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Properties

Label 16.8.69833752984...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{40}\cdot 5^{14}\cdot 101^{4}$
Root discriminant $73.33$
Ramified primes $2, 5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5279, -313580, -23276, 178840, -388788, -33880, 304472, 11200, -81135, -7480, 9668, 1120, -318, -20, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 - 20*x^13 - 318*x^12 + 1120*x^11 + 9668*x^10 - 7480*x^9 - 81135*x^8 + 11200*x^7 + 304472*x^6 - 33880*x^5 - 388788*x^4 + 178840*x^3 - 23276*x^2 - 313580*x - 5279)
 
gp: K = bnfinit(x^16 - 24*x^14 - 20*x^13 - 318*x^12 + 1120*x^11 + 9668*x^10 - 7480*x^9 - 81135*x^8 + 11200*x^7 + 304472*x^6 - 33880*x^5 - 388788*x^4 + 178840*x^3 - 23276*x^2 - 313580*x - 5279, 1)
 

Normalized defining polynomial

\( x^{16} - 24 x^{14} - 20 x^{13} - 318 x^{12} + 1120 x^{11} + 9668 x^{10} - 7480 x^{9} - 81135 x^{8} + 11200 x^{7} + 304472 x^{6} - 33880 x^{5} - 388788 x^{4} + 178840 x^{3} - 23276 x^{2} - 313580 x - 5279 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(698337529849446400000000000000=2^{40}\cdot 5^{14}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.33$
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:  $2, 5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3367019425328217358814274089713186666697133207} a^{15} - \frac{194407743945435903797211643753454047230699323}{3367019425328217358814274089713186666697133207} a^{14} + \frac{1096577962894954522326145091691998988715959524}{3367019425328217358814274089713186666697133207} a^{13} + \frac{204424533211382175311462554117821969138500873}{1122339808442739119604758029904395555565711069} a^{12} - \frac{369741932248729978360138625302700135911620973}{3367019425328217358814274089713186666697133207} a^{11} - \frac{221408700563034539044071416492842131158170763}{3367019425328217358814274089713186666697133207} a^{10} + \frac{1008187889448397976989781848842077248232778733}{3367019425328217358814274089713186666697133207} a^{9} - \frac{305342486027374714339747536489901822554504405}{1122339808442739119604758029904395555565711069} a^{8} - \frac{952887076501093614082193589269106210203354429}{3367019425328217358814274089713186666697133207} a^{7} - \frac{159216978693484586606731848491989244147161151}{3367019425328217358814274089713186666697133207} a^{6} - \frac{6936026078627450591764458179566832796500413}{3367019425328217358814274089713186666697133207} a^{5} + \frac{110006096048349746587115710888291833336920497}{1122339808442739119604758029904395555565711069} a^{4} + \frac{147277160031519364656248093035458850118600636}{1122339808442739119604758029904395555565711069} a^{3} - \frac{1045792356115458115996756451676173949836210069}{3367019425328217358814274089713186666697133207} a^{2} - \frac{475616483073719141715047592218680748651204755}{3367019425328217358814274089713186666697133207} a + \frac{1443389676475220694146242008985578517239696763}{3367019425328217358814274089713186666697133207}$

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:  $11$
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:  \( 1107580346.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.4.517120000000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
101Data not computed