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Properties

Label 16.0.24471271802...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{14}\cdot 9929^{3}$
Root discriminant $38.62$
Ramified primes $2, 5, 9929$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group 16T1872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51851, -72795, 129214, -138940, 141942, -116005, 78422, -49895, 26010, -13220, 5678, -1955, 672, -155, 41, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 41*x^14 - 155*x^13 + 672*x^12 - 1955*x^11 + 5678*x^10 - 13220*x^9 + 26010*x^8 - 49895*x^7 + 78422*x^6 - 116005*x^5 + 141942*x^4 - 138940*x^3 + 129214*x^2 - 72795*x + 51851)
 
gp: K = bnfinit(x^16 - 5*x^15 + 41*x^14 - 155*x^13 + 672*x^12 - 1955*x^11 + 5678*x^10 - 13220*x^9 + 26010*x^8 - 49895*x^7 + 78422*x^6 - 116005*x^5 + 141942*x^4 - 138940*x^3 + 129214*x^2 - 72795*x + 51851, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 41 x^{14} - 155 x^{13} + 672 x^{12} - 1955 x^{11} + 5678 x^{10} - 13220 x^{9} + 26010 x^{8} - 49895 x^{7} + 78422 x^{6} - 116005 x^{5} + 141942 x^{4} - 138940 x^{3} + 129214 x^{2} - 72795 x + 51851 \)

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:  \(24471271802225000000000000=2^{12}\cdot 5^{14}\cdot 9929^{3}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{20900863468362448251006058957790042021} a^{15} + \frac{4104012123775582619512207841273097363}{20900863468362448251006058957790042021} a^{14} - \frac{4912987777820571610140070577495867373}{20900863468362448251006058957790042021} a^{13} + \frac{8528698957395382417814183926006832085}{20900863468362448251006058957790042021} a^{12} + \frac{2194496242418790013356905964093281548}{20900863468362448251006058957790042021} a^{11} - \frac{4177842987428490793899906322061698274}{20900863468362448251006058957790042021} a^{10} - \frac{3301034861362439940749882241194503973}{20900863468362448251006058957790042021} a^{9} - \frac{8997556790393282287762974570873210968}{20900863468362448251006058957790042021} a^{8} + \frac{3986727157176468001396863614010444365}{20900863468362448251006058957790042021} a^{7} - \frac{6029020833509922103337967457494930905}{20900863468362448251006058957790042021} a^{6} - \frac{6857201824117636839348385436531624554}{20900863468362448251006058957790042021} a^{5} + \frac{9204233622822837100446879320704732368}{20900863468362448251006058957790042021} a^{4} - \frac{6282499462876823414972085398319762}{20900863468362448251006058957790042021} a^{3} + \frac{3612206630313005580080834719482743116}{20900863468362448251006058957790042021} a^{2} + \frac{4494302736816012682351859425508801976}{20900863468362448251006058957790042021} a + \frac{1994168382119101587666101518450816375}{20900863468362448251006058957790042021}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 197884.814899 \) (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 $16$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.12.12.17$x^{12} + 22 x^{10} + 75 x^{8} - 12 x^{6} - 89 x^{4} + 54 x^{2} - 115$$2$$6$$12$12T29$[2, 2]^{12}$
5Data not computed
9929Data not computed