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Properties

Label 16.0.97190103089...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{12}\cdot 9929^{4}$
Root discriminant $56.13$
Ramified primes $2, 5, 9929$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10180861, -997033, 9216181, -1463547, 3965073, -698966, 1019176, -176611, 169521, -26109, 18491, -2290, 1290, -107, 52, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 52*x^14 - 107*x^13 + 1290*x^12 - 2290*x^11 + 18491*x^10 - 26109*x^9 + 169521*x^8 - 176611*x^7 + 1019176*x^6 - 698966*x^5 + 3965073*x^4 - 1463547*x^3 + 9216181*x^2 - 997033*x + 10180861)
 
gp: K = bnfinit(x^16 - 2*x^15 + 52*x^14 - 107*x^13 + 1290*x^12 - 2290*x^11 + 18491*x^10 - 26109*x^9 + 169521*x^8 - 176611*x^7 + 1019176*x^6 - 698966*x^5 + 3965073*x^4 - 1463547*x^3 + 9216181*x^2 - 997033*x + 10180861, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 52 x^{14} - 107 x^{13} + 1290 x^{12} - 2290 x^{11} + 18491 x^{10} - 26109 x^{9} + 169521 x^{8} - 176611 x^{7} + 1019176 x^{6} - 698966 x^{5} + 3965073 x^{4} - 1463547 x^{3} + 9216181 x^{2} - 997033 x + 10180861 \)

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:  \(9719010308971681000000000000=2^{12}\cdot 5^{12}\cdot 9929^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.13$
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:  $2, 5, 9929$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1171707686450785967172766206025370086428541} a^{15} + \frac{145772569114289994394367875114902530551128}{1171707686450785967172766206025370086428541} a^{14} + \frac{79388545682802519818420853678033361046942}{1171707686450785967172766206025370086428541} a^{13} - \frac{82540578096057327133946619368327749271936}{1171707686450785967172766206025370086428541} a^{12} - \frac{71121666003839659752786669813874187467270}{1171707686450785967172766206025370086428541} a^{11} - \frac{29841117294920300387822118055897298700302}{68923981555928586304280365060315887436973} a^{10} + \frac{7971977592760680224990917960007227606912}{1171707686450785967172766206025370086428541} a^{9} - \frac{481015437938666436629000646517802759396789}{1171707686450785967172766206025370086428541} a^{8} + \frac{374789622169240777204623767925571958297073}{1171707686450785967172766206025370086428541} a^{7} + \frac{352260884277407294108312550659781347311855}{1171707686450785967172766206025370086428541} a^{6} - \frac{4628468820324628824578607851848507420734}{1171707686450785967172766206025370086428541} a^{5} + \frac{55529174617751397628368507606883833928732}{1171707686450785967172766206025370086428541} a^{4} - \frac{94924870519004418521921363807151895804679}{1171707686450785967172766206025370086428541} a^{3} - \frac{138722434491949730136672025254023988515268}{1171707686450785967172766206025370086428541} a^{2} + \frac{75988362260889230735964565809736117609904}{1171707686450785967172766206025370086428541} a - \frac{509494776782943042002078790636125591528666}{1171707686450785967172766206025370086428541}$

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:  $7$
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:  \( 3287689.12716 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1869 are not computed
Character table for t16n1869 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.4.155140625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.12.12.19$x^{12} - 6 x^{10} + 27 x^{8} - 4 x^{6} + 7 x^{4} + 10 x^{2} + 29$$2$$6$$12$12T105$[2, 2, 2, 2]^{12}$
5Data not computed
9929Data not computed