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Properties

Label 16.16.6896992068...1168.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{62}\cdot 113^{3}\cdot 1009^{4}$
Root discriminant $200.64$
Ramified primes $2, 113, 1009$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1354

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![448136625922, 496411645008, -336014595440, -125162855440, 56738103992, 13083333328, -4199520744, -700013280, 166409360, 20439024, -3800032, -324544, 50132, 2592, -352, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 352*x^14 + 2592*x^13 + 50132*x^12 - 324544*x^11 - 3800032*x^10 + 20439024*x^9 + 166409360*x^8 - 700013280*x^7 - 4199520744*x^6 + 13083333328*x^5 + 56738103992*x^4 - 125162855440*x^3 - 336014595440*x^2 + 496411645008*x + 448136625922)
 
gp: K = bnfinit(x^16 - 8*x^15 - 352*x^14 + 2592*x^13 + 50132*x^12 - 324544*x^11 - 3800032*x^10 + 20439024*x^9 + 166409360*x^8 - 700013280*x^7 - 4199520744*x^6 + 13083333328*x^5 + 56738103992*x^4 - 125162855440*x^3 - 336014595440*x^2 + 496411645008*x + 448136625922, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 352 x^{14} + 2592 x^{13} + 50132 x^{12} - 324544 x^{11} - 3800032 x^{10} + 20439024 x^{9} + 166409360 x^{8} - 700013280 x^{7} - 4199520744 x^{6} + 13083333328 x^{5} + 56738103992 x^{4} - 125162855440 x^{3} - 336014595440 x^{2} + 496411645008 x + 448136625922 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6896992068684009105950840251291271168=2^{62}\cdot 113^{3}\cdot 1009^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.64$
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:  $2, 113, 1009$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{204509091869108509215444123217100849315196921566945383782080457262551031} a^{15} - \frac{516517524845140074406913866395203622046159107310411479422618550007459}{6597067479648661587594971716680672558554739405385334960712272814921001} a^{14} + \frac{2549603587811244877097849726228277063291991893105997343016044869102407}{204509091869108509215444123217100849315196921566945383782080457262551031} a^{13} - \frac{3227874433081713033477151990962960581326050307224158282221681120161583}{204509091869108509215444123217100849315196921566945383782080457262551031} a^{12} - \frac{23130298094332898428079986266708598306601170730939335614754448642224104}{204509091869108509215444123217100849315196921566945383782080457262551031} a^{11} + \frac{31105753799088969611628575063771410579695799512769440629677718601973289}{204509091869108509215444123217100849315196921566945383782080457262551031} a^{10} + \frac{84843954814410377772872095901001886364135519785345117733147304525096719}{204509091869108509215444123217100849315196921566945383782080457262551031} a^{9} + \frac{81815642033097386471693208863581343700304635167387689722417388718166380}{204509091869108509215444123217100849315196921566945383782080457262551031} a^{8} + \frac{20171317690959659109063702485199147789851819119608919731703169600728048}{204509091869108509215444123217100849315196921566945383782080457262551031} a^{7} + \frac{1963469877790397633811019823788335703822897763041147059257801624548232}{6597067479648661587594971716680672558554739405385334960712272814921001} a^{6} - \frac{254801036890694919505069204935140782468600932908662678624849641037600}{795755221280577856869432386058758168541622262906402271525604892072183} a^{5} - \frac{84410695576184692625465427491851968197805246234135411811421595383392678}{204509091869108509215444123217100849315196921566945383782080457262551031} a^{4} - \frac{90235586074990520055715611692776487499616842334570994817221598400049067}{204509091869108509215444123217100849315196921566945383782080457262551031} a^{3} + \frac{42806967957691626614039280374248147220821773973894022352548275338201637}{204509091869108509215444123217100849315196921566945383782080457262551031} a^{2} + \frac{2392934344407132161480353171170913160936725200977527096024089941356686}{204509091869108509215444123217100849315196921566945383782080457262551031} a - \frac{677052471733307728683684263643128034008041357197787839849045655621496}{2588722681887449483739799028064567712850593943885384604836461484336089}$

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:  $15$
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:  \( 31042889746300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1354:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 59 conjugacy class representatives for t16n1354 are not computed
Character table for t16n1354 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7583301632.1

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 R $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
2Data not computed
113Data not computed
1009Data not computed