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Properties

Label 16.8.21634605854...1424.5
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 223^{6}$
Root discriminant $51.10$
Ramified primes $2, 223$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1048

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49729, 0, 0, 0, -82956, 0, -49952, 0, -3738, 0, 1984, 0, 116, 0, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 + 116*x^12 + 1984*x^10 - 3738*x^8 - 49952*x^6 - 82956*x^4 + 49729)
 
gp: K = bnfinit(x^16 - 32*x^14 + 116*x^12 + 1984*x^10 - 3738*x^8 - 49952*x^6 - 82956*x^4 + 49729, 1)
 

Normalized defining polynomial

\( x^{16} - 32 x^{14} + 116 x^{12} + 1984 x^{10} - 3738 x^{8} - 49952 x^{6} - 82956 x^{4} + 49729 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2163460585448341410282471424=2^{44}\cdot 223^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.10$
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:  $2, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{8} + \frac{1}{16} a^{4} - \frac{7}{32}$, $\frac{1}{64} a^{9} - \frac{1}{64} a^{8} + \frac{1}{32} a^{5} - \frac{1}{32} a^{4} + \frac{25}{64} a - \frac{25}{64}$, $\frac{1}{64} a^{10} - \frac{1}{64} a^{8} + \frac{1}{32} a^{6} - \frac{1}{32} a^{4} + \frac{25}{64} a^{2} - \frac{25}{64}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{8} + \frac{1}{32} a^{7} - \frac{1}{32} a^{4} + \frac{25}{64} a^{3} - \frac{25}{64}$, $\frac{1}{28544} a^{12} - \frac{1}{892} a^{10} + \frac{339}{28544} a^{8} + \frac{31}{446} a^{6} - \frac{1285}{28544} a^{4} + \frac{1}{4} a^{2} - \frac{9}{128}$, $\frac{1}{57088} a^{13} - \frac{1}{57088} a^{12} + \frac{207}{28544} a^{11} - \frac{207}{28544} a^{10} - \frac{107}{57088} a^{9} + \frac{107}{57088} a^{8} + \frac{719}{14272} a^{7} - \frac{719}{14272} a^{6} - \frac{2177}{57088} a^{5} + \frac{2177}{57088} a^{4} - \frac{23}{128} a^{3} + \frac{23}{128} a^{2} + \frac{69}{256} a - \frac{69}{256}$, $\frac{1}{617178368} a^{14} - \frac{251}{32483072} a^{12} - \frac{4155545}{617178368} a^{10} - \frac{9407075}{617178368} a^{8} - \frac{22410205}{617178368} a^{6} - \frac{42994363}{617178368} a^{4} + \frac{238723}{2767616} a^{2} + \frac{648409}{2767616}$, $\frac{1}{617178368} a^{15} - \frac{251}{32483072} a^{13} - \frac{4155545}{617178368} a^{11} + \frac{236337}{617178368} a^{9} - \frac{1}{64} a^{8} - \frac{22410205}{617178368} a^{7} - \frac{23707539}{617178368} a^{5} - \frac{1}{32} a^{4} + \frac{238723}{2767616} a^{3} - \frac{1038107}{2767616} a - \frac{25}{64}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 29489199.4911 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1048:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 31 conjugacy class representatives for t16n1048
Character table for t16n1048 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.14272.1, 8.8.3259039744.1

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
2Data not computed
223Data not computed