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Properties

Label 16.4.77740662152...1872.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{8}\cdot 17^{15}\cdot 103^{2}$
Root discriminant $35.95$
Ramified primes $2, 17, 103$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1291, -9918, -17852, -4627, -10621, -2646, -1963, 1535, -1616, 1639, -491, 207, 64, -40, 25, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 25*x^14 - 40*x^13 + 64*x^12 + 207*x^11 - 491*x^10 + 1639*x^9 - 1616*x^8 + 1535*x^7 - 1963*x^6 - 2646*x^5 - 10621*x^4 - 4627*x^3 - 17852*x^2 - 9918*x - 1291)
 
gp: K = bnfinit(x^16 - 5*x^15 + 25*x^14 - 40*x^13 + 64*x^12 + 207*x^11 - 491*x^10 + 1639*x^9 - 1616*x^8 + 1535*x^7 - 1963*x^6 - 2646*x^5 - 10621*x^4 - 4627*x^3 - 17852*x^2 - 9918*x - 1291, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 25 x^{14} - 40 x^{13} + 64 x^{12} + 207 x^{11} - 491 x^{10} + 1639 x^{9} - 1616 x^{8} + 1535 x^{7} - 1963 x^{6} - 2646 x^{5} - 10621 x^{4} - 4627 x^{3} - 17852 x^{2} - 9918 x - 1291 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7774066215287714751471872=2^{8}\cdot 17^{15}\cdot 103^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.95$
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:  $2, 17, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{67} a^{13} - \frac{18}{67} a^{12} + \frac{2}{67} a^{10} - \frac{16}{67} a^{9} + \frac{31}{67} a^{8} - \frac{27}{67} a^{7} + \frac{10}{67} a^{6} - \frac{20}{67} a^{5} - \frac{5}{67} a^{4} + \frac{29}{67} a^{3} + \frac{21}{67} a^{2} + \frac{5}{67} a + \frac{6}{67}$, $\frac{1}{67} a^{14} + \frac{11}{67} a^{12} + \frac{2}{67} a^{11} + \frac{20}{67} a^{10} + \frac{11}{67} a^{9} - \frac{5}{67} a^{8} - \frac{7}{67} a^{7} + \frac{26}{67} a^{6} - \frac{30}{67} a^{5} + \frac{6}{67} a^{4} + \frac{7}{67} a^{3} - \frac{19}{67} a^{2} + \frac{29}{67} a - \frac{26}{67}$, $\frac{1}{5088926267038173608691179} a^{15} + \frac{14306960358978824269622}{5088926267038173608691179} a^{14} - \frac{5285888942685682830433}{5088926267038173608691179} a^{13} - \frac{2268625753468079893809267}{5088926267038173608691179} a^{12} + \frac{1995992884038089214540823}{5088926267038173608691179} a^{11} + \frac{1286206375727016645448420}{5088926267038173608691179} a^{10} + \frac{1756668661406520776563386}{5088926267038173608691179} a^{9} - \frac{1751269749395532596058935}{5088926267038173608691179} a^{8} - \frac{1816044500253501173025772}{5088926267038173608691179} a^{7} + \frac{559031046083520400312202}{5088926267038173608691179} a^{6} + \frac{2093911139174481915644404}{5088926267038173608691179} a^{5} + \frac{1999620983027469003428884}{5088926267038173608691179} a^{4} - \frac{2232967427380029074665698}{5088926267038173608691179} a^{3} - \frac{1770718853665472923496217}{5088926267038173608691179} a^{2} - \frac{26357463894242680156392}{75954123388629456846137} a - \frac{1823080519159172869090749}{5088926267038173608691179}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 508837.60069 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed
$103$$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$