↑

Properties

Label 16.4.24471271802...0000.4
Degree $16$
Signature $[4, 6]$
Discriminant $2^{12}\cdot 5^{14}\cdot 9929^{3}$
Root discriminant $38.62$
Ramified primes $2, 5, 9929$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1872

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2621, 10070, 10364, -6895, -29888, 32000, -30383, 25905, -14215, 5525, -1152, -70, 227, -85, 31, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 31*x^14 - 85*x^13 + 227*x^12 - 70*x^11 - 1152*x^10 + 5525*x^9 - 14215*x^8 + 25905*x^7 - 30383*x^6 + 32000*x^5 - 29888*x^4 - 6895*x^3 + 10364*x^2 + 10070*x + 2621)
 
gp: K = bnfinit(x^16 - 5*x^15 + 31*x^14 - 85*x^13 + 227*x^12 - 70*x^11 - 1152*x^10 + 5525*x^9 - 14215*x^8 + 25905*x^7 - 30383*x^6 + 32000*x^5 - 29888*x^4 - 6895*x^3 + 10364*x^2 + 10070*x + 2621, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 31 x^{14} - 85 x^{13} + 227 x^{12} - 70 x^{11} - 1152 x^{10} + 5525 x^{9} - 14215 x^{8} + 25905 x^{7} - 30383 x^{6} + 32000 x^{5} - 29888 x^{4} - 6895 x^{3} + 10364 x^{2} + 10070 x + 2621 \)

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:  \(24471271802225000000000000=2^{12}\cdot 5^{14}\cdot 9929^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.62$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{23267428462245731721786088350874931599} a^{15} + \frac{1581679600287499242776429149042831557}{23267428462245731721786088350874931599} a^{14} - \frac{7184732093129071324893078764234195600}{23267428462245731721786088350874931599} a^{13} + \frac{5786608531795734801574244048050037141}{23267428462245731721786088350874931599} a^{12} + \frac{7687547638980667935246170355743160820}{23267428462245731721786088350874931599} a^{11} + \frac{6979851988226112500742709648468195622}{23267428462245731721786088350874931599} a^{10} + \frac{4914820002072946791602485097810285304}{23267428462245731721786088350874931599} a^{9} - \frac{5812942652104172627272233581302397217}{23267428462245731721786088350874931599} a^{8} + \frac{5427072320856473083878042372446480466}{23267428462245731721786088350874931599} a^{7} - \frac{10874452587042794633641719232438255196}{23267428462245731721786088350874931599} a^{6} - \frac{5963984882274882236411083903406333882}{23267428462245731721786088350874931599} a^{5} - \frac{2967803792847316272516040390787242007}{23267428462245731721786088350874931599} a^{4} - \frac{3665303641932206563611843394752207568}{23267428462245731721786088350874931599} a^{3} + \frac{4884029857219298838171131962347911696}{23267428462245731721786088350874931599} a^{2} + \frac{7695091138704215577750652427794190203}{23267428462245731721786088350874931599} a - \frac{9652152670508440735412412456555339915}{23267428462245731721786088350874931599}$

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:  $9$
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:  \( 1937913.38623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1872:

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 t16n1872 are not computed
Character table for t16n1872 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 $16$ R $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.12.12.6$x^{12} - 18 x^{10} + 11 x^{8} - 52 x^{6} - x^{4} + 6 x^{2} - 11$$2$$6$$12$12T105$[2, 2, 2, 2]^{12}$
5Data not computed
9929Data not computed