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Properties

Label 16.0.23806000582...1584.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{18}\cdot 3^{8}\cdot 7^{12}$
Root discriminant $16.26$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $D_8:C_2$ (as 16T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![127, -26, -197, -77, 174, 233, -152, -189, 62, 85, 17, -39, -20, 18, 4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 4*x^14 + 18*x^13 - 20*x^12 - 39*x^11 + 17*x^10 + 85*x^9 + 62*x^8 - 189*x^7 - 152*x^6 + 233*x^5 + 174*x^4 - 77*x^3 - 197*x^2 - 26*x + 127)
 
gp: K = bnfinit(x^16 - 5*x^15 + 4*x^14 + 18*x^13 - 20*x^12 - 39*x^11 + 17*x^10 + 85*x^9 + 62*x^8 - 189*x^7 - 152*x^6 + 233*x^5 + 174*x^4 - 77*x^3 - 197*x^2 - 26*x + 127, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 4 x^{14} + 18 x^{13} - 20 x^{12} - 39 x^{11} + 17 x^{10} + 85 x^{9} + 62 x^{8} - 189 x^{7} - 152 x^{6} + 233 x^{5} + 174 x^{4} - 77 x^{3} - 197 x^{2} - 26 x + 127 \)

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:  \(23806000582036291584=2^{18}\cdot 3^{8}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.26$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{90379212575224} a^{15} - \frac{476810967511}{11297401571903} a^{14} + \frac{1602905177769}{22594803143806} a^{13} + \frac{9621975463691}{45189606287612} a^{12} + \frac{18899282555041}{45189606287612} a^{11} - \frac{13530704684013}{90379212575224} a^{10} + \frac{3849018415404}{11297401571903} a^{9} - \frac{39126456013443}{90379212575224} a^{8} - \frac{40272864789121}{90379212575224} a^{7} - \frac{19195758281177}{45189606287612} a^{6} - \frac{9465685397937}{45189606287612} a^{5} - \frac{18602732065769}{90379212575224} a^{4} - \frac{23429843212559}{90379212575224} a^{3} - \frac{55839139773}{305335177619} a^{2} + \frac{38818943079219}{90379212575224} a - \frac{303785687965}{711647343112}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{432397}{236989366} a^{15} + \frac{25932203}{473978732} a^{14} - \frac{139566741}{473978732} a^{13} + \frac{138680331}{473978732} a^{12} + \frac{355848381}{473978732} a^{11} - \frac{326444607}{473978732} a^{10} - \frac{192268838}{118494683} a^{9} - \frac{57511639}{236989366} a^{8} + \frac{1559066989}{473978732} a^{7} + \frac{476755074}{118494683} a^{6} - \frac{1561725687}{236989366} a^{5} - \frac{1196120357}{236989366} a^{4} + \frac{1999290703}{473978732} a^{3} + \frac{16788540}{3202559} a^{2} + \frac{343371179}{236989366} a - \frac{18260115}{3732116} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3314.66450426 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8:C_2$ (as 16T44):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), 4.0.392.1, 4.0.3528.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.12446784.1

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 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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
3Data not computed
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$