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Properties

Label 16.0.82163008108...3649.3
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 47^{4}$
Root discriminant $31.24$
Ramified primes $17, 47$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T157)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 0, 64, 0, 160, -272, 44, -68, 117, -34, 11, -34, 10, 0, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + x^14 + 10*x^12 - 34*x^11 + 11*x^10 - 34*x^9 + 117*x^8 - 68*x^7 + 44*x^6 - 272*x^5 + 160*x^4 + 64*x^2 + 256)
 
gp: K = bnfinit(x^16 + x^14 + 10*x^12 - 34*x^11 + 11*x^10 - 34*x^9 + 117*x^8 - 68*x^7 + 44*x^6 - 272*x^5 + 160*x^4 + 64*x^2 + 256, 1)
 

Normalized defining polynomial

\( x^{16} + x^{14} + 10 x^{12} - 34 x^{11} + 11 x^{10} - 34 x^{9} + 117 x^{8} - 68 x^{7} + 44 x^{6} - 272 x^{5} + 160 x^{4} + 64 x^{2} + 256 \)

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:  \(821630081083204084623649=17^{14}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.24$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{3}{16} a^{4} - \frac{5}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{8} a^{9} + \frac{1}{16} a^{8} - \frac{3}{16} a^{6} + \frac{1}{32} a^{5} + \frac{1}{32} a^{4} - \frac{1}{16} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{10} + \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{5}{32} a^{6} - \frac{3}{16} a^{5} + \frac{3}{32} a^{4} + \frac{5}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{1}{32} a^{10} + \frac{3}{32} a^{9} - \frac{3}{32} a^{8} + \frac{3}{64} a^{7} + \frac{5}{32} a^{6} + \frac{11}{64} a^{5} - \frac{5}{32} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{107776} a^{14} - \frac{191}{26944} a^{13} + \frac{189}{107776} a^{12} - \frac{353}{26944} a^{11} + \frac{1583}{53888} a^{10} - \frac{589}{53888} a^{9} - \frac{7269}{107776} a^{8} - \frac{9943}{53888} a^{7} - \frac{14959}{107776} a^{6} - \frac{505}{6736} a^{5} - \frac{101}{6736} a^{4} + \frac{1399}{3368} a^{3} + \frac{305}{1684} a^{2} + \frac{39}{842} a + \frac{423}{1684}$, $\frac{1}{1724416} a^{15} + \frac{1}{862208} a^{14} - \frac{12475}{1724416} a^{13} + \frac{12741}{862208} a^{12} + \frac{4719}{862208} a^{11} + \frac{9613}{862208} a^{10} + \frac{188351}{1724416} a^{9} - \frac{29577}{431104} a^{8} + \frac{379885}{1724416} a^{7} - \frac{174453}{862208} a^{6} - \frac{10589}{215552} a^{5} - \frac{8235}{53888} a^{4} + \frac{17639}{53888} a^{3} + \frac{10999}{26944} a^{2} - \frac{5505}{26944} a - \frac{497}{13472}$

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:  $7$
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:  \( 678126.261537 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.2.19285917631.1 x2, 8.4.53319889921.2

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$