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Properties

Label 16.0.11623374807...1313.7
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 67^{8}$
Root discriminant $116.57$
Ramified primes $17, 67$
Class number $2304$ (GRH)
Class group $[2, 4, 12, 24]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2941783, 762645, 2190191, 255478, 1006059, 167323, 262416, 12272, 46394, 1555, 217, -661, 220, -42, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 42*x^13 + 220*x^12 - 661*x^11 + 217*x^10 + 1555*x^9 + 46394*x^8 + 12272*x^7 + 262416*x^6 + 167323*x^5 + 1006059*x^4 + 255478*x^3 + 2190191*x^2 + 762645*x + 2941783)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 - 42*x^13 + 220*x^12 - 661*x^11 + 217*x^10 + 1555*x^9 + 46394*x^8 + 12272*x^7 + 262416*x^6 + 167323*x^5 + 1006059*x^4 + 255478*x^3 + 2190191*x^2 + 762645*x + 2941783, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} - 42 x^{13} + 220 x^{12} - 661 x^{11} + 217 x^{10} + 1555 x^{9} + 46394 x^{8} + 12272 x^{7} + 262416 x^{6} + 167323 x^{5} + 1006059 x^{4} + 255478 x^{3} + 2190191 x^{2} + 762645 x + 2941783 \)

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:  \(1162337480711184271576898233831313=17^{15}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.57$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3263} a^{14} - \frac{1608}{3263} a^{13} - \frac{830}{3263} a^{12} - \frac{641}{3263} a^{11} - \frac{1208}{3263} a^{10} + \frac{528}{3263} a^{9} - \frac{658}{3263} a^{8} + \frac{1351}{3263} a^{7} - \frac{659}{3263} a^{6} - \frac{184}{3263} a^{5} + \frac{636}{3263} a^{4} + \frac{228}{3263} a^{3} - \frac{827}{3263} a^{2} + \frac{604}{3263} a - \frac{14}{251}$, $\frac{1}{9708851429195963609746063720323616927084070029} a^{15} + \frac{199074451254129075160116514222375269898433}{9708851429195963609746063720323616927084070029} a^{14} + \frac{4677847745746889987462052127144194425881818804}{9708851429195963609746063720323616927084070029} a^{13} + \frac{3000809726635130714935188599634538660999178930}{9708851429195963609746063720323616927084070029} a^{12} - \frac{3669399343650178840712338425529236218227648327}{9708851429195963609746063720323616927084070029} a^{11} - \frac{1923002343912046906660168366171997772152206046}{9708851429195963609746063720323616927084070029} a^{10} + \frac{117870798455700998385221051869185449600192859}{9708851429195963609746063720323616927084070029} a^{9} + \frac{661249754918737567900289671944383009501074288}{9708851429195963609746063720323616927084070029} a^{8} + \frac{3650888933131225500486331865812806679147987210}{9708851429195963609746063720323616927084070029} a^{7} + \frac{514372093230578636465651951688163671145098006}{9708851429195963609746063720323616927084070029} a^{6} - \frac{4714903429846292820836493671097036232087537322}{9708851429195963609746063720323616927084070029} a^{5} + \frac{2817567722395231626721759817690627272204871921}{9708851429195963609746063720323616927084070029} a^{4} + \frac{3830484294668721569224897737994972498233673715}{9708851429195963609746063720323616927084070029} a^{3} + \frac{4608134216082125122807376744109074163940516988}{9708851429195963609746063720323616927084070029} a^{2} + \frac{75081517785240021296099466611869774763795390}{746834725322766431518927978486432071314159233} a - \frac{28595834233441103541199398563163544314994047}{57448825024828187039917536806648620870319941}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{12}\times C_{24}$, which has order $2304$ (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:  \( 3171574.24527 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.0.8268784250602433.5

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$