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Properties

Label 16.16.7700400667...1424.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{56}\cdot 7^{4}\cdot 17^{4}\cdot 73^{2}$
Root discriminant $63.89$
Ramified primes $2, 7, 17, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-114206, 27744, 600216, 278896, -721408, -583440, 219304, 327552, 29860, -62168, -16144, 4488, 1792, -96, -72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 72*x^14 - 96*x^13 + 1792*x^12 + 4488*x^11 - 16144*x^10 - 62168*x^9 + 29860*x^8 + 327552*x^7 + 219304*x^6 - 583440*x^5 - 721408*x^4 + 278896*x^3 + 600216*x^2 + 27744*x - 114206)
 
gp: K = bnfinit(x^16 - 72*x^14 - 96*x^13 + 1792*x^12 + 4488*x^11 - 16144*x^10 - 62168*x^9 + 29860*x^8 + 327552*x^7 + 219304*x^6 - 583440*x^5 - 721408*x^4 + 278896*x^3 + 600216*x^2 + 27744*x - 114206, 1)
 

Normalized defining polynomial

\( x^{16} - 72 x^{14} - 96 x^{13} + 1792 x^{12} + 4488 x^{11} - 16144 x^{10} - 62168 x^{9} + 29860 x^{8} + 327552 x^{7} + 219304 x^{6} - 583440 x^{5} - 721408 x^{4} + 278896 x^{3} + 600216 x^{2} + 27744 x - 114206 \)

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:  \(77004006676572437563964391424=2^{56}\cdot 7^{4}\cdot 17^{4}\cdot 73^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.89$
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:  $2, 7, 17, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{80423484981767527251261387} a^{15} - \frac{9941907132129237480434428}{80423484981767527251261387} a^{14} - \frac{26801364445786751012378542}{80423484981767527251261387} a^{13} + \frac{5840645660243173491526459}{80423484981767527251261387} a^{12} + \frac{11539853773389084158632208}{80423484981767527251261387} a^{11} + \frac{10369973848660878423041212}{80423484981767527251261387} a^{10} - \frac{10359130922782596570280858}{26807828327255842417087129} a^{9} + \frac{2111321844992895823146523}{80423484981767527251261387} a^{8} - \frac{12068008761624086604931751}{26807828327255842417087129} a^{7} - \frac{32023798062483476839344467}{80423484981767527251261387} a^{6} - \frac{29527049190365299147738139}{80423484981767527251261387} a^{5} - \frac{589517608131111574684999}{1961548414189451884177107} a^{4} + \frac{6114901374031193197573258}{80423484981767527251261387} a^{3} + \frac{9132017163536520734781064}{80423484981767527251261387} a^{2} + \frac{167343681949638402131591}{587032737093193629571251} a - \frac{37879216783680105938423743}{80423484981767527251261387}$

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:  \( 2956296158.56 \) (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

\(\Q(\sqrt{2}) \), 4.4.7168.1, \(\Q(\zeta_{16})^+\), 4.4.14336.1, 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$