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Properties

Label 16.0.54754869292...4489.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{14}\cdot 157^{4}$
Root discriminant $83.40$
Ramified primes $37, 157$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![463589, -527756, 843018, -609071, 635768, -324957, 250634, -83761, 60260, -13208, 9225, -1005, 1046, -55, 63, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 63*x^14 - 55*x^13 + 1046*x^12 - 1005*x^11 + 9225*x^10 - 13208*x^9 + 60260*x^8 - 83761*x^7 + 250634*x^6 - 324957*x^5 + 635768*x^4 - 609071*x^3 + 843018*x^2 - 527756*x + 463589)
 
gp: K = bnfinit(x^16 - 4*x^15 + 63*x^14 - 55*x^13 + 1046*x^12 - 1005*x^11 + 9225*x^10 - 13208*x^9 + 60260*x^8 - 83761*x^7 + 250634*x^6 - 324957*x^5 + 635768*x^4 - 609071*x^3 + 843018*x^2 - 527756*x + 463589, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 63 x^{14} - 55 x^{13} + 1046 x^{12} - 1005 x^{11} + 9225 x^{10} - 13208 x^{9} + 60260 x^{8} - 83761 x^{7} + 250634 x^{6} - 324957 x^{5} + 635768 x^{4} - 609071 x^{3} + 843018 x^{2} - 527756 x + 463589 \)

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:  \(5475486929215895672663613034489=37^{14}\cdot 157^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.40$
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:  $37, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{37} a^{8} - \frac{2}{37} a^{7} + \frac{11}{37} a^{6} + \frac{13}{37} a^{5} - \frac{11}{37} a^{4} + \frac{17}{37} a^{3} - \frac{16}{37} a^{2} + \frac{17}{37} a - \frac{4}{37}$, $\frac{1}{37} a^{9} + \frac{7}{37} a^{7} - \frac{2}{37} a^{6} + \frac{15}{37} a^{5} - \frac{5}{37} a^{4} + \frac{18}{37} a^{3} - \frac{15}{37} a^{2} - \frac{7}{37} a - \frac{8}{37}$, $\frac{1}{37} a^{10} + \frac{12}{37} a^{7} + \frac{12}{37} a^{6} + \frac{15}{37} a^{5} - \frac{16}{37} a^{4} + \frac{14}{37} a^{3} - \frac{6}{37} a^{2} - \frac{16}{37} a - \frac{9}{37}$, $\frac{1}{37} a^{11} - \frac{1}{37} a^{7} - \frac{6}{37} a^{6} + \frac{13}{37} a^{5} - \frac{2}{37} a^{4} + \frac{12}{37} a^{3} - \frac{9}{37} a^{2} + \frac{9}{37} a + \frac{11}{37}$, $\frac{1}{259} a^{12} - \frac{3}{259} a^{11} + \frac{2}{259} a^{10} + \frac{1}{259} a^{9} - \frac{1}{259} a^{8} - \frac{46}{259} a^{7} + \frac{16}{259} a^{6} + \frac{78}{259} a^{5} - \frac{19}{259} a^{4} + \frac{1}{259} a^{3} + \frac{120}{259} a^{2} - \frac{129}{259} a + \frac{18}{37}$, $\frac{1}{259} a^{13} + \frac{2}{259} a^{9} - \frac{52}{259} a^{7} + \frac{3}{37} a^{6} + \frac{61}{259} a^{5} + \frac{3}{37} a^{4} - \frac{94}{259} a^{3} - \frac{8}{37} a^{2} - \frac{30}{259} a + \frac{9}{37}$, $\frac{1}{259} a^{14} + \frac{2}{259} a^{10} - \frac{3}{259} a^{8} - \frac{11}{37} a^{7} + \frac{82}{259} a^{6} - \frac{17}{37} a^{5} - \frac{115}{259} a^{4} - \frac{1}{7} a^{2} + \frac{17}{37} a + \frac{9}{37}$, $\frac{1}{1665964083343730418972737649839} a^{15} + \frac{374832420871999053119955019}{1665964083343730418972737649839} a^{14} + \frac{253564495793350011496261561}{237994869049104345567533949977} a^{13} + \frac{1411496743652596644565428631}{1665964083343730418972737649839} a^{12} - \frac{15115851542354452707887048388}{1665964083343730418972737649839} a^{11} - \frac{5095296845445971996535606635}{1665964083343730418972737649839} a^{10} - \frac{21841435391112110987569121611}{1665964083343730418972737649839} a^{9} - \frac{20687648307275706039857875884}{1665964083343730418972737649839} a^{8} - \frac{621566233818204362786005868174}{1665964083343730418972737649839} a^{7} + \frac{59381790371636264303670864152}{237994869049104345567533949977} a^{6} + \frac{455534654491621679509315660146}{1665964083343730418972737649839} a^{5} + \frac{92312879850453862531304786906}{1665964083343730418972737649839} a^{4} + \frac{6600191850223992798148470997}{1665964083343730418972737649839} a^{3} - \frac{566794327832444073669832810443}{1665964083343730418972737649839} a^{2} - \frac{579570547247434982354278139898}{1665964083343730418972737649839} a - \frac{94673990773731685309393186853}{237994869049104345567533949977}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 839406602.571 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.50653.1, 8.0.402819046213.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed
$157$$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
157.2.1.2$x^{2} + 785$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.4.0.1$x^{4} - x + 15$$1$$4$$0$$C_4$$[\ ]^{4}$
157.4.3.4$x^{4} + 19625$$4$$1$$3$$C_4$$[\ ]_{4}$