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Properties

Label 16.0.38592250557...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{6}\cdot 89^{12}$
Root discriminant $52.99$
Ramified primes $5, 89$
Class number $37$ (GRH)
Class group $[37]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, -1552, 5176, -3864, 4266, -317, 640, -815, 539, -620, 169, -95, 72, -29, 17, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 17*x^14 - 29*x^13 + 72*x^12 - 95*x^11 + 169*x^10 - 620*x^9 + 539*x^8 - 815*x^7 + 640*x^6 - 317*x^5 + 4266*x^4 - 3864*x^3 + 5176*x^2 - 1552*x + 512)
 
gp: K = bnfinit(x^16 - 4*x^15 + 17*x^14 - 29*x^13 + 72*x^12 - 95*x^11 + 169*x^10 - 620*x^9 + 539*x^8 - 815*x^7 + 640*x^6 - 317*x^5 + 4266*x^4 - 3864*x^3 + 5176*x^2 - 1552*x + 512, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 17 x^{14} - 29 x^{13} + 72 x^{12} - 95 x^{11} + 169 x^{10} - 620 x^{9} + 539 x^{8} - 815 x^{7} + 640 x^{6} - 317 x^{5} + 4266 x^{4} - 3864 x^{3} + 5176 x^{2} - 1552 x + 512 \)

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:  \(3859225055707220942242515625=5^{6}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.99$
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 a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{13} + \frac{1}{32} a^{11} + \frac{3}{32} a^{10} - \frac{1}{8} a^{9} + \frac{3}{32} a^{8} - \frac{3}{32} a^{7} - \frac{1}{4} a^{6} + \frac{3}{32} a^{5} + \frac{1}{32} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{1371384793759674208384} a^{15} - \frac{4600174096053022751}{1371384793759674208384} a^{14} - \frac{9908676123123899827}{685692396879837104192} a^{13} + \frac{30129858153477946545}{1371384793759674208384} a^{12} + \frac{144441289335854578397}{1371384793759674208384} a^{11} + \frac{12779828303256838923}{685692396879837104192} a^{10} - \frac{3757186647774113613}{1371384793759674208384} a^{9} + \frac{109373626115731255587}{1371384793759674208384} a^{8} + \frac{107278150747938829995}{685692396879837104192} a^{7} + \frac{130656189142954232611}{1371384793759674208384} a^{6} + \frac{112278900150726910319}{1371384793759674208384} a^{5} + \frac{51137939077809471021}{685692396879837104192} a^{4} + \frac{1638920092851397311}{42855774804989819012} a^{3} - \frac{70351367222533233249}{171423099219959276048} a^{2} + \frac{26900314806775830631}{85711549609979638024} a - \frac{287188187505528610}{10713943701247454753}$

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

Class group and class number

$C_{37}$, which has order $37$ (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:  \( 15350212.3839 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 4.4.39605.1, 4.4.3524845.1, 8.8.12424532274025.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
89Data not computed