↑

Properties

Label 16.16.1062747667...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{14}\cdot 89^{15}$
Root discriminant $274.89$
Ramified primes $5, 89$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10313939, 29665257, -3024783, -53976394, 36131021, 17914002, -17580825, -1649977, 3149729, -11771, -264375, 5579, 10926, -102, -197, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 197*x^14 - 102*x^13 + 10926*x^12 + 5579*x^11 - 264375*x^10 - 11771*x^9 + 3149729*x^8 - 1649977*x^7 - 17580825*x^6 + 17914002*x^5 + 36131021*x^4 - 53976394*x^3 - 3024783*x^2 + 29665257*x - 10313939)
 
gp: K = bnfinit(x^16 - x^15 - 197*x^14 - 102*x^13 + 10926*x^12 + 5579*x^11 - 264375*x^10 - 11771*x^9 + 3149729*x^8 - 1649977*x^7 - 17580825*x^6 + 17914002*x^5 + 36131021*x^4 - 53976394*x^3 - 3024783*x^2 + 29665257*x - 10313939, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 197 x^{14} - 102 x^{13} + 10926 x^{12} + 5579 x^{11} - 264375 x^{10} - 11771 x^{9} + 3149729 x^{8} - 1649977 x^{7} - 17580825 x^{6} + 17914002 x^{5} + 36131021 x^{4} - 53976394 x^{3} - 3024783 x^{2} + 29665257 x - 10313939 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1062747667303462437668657811578369140625=5^{14}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $274.89$
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:  $5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17} a^{13} - \frac{6}{17} a^{12} - \frac{6}{17} a^{11} + \frac{5}{17} a^{10} + \frac{5}{17} a^{9} - \frac{1}{17} a^{8} - \frac{1}{17} a^{7} + \frac{3}{17} a^{6} + \frac{8}{17} a^{5} + \frac{4}{17} a^{4} + \frac{8}{17} a^{3} + \frac{3}{17} a^{2} + \frac{8}{17} a - \frac{2}{17}$, $\frac{1}{17} a^{14} - \frac{8}{17} a^{12} + \frac{3}{17} a^{11} + \frac{1}{17} a^{10} - \frac{5}{17} a^{9} - \frac{7}{17} a^{8} - \frac{3}{17} a^{7} - \frac{8}{17} a^{6} + \frac{1}{17} a^{5} - \frac{2}{17} a^{4} - \frac{8}{17} a^{2} - \frac{5}{17} a + \frac{5}{17}$, $\frac{1}{628293625277911600761496885302156324706979663869} a^{15} - \frac{10226234306101834727210450016669007541111457667}{628293625277911600761496885302156324706979663869} a^{14} + \frac{4228134111895390325685571332278041840435530994}{628293625277911600761496885302156324706979663869} a^{13} - \frac{166853441009303414078344174806178995652607935897}{628293625277911600761496885302156324706979663869} a^{12} - \frac{291334423862109367075858314856601927223843953277}{628293625277911600761496885302156324706979663869} a^{11} + \frac{267751490914963101574610901729544752192421904836}{628293625277911600761496885302156324706979663869} a^{10} + \frac{4355685778168745571079277257407256000904228497}{628293625277911600761496885302156324706979663869} a^{9} + \frac{106509999408658956769682259534801724188735111803}{628293625277911600761496885302156324706979663869} a^{8} - \frac{38344783615754842234405497172177320875258979048}{628293625277911600761496885302156324706979663869} a^{7} - \frac{31988919383453222024467290407751367923149968925}{628293625277911600761496885302156324706979663869} a^{6} + \frac{82209305392402706833675548630492529955157970235}{628293625277911600761496885302156324706979663869} a^{5} + \frac{206366203387184128885799860260402454434101950240}{628293625277911600761496885302156324706979663869} a^{4} + \frac{208515905974459142208297605667650992258736970101}{628293625277911600761496885302156324706979663869} a^{3} + \frac{116814989006520778193204307112574701224670952424}{628293625277911600761496885302156324706979663869} a^{2} + \frac{265618393770588474276708261811177258110519865101}{628293625277911600761496885302156324706979663869} a - \frac{123015775515425388176409793025975627441901816319}{628293625277911600761496885302156324706979663869}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $15$
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:  \( 100217027695000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.17624225.2, 8.8.691114607742640625.2

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
$5$5.8.7.4$x^{8} + 40$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.4$x^{8} + 40$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
89Data not computed