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Properties

Label 16.0.11757829884...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 19^{2}\cdot 89^{6}$
Root discriminant $49.19$
Ramified primes $2, 5, 19, 89$
Class number $32$ (GRH)
Class group $[2, 2, 8]$ (GRH)
Galois group 16T1177

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![970939, 490946, 514611, 311462, 397969, -129216, 229594, -82660, 52262, -13364, 6210, -1088, 487, -54, 27, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 27*x^14 - 54*x^13 + 487*x^12 - 1088*x^11 + 6210*x^10 - 13364*x^9 + 52262*x^8 - 82660*x^7 + 229594*x^6 - 129216*x^5 + 397969*x^4 + 311462*x^3 + 514611*x^2 + 490946*x + 970939)
 
gp: K = bnfinit(x^16 - 2*x^15 + 27*x^14 - 54*x^13 + 487*x^12 - 1088*x^11 + 6210*x^10 - 13364*x^9 + 52262*x^8 - 82660*x^7 + 229594*x^6 - 129216*x^5 + 397969*x^4 + 311462*x^3 + 514611*x^2 + 490946*x + 970939, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 27 x^{14} - 54 x^{13} + 487 x^{12} - 1088 x^{11} + 6210 x^{10} - 13364 x^{9} + 52262 x^{8} - 82660 x^{7} + 229594 x^{6} - 129216 x^{5} + 397969 x^{4} + 311462 x^{3} + 514611 x^{2} + 490946 x + 970939 \)

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:  \(1175782988427565465600000000=2^{24}\cdot 5^{8}\cdot 19^{2}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.19$
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:  $2, 5, 19, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3} + \frac{1}{8} a$, $\frac{1}{304} a^{14} + \frac{5}{304} a^{13} + \frac{7}{304} a^{12} - \frac{7}{152} a^{11} - \frac{17}{152} a^{10} + \frac{7}{152} a^{9} + \frac{1}{38} a^{8} - \frac{7}{38} a^{7} + \frac{1}{19} a^{6} + \frac{33}{152} a^{5} - \frac{39}{152} a^{4} - \frac{63}{152} a^{3} + \frac{135}{304} a^{2} - \frac{129}{304} a - \frac{67}{304}$, $\frac{1}{22381637252779333792774388286831653488} a^{15} - \frac{32213674736490939469476602101797721}{22381637252779333792774388286831653488} a^{14} - \frac{619072867975666400372345527194438749}{22381637252779333792774388286831653488} a^{13} - \frac{59357591343458689834031653610727729}{11190818626389666896387194143415826744} a^{12} - \frac{172847824402999952432262417331567759}{5595409313194833448193597071707913372} a^{11} - \frac{31722757162022204253715557418922847}{294495227010254392010189319563574388} a^{10} + \frac{932558821582764982543942153489956455}{11190818626389666896387194143415826744} a^{9} + \frac{1073722278146159319718989070806484049}{11190818626389666896387194143415826744} a^{8} + \frac{939392176605500333633439727324398941}{11190818626389666896387194143415826744} a^{7} + \frac{92251677008607034800805914030586095}{1398852328298708362048399267926978343} a^{6} + \frac{529373201289983245419648883012436137}{2797704656597416724096798535853956686} a^{5} + \frac{1126769384125440246335891152394361057}{2797704656597416724096798535853956686} a^{4} - \frac{10578996229181014317483853681347617555}{22381637252779333792774388286831653488} a^{3} + \frac{9617079812644286602861893232207220455}{22381637252779333792774388286831653488} a^{2} - \frac{822699127269345412325155489004211061}{22381637252779333792774388286831653488} a + \frac{2599529528489607770726735108626639137}{11190818626389666896387194143415826744}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (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:  \( 374709.809098 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1177:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 76 conjugacy class representatives for t16n1177 are not computed
Character table for t16n1177 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.0.451180160000.1, 8.4.1504990000.1, 8.4.8572423040000.2

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.6.6$x^{4} - 20$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
2.4.6.6$x^{4} - 20$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$