↑

Properties

Label 16.8.17142224957...4976.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{58}\cdot 3^{4}\cdot 7^{6}\cdot 79^{2}$
Root discriminant $58.16$
Ramified primes $2, 3, 7, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T781

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![71138, 188176, 98064, -14832, 24088, 11248, -30320, -14368, 64, 352, 1576, 648, -252, -24, 16, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 16*x^14 - 24*x^13 - 252*x^12 + 648*x^11 + 1576*x^10 + 352*x^9 + 64*x^8 - 14368*x^7 - 30320*x^6 + 11248*x^5 + 24088*x^4 - 14832*x^3 + 98064*x^2 + 188176*x + 71138)
 
gp: K = bnfinit(x^16 - 8*x^15 + 16*x^14 - 24*x^13 - 252*x^12 + 648*x^11 + 1576*x^10 + 352*x^9 + 64*x^8 - 14368*x^7 - 30320*x^6 + 11248*x^5 + 24088*x^4 - 14832*x^3 + 98064*x^2 + 188176*x + 71138, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 16 x^{14} - 24 x^{13} - 252 x^{12} + 648 x^{11} + 1576 x^{10} + 352 x^{9} + 64 x^{8} - 14368 x^{7} - 30320 x^{6} + 11248 x^{5} + 24088 x^{4} - 14832 x^{3} + 98064 x^{2} + 188176 x + 71138 \)

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:  \(17142224957643668854696574976=2^{58}\cdot 3^{4}\cdot 7^{6}\cdot 79^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.16$
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:  $2, 3, 7, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} + \frac{4}{11} a^{13} + \frac{1}{11} a^{12} + \frac{4}{11} a^{10} + \frac{3}{11} a^{9} - \frac{4}{11} a^{8} + \frac{5}{11} a^{7} + \frac{2}{11} a^{6} + \frac{4}{11} a^{5} - \frac{5}{11} a^{4} + \frac{2}{11} a^{3} - \frac{4}{11} a^{2} - \frac{2}{11} a - \frac{4}{11}$, $\frac{1}{8487569983119822504312569759908108013} a^{15} - \frac{56832541137092295713828508993267171}{8487569983119822504312569759908108013} a^{14} + \frac{2363409817544881756898057469539917655}{8487569983119822504312569759908108013} a^{13} - \frac{1014260440042387678903945461357299446}{8487569983119822504312569759908108013} a^{12} - \frac{110875172392204129532172654610521787}{273792580100639435622986121287358323} a^{11} - \frac{3932688639759227329697771295433364848}{8487569983119822504312569759908108013} a^{10} - \frac{25065167225844240143847589760678504}{8487569983119822504312569759908108013} a^{9} - \frac{1044996179459561091147859628395857030}{8487569983119822504312569759908108013} a^{8} - \frac{2897217977669235175201580167154828475}{8487569983119822504312569759908108013} a^{7} + \frac{1970172121901031269652212751811052612}{8487569983119822504312569759908108013} a^{6} - \frac{64584909808859473192662614305884452}{273792580100639435622986121287358323} a^{5} - \frac{2971534079725212015991645871881106445}{8487569983119822504312569759908108013} a^{4} - \frac{1183639662264109727451349606649271525}{8487569983119822504312569759908108013} a^{3} - \frac{3385567308364278745062452076469075755}{8487569983119822504312569759908108013} a^{2} - \frac{54296716818460446979545685823521941}{273792580100639435622986121287358323} a - \frac{2436590371312176052527902574156742473}{8487569983119822504312569759908108013}$

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:  $11$
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:  \( 234536545.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n781 are not computed
Character table for t16n781 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.14336.1, 4.4.7168.1, 8.8.3288334336.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
79Data not computed