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Properties

Label 16.0.18666755934...0161.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 37^{8}$
Root discriminant $13.87$
Ramified primes $3, 37$
Class number $1$
Class group Trivial
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -9, 0, -18, 24, -24, 84, -54, 76, -61, 14, -34, 31, -11, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 - 11*x^13 + 31*x^12 - 34*x^11 + 14*x^10 - 61*x^9 + 76*x^8 - 54*x^7 + 84*x^6 - 24*x^5 + 24*x^4 - 18*x^3 - 9*x + 9)
 
gp: K = bnfinit(x^16 - 5*x^15 + 8*x^14 - 11*x^13 + 31*x^12 - 34*x^11 + 14*x^10 - 61*x^9 + 76*x^8 - 54*x^7 + 84*x^6 - 24*x^5 + 24*x^4 - 18*x^3 - 9*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 8 x^{14} - 11 x^{13} + 31 x^{12} - 34 x^{11} + 14 x^{10} - 61 x^{9} + 76 x^{8} - 54 x^{7} + 84 x^{6} - 24 x^{5} + 24 x^{4} - 18 x^{3} - 9 x + 9 \)

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:  \(1866675593471230161=3^{12}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.87$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{3} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{72363161142} a^{15} - \frac{49870163}{72363161142} a^{14} - \frac{4516466779}{72363161142} a^{13} - \frac{85565497}{36181580571} a^{12} - \frac{303853771}{36181580571} a^{11} - \frac{28804237417}{72363161142} a^{10} + \frac{10855043038}{36181580571} a^{9} + \frac{505115752}{36181580571} a^{8} + \frac{22058161687}{72363161142} a^{7} + \frac{1221525167}{8040351238} a^{6} + \frac{5628946079}{12060526857} a^{5} - \frac{5281805672}{12060526857} a^{4} - \frac{4223206817}{24121053714} a^{3} - \frac{51686941}{4020175619} a^{2} - \frac{188305647}{4020175619} a - \frac{769611937}{8040351238}$

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:  \( -\frac{214759}{32862471} a^{15} + \frac{816148}{32862471} a^{14} - \frac{455967}{10954157} a^{13} + \frac{477028}{10954157} a^{12} + \frac{387715}{32862471} a^{11} + \frac{198892}{10954157} a^{10} + \frac{2747960}{32862471} a^{9} - \frac{26690656}{32862471} a^{8} - \frac{2007892}{10954157} a^{7} - \frac{518887}{32862471} a^{6} + \frac{21031620}{10954157} a^{5} + \frac{3256218}{10954157} a^{4} + \frac{28966707}{10954157} a^{3} - \frac{2390727}{10954157} a^{2} + \frac{1150149}{10954157} a + \frac{1357507}{10954157} \) (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:  \( 917.718081434 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.333.1, 8.0.36926037.1, 8.0.4102893.1, 8.0.1366263369.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$