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Properties

Label 16.0.75482384716...209.11
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 61^{8}$
Root discriminant $73.68$
Ramified primes $13, 61$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![134139983, -81236989, 22641411, 29579481, -6362725, -1701847, -1006648, -120954, 194597, 11629, 1168, -5490, 1233, -179, 26, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 26*x^14 - 179*x^13 + 1233*x^12 - 5490*x^11 + 1168*x^10 + 11629*x^9 + 194597*x^8 - 120954*x^7 - 1006648*x^6 - 1701847*x^5 - 6362725*x^4 + 29579481*x^3 + 22641411*x^2 - 81236989*x + 134139983)
 
gp: K = bnfinit(x^16 - 6*x^15 + 26*x^14 - 179*x^13 + 1233*x^12 - 5490*x^11 + 1168*x^10 + 11629*x^9 + 194597*x^8 - 120954*x^7 - 1006648*x^6 - 1701847*x^5 - 6362725*x^4 + 29579481*x^3 + 22641411*x^2 - 81236989*x + 134139983, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 26 x^{14} - 179 x^{13} + 1233 x^{12} - 5490 x^{11} + 1168 x^{10} + 11629 x^{9} + 194597 x^{8} - 120954 x^{7} - 1006648 x^{6} - 1701847 x^{5} - 6362725 x^{4} + 29579481 x^{3} + 22641411 x^{2} - 81236989 x + 134139983 \)

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:  \(754823847161356593807740633209=13^{14}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.68$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{45} a^{14} - \frac{7}{45} a^{13} + \frac{7}{45} a^{12} - \frac{4}{45} a^{11} + \frac{1}{9} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{14}{45} a^{7} - \frac{8}{45} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{8}{45} a^{3} + \frac{1}{3} a^{2} - \frac{7}{45} a - \frac{2}{45}$, $\frac{1}{2674773898556367988763350223959422404729546220500938637265} a^{15} + \frac{5482113701008365107274151061897960302704212315513740472}{2674773898556367988763350223959422404729546220500938637265} a^{14} - \frac{405602445047463515776697903934415099800675025700261354111}{2674773898556367988763350223959422404729546220500938637265} a^{13} + \frac{247827810793233948518925365609541605473459370102493982484}{2674773898556367988763350223959422404729546220500938637265} a^{12} + \frac{350148673024920618782632103032995822766103794168034165234}{2674773898556367988763350223959422404729546220500938637265} a^{11} + \frac{91623828252767903698039610781679209314648942134174813393}{891591299518789329587783407986474134909848740166979545755} a^{10} - \frac{78043121703355583043288176806666324849945194832043795771}{178318259903757865917556681597294826981969748033395909151} a^{9} - \frac{681108475841892505635712134931512083247585459582114884018}{2674773898556367988763350223959422404729546220500938637265} a^{8} + \frac{1020427273634295504339773287373773395171395761773887723431}{2674773898556367988763350223959422404729546220500938637265} a^{7} - \frac{41044761496932648228120985099259691201306966160849269658}{891591299518789329587783407986474134909848740166979545755} a^{6} - \frac{55257282519981079244735537471638439120436298877859901099}{297197099839596443195927802662158044969949580055659848585} a^{5} - \frac{791918224416968120050046422527845104920035485687726872964}{2674773898556367988763350223959422404729546220500938637265} a^{4} + \frac{63238638950030869568629688317969665182107369345784217478}{297197099839596443195927802662158044969949580055659848585} a^{3} + \frac{667659639318185313501558766122164557400337056392151882843}{2674773898556367988763350223959422404729546220500938637265} a^{2} + \frac{237398457808064443475854374810829309144085074271168662262}{534954779711273597752670044791884480945909244100187727453} a + \frac{101074322110384871626732702907515493619905842858301015433}{297197099839596443195927802662158044969949580055659848585}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 55915267.1953 \) (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

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.294435349.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed
61Data not computed