^{3} + \frac{42}{289} a^{2} - \frac{78}{289} a + \frac{101}{289}$, $\frac{1}{289} a^{13} + \frac{6}{289} a^{11} - \frac{5}{289} a^{10} + \frac{3}{289} a^{9} - \frac{7}{289} a^{8} - \frac{65}{289} a^{7} + \frac{56}{289} a^{6} - \frac{33}{289} a^{5} - \frac{98}{289} a^{4} + \frac{79}{289} a^{3} + \frac{123}{289} a^{2} - \frac{129}{289} a - \frac{142}{289}$, $\frac{1}{1327884011425177} a^{14} - \frac{7}{1327884011425177} a^{13} - \frac{961584650892}{1327884011425177} a^{12} + \frac{5769507905443}{1327884011425177} a^{11} - \frac{23020146191414}{1327884011425177} a^{10} - \frac{15897249044472}{1327884011425177} a^{9} - \frac{27981566374437}{1327884011425177} a^{8} + \frac{505106819329182}{1327884011425177} a^{7} + \frac{37499511870978}{78110824201481} a^{6} + \frac{383969253600559}{1327884011425177} a^{5} - \frac{22967936663833}{78110824201481} a^{4} + \frac{364169638247656}{1327884011425177} a^{3} + \frac{358461125175984}{1327884011425177} a^{2} - \frac{546879389295372}{1327884011425177} a + \frac{421717411512}{1028570109547}$, $\frac{1}{5753721421505291941} a^{15} + \frac{127}{338454201265017173} a^{14} - \frac{463551166601416}{821960203072184563} a^{13} - \frac{128680818362803}{821960203072184563} a^{12} + \frac{76092702911922002}{5753721421505291941} a^{11} - \frac{133915591231711166}{5753721421505291941} a^{10} + \frac{37579730688888658}{5753721421505291941} a^{9} - \frac{18298792326220483}{821960203072184563} a^{8} - \frac{1675055826763323906}{5753721421505291941} a^{7} - \frac{304986860608024633}{821960203072184563} a^{6} - \frac{673729438566786310}{5753721421505291941} a^{5} - \frac{485895181222195508}{5753721421505291941} a^{4} + \frac{360142277283241263}{5753721421505291941} a^{3} + \frac{82517699831527893}{5753721421505291941} a^{2} - \frac{128585837196926453}{5753721421505291941} a + \frac{1480370674199908}{4456794284667151}$

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

Class group and class number

$C_{4}\times C_{20}$, which has order $80$ (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:  \( 1324428.56403 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-47}) \), 4.0.37553.1, 8.0.6928449225617.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$17$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
$47$47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$