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Properties

Label 16.0.83680629612...2769.6
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 89^{6}$
Root discriminant $64.22$
Ramified primes $17, 89$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10214416, 0, 2901560, 0, 727413, 0, 218195, 0, 31042, 0, 4471, 0, 714, 0, 51, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 51*x^14 + 714*x^12 + 4471*x^10 + 31042*x^8 + 218195*x^6 + 727413*x^4 + 2901560*x^2 + 10214416)
 
gp: K = bnfinit(x^16 + 51*x^14 + 714*x^12 + 4471*x^10 + 31042*x^8 + 218195*x^6 + 727413*x^4 + 2901560*x^2 + 10214416, 1)
 

Normalized defining polynomial

\( x^{16} + 51 x^{14} + 714 x^{12} + 4471 x^{10} + 31042 x^{8} + 218195 x^{6} + 727413 x^{4} + 2901560 x^{2} + 10214416 \)

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:  \(83680629612698426645202702769=17^{14}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.22$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{6} + \frac{1}{8} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{272} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{5} + \frac{3}{16} a^{3} - \frac{1}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{272} a^{9} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3} + \frac{7}{16} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{272} a^{10} - \frac{1}{16} a^{6} + \frac{1}{8} a^{4} - \frac{5}{16} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{1088} a^{11} - \frac{1}{544} a^{10} - \frac{1}{544} a^{9} - \frac{1}{64} a^{7} + \frac{1}{32} a^{6} - \frac{3}{32} a^{5} + \frac{3}{16} a^{4} + \frac{13}{64} a^{3} - \frac{11}{32} a^{2} + \frac{3}{16} a + \frac{3}{8}$, $\frac{1}{1088} a^{12} - \frac{1}{544} a^{10} - \frac{1}{1088} a^{8} - \frac{1}{16} a^{7} - \frac{1}{32} a^{6} - \frac{1}{16} a^{5} + \frac{1}{64} a^{4} + \frac{3}{16} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{1088} a^{13} - \frac{1}{1088} a^{9} - \frac{1}{16} a^{7} + \frac{5}{64} a^{5} + \frac{3}{32} a^{3} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{73925373966464} a^{14} - \frac{1}{2176} a^{13} - \frac{23182991893}{73925373966464} a^{12} + \frac{40530508633}{73925373966464} a^{10} + \frac{1}{2176} a^{9} - \frac{29396696155}{73925373966464} a^{8} - \frac{1}{32} a^{7} + \frac{57290774311}{4348551409792} a^{6} + \frac{3}{128} a^{5} + \frac{428014151001}{4348551409792} a^{4} + \frac{1}{64} a^{3} + \frac{109838003411}{1087137852448} a^{2} + \frac{1}{16} a - \frac{4224131283}{33973057889}$, $\frac{1}{3474492576423808} a^{15} + \frac{39368090887}{1737246288211904} a^{13} - \frac{1}{2176} a^{12} - \frac{231253954479}{3474492576423808} a^{11} + \frac{1}{1088} a^{10} + \frac{1293264380649}{1737246288211904} a^{9} - \frac{3}{2176} a^{8} + \frac{1280320858315}{204381916260224} a^{7} + \frac{1}{64} a^{6} - \frac{1351281276185}{12773869766264} a^{5} + \frac{31}{128} a^{4} - \frac{22032676910473}{102190958130112} a^{3} - \frac{7}{32} a^{2} - \frac{11176776030231}{25547739532528} a$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( 4485613.46046 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.36520141897.1, 8.0.3250292628833.2, 8.4.17016237880361.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
89Data not computed