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Properties

Label 16.8.30900637958...4833.3
Degree $16$
Signature $[8, 4]$
Discriminant $47^{4}\cdot 97^{15}$
Root discriminant $190.82$
Ramified primes $47, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![631471, 5095648, -10934242, -86630330, 15633220, -8766217, 4209376, 4917135, 166968, 577894, -2603, 4292, -2640, -652, -17, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 17*x^14 - 652*x^13 - 2640*x^12 + 4292*x^11 - 2603*x^10 + 577894*x^9 + 166968*x^8 + 4917135*x^7 + 4209376*x^6 - 8766217*x^5 + 15633220*x^4 - 86630330*x^3 - 10934242*x^2 + 5095648*x + 631471)
 
gp: K = bnfinit(x^16 - 3*x^15 - 17*x^14 - 652*x^13 - 2640*x^12 + 4292*x^11 - 2603*x^10 + 577894*x^9 + 166968*x^8 + 4917135*x^7 + 4209376*x^6 - 8766217*x^5 + 15633220*x^4 - 86630330*x^3 - 10934242*x^2 + 5095648*x + 631471, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 17 x^{14} - 652 x^{13} - 2640 x^{12} + 4292 x^{11} - 2603 x^{10} + 577894 x^{9} + 166968 x^{8} + 4917135 x^{7} + 4209376 x^{6} - 8766217 x^{5} + 15633220 x^{4} - 86630330 x^{3} - 10934242 x^{2} + 5095648 x + 631471 \)

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:  \(3090063795858197568077038853384324833=47^{4}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $190.82$
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:  $47, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{61} a^{13} + \frac{16}{61} a^{12} + \frac{2}{61} a^{11} - \frac{3}{61} a^{10} - \frac{14}{61} a^{9} - \frac{27}{61} a^{8} + \frac{1}{61} a^{7} + \frac{21}{61} a^{6} + \frac{15}{61} a^{5} + \frac{6}{61} a^{4} + \frac{8}{61} a^{3} - \frac{28}{61} a^{2} - \frac{11}{61} a + \frac{13}{61}$, $\frac{1}{61} a^{14} - \frac{10}{61} a^{12} + \frac{26}{61} a^{11} - \frac{27}{61} a^{10} + \frac{14}{61} a^{9} + \frac{6}{61} a^{8} + \frac{5}{61} a^{7} - \frac{16}{61} a^{6} + \frac{10}{61} a^{5} - \frac{27}{61} a^{4} + \frac{27}{61} a^{3} + \frac{10}{61} a^{2} + \frac{6}{61} a - \frac{25}{61}$, $\frac{1}{228171636724849016068111039138049681318387982032249167994072003} a^{15} + \frac{1521370723672449070358662317646190857432849124204439024442770}{228171636724849016068111039138049681318387982032249167994072003} a^{14} - \frac{1713210725112293480614126857414723093456242584687688647650638}{228171636724849016068111039138049681318387982032249167994072003} a^{13} + \frac{21327473636967104282057528750028691195809246774960974156453484}{228171636724849016068111039138049681318387982032249167994072003} a^{12} - \frac{113278927386581548301563814906103305134432822208892960529200325}{228171636724849016068111039138049681318387982032249167994072003} a^{11} - \frac{38586841136810171270980326793589878912249710142612462929264151}{228171636724849016068111039138049681318387982032249167994072003} a^{10} - \frac{256757145353900623788415778167876591916245916791273872386423}{3740518634833590427346082608820486578989966918561461770394623} a^{9} + \frac{42543886782671722397631396162063862165255050487030618882106422}{228171636724849016068111039138049681318387982032249167994072003} a^{8} - \frac{92436706473428124504897389161505323549487950394361819340414665}{228171636724849016068111039138049681318387982032249167994072003} a^{7} + \frac{56143314892825520506591480773711319098836372798942940981300033}{228171636724849016068111039138049681318387982032249167994072003} a^{6} - \frac{104635912362486475218358583291652469502381313150286570520685774}{228171636724849016068111039138049681318387982032249167994072003} a^{5} + \frac{91049024232322319975552551946161462406824747822344750297962266}{228171636724849016068111039138049681318387982032249167994072003} a^{4} - \frac{102936846019248948879947970408934082120976219821473171826304942}{228171636724849016068111039138049681318387982032249167994072003} a^{3} + \frac{31911309651124766918790601455885192338998226281879230967716127}{228171636724849016068111039138049681318387982032249167994072003} a^{2} - \frac{49029906601951512682083640422176389563559573245767346664733645}{228171636724849016068111039138049681318387982032249167994072003} a + \frac{15863830000518598785272084681124086276006075886123544768610275}{228171636724849016068111039138049681318387982032249167994072003}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 378443100977 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$47$47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
97Data not computed