↑

Properties

Label 16.0.70917653264...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 47^{12}$
Root discriminant $73.40$
Ramified primes $5, 47$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![272283001, 96448345, -158984011, -62209555, 33865162, 15873125, 397857, 476460, 252315, -93940, -37077, 2365, 2022, 120, -54, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 54*x^14 + 120*x^13 + 2022*x^12 + 2365*x^11 - 37077*x^10 - 93940*x^9 + 252315*x^8 + 476460*x^7 + 397857*x^6 + 15873125*x^5 + 33865162*x^4 - 62209555*x^3 - 158984011*x^2 + 96448345*x + 272283001)
 
gp: K = bnfinit(x^16 - 5*x^15 - 54*x^14 + 120*x^13 + 2022*x^12 + 2365*x^11 - 37077*x^10 - 93940*x^9 + 252315*x^8 + 476460*x^7 + 397857*x^6 + 15873125*x^5 + 33865162*x^4 - 62209555*x^3 - 158984011*x^2 + 96448345*x + 272283001, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 54 x^{14} + 120 x^{13} + 2022 x^{12} + 2365 x^{11} - 37077 x^{10} - 93940 x^{9} + 252315 x^{8} + 476460 x^{7} + 397857 x^{6} + 15873125 x^{5} + 33865162 x^{4} - 62209555 x^{3} - 158984011 x^{2} + 96448345 x + 272283001 \)

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:  \(709176532647391224615478515625=5^{14}\cdot 47^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.40$
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:  $5, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{131} a^{13} + \frac{11}{131} a^{12} - \frac{20}{131} a^{11} - \frac{17}{131} a^{10} - \frac{57}{131} a^{9} + \frac{14}{131} a^{8} + \frac{2}{131} a^{7} - \frac{40}{131} a^{6} - \frac{65}{131} a^{5} + \frac{21}{131} a^{4} + \frac{36}{131} a^{3} - \frac{41}{131} a^{2} + \frac{51}{131} a + \frac{15}{131}$, $\frac{1}{2489} a^{14} - \frac{8}{2489} a^{13} - \frac{491}{2489} a^{12} - \frac{685}{2489} a^{11} + \frac{528}{2489} a^{10} + \frac{311}{2489} a^{9} - \frac{788}{2489} a^{8} + \frac{577}{2489} a^{7} + \frac{9}{131} a^{6} - \frac{58}{131} a^{5} + \frac{947}{2489} a^{4} + \frac{192}{2489} a^{3} - \frac{87}{2489} a^{2} + \frac{1142}{2489} a + \frac{894}{2489}$, $\frac{1}{2213168819003997566896561816990254369753410799275184466561} a^{15} - \frac{405192170699617245331213377785533523664940723310458962}{2213168819003997566896561816990254369753410799275184466561} a^{14} + \frac{7598588016032501601771341395444075238488965369420836617}{2213168819003997566896561816990254369753410799275184466561} a^{13} - \frac{140098063504670375615953376282013352498794369628908803087}{2213168819003997566896561816990254369753410799275184466561} a^{12} + \frac{910747869134505902958507404781890578722385275187702802907}{2213168819003997566896561816990254369753410799275184466561} a^{11} - \frac{614251227474765110184244474349093394496209111518557762907}{2213168819003997566896561816990254369753410799275184466561} a^{10} - \frac{519943643650427656742782967479069634909323991529550704380}{2213168819003997566896561816990254369753410799275184466561} a^{9} - \frac{142229657372785855156522666284926175589450276608175307121}{2213168819003997566896561816990254369753410799275184466561} a^{8} + \frac{390807278577699183863591423666061292147370923467622910}{116482569421263029836661148262644966829126884172378129819} a^{7} + \frac{32782065915922320450346998387083132360445298845024918812}{116482569421263029836661148262644966829126884172378129819} a^{6} - \frac{378951709829609170453764219184391097840211017261207936600}{2213168819003997566896561816990254369753410799275184466561} a^{5} + \frac{673964342134744141895378800828804246842975923741796621320}{2213168819003997566896561816990254369753410799275184466561} a^{4} + \frac{714610259862292062400127669726913769650997972411635046314}{2213168819003997566896561816990254369753410799275184466561} a^{3} - \frac{233239112986548714011124691011771320721949512334808359913}{2213168819003997566896561816990254369753410799275184466561} a^{2} + \frac{108391530459133993842588447910769304898378325555027889656}{2213168819003997566896561816990254369753410799275184466561} a + \frac{2623862991017439904479955452862367034112180373463192}{7059121836328890966405741970949940417497538583866319}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( \frac{9346917821637279914437863667742831244028977182530}{116482569421263029836661148262644966829126884172378129819} a^{15} - \frac{75384939091075986431901586847519823670591840329268}{116482569421263029836661148262644966829126884172378129819} a^{14} - \frac{275174354171542532854620871313188791157425610839359}{116482569421263029836661148262644966829126884172378129819} a^{13} + \frac{1964239754221651528712674874777616725544935086508153}{116482569421263029836661148262644966829126884172378129819} a^{12} + \frac{13033663781371407451521653581569344856471843163267447}{116482569421263029836661148262644966829126884172378129819} a^{11} - \frac{17810537668580881700055397470999671361506883587372385}{116482569421263029836661148262644966829126884172378129819} a^{10} - \frac{296887038132011278965657329537970584483206003593253449}{116482569421263029836661148262644966829126884172378129819} a^{9} + \frac{16514209788267707384412054368597250788129806693955895}{116482569421263029836661148262644966829126884172378129819} a^{8} + \frac{2384075391077359618243836391782503959172876128116271173}{116482569421263029836661148262644966829126884172378129819} a^{7} - \frac{2429315281178465402237760076119567002688354052365352933}{116482569421263029836661148262644966829126884172378129819} a^{6} + \frac{10816050927900297775217613380541917546106952251238481171}{116482569421263029836661148262644966829126884172378129819} a^{5} + \frac{112331743432806462818978168351836697819912548582430848701}{116482569421263029836661148262644966829126884172378129819} a^{4} - \frac{28155882734483704598120979631696687344798835131325479323}{116482569421263029836661148262644966829126884172378129819} a^{3} - \frac{531151217323157386942110203110033837838771626153329403535}{116482569421263029836661148262644966829126884172378129819} a^{2} + \frac{10453740307769477256797209567535721246208073145952558823}{116482569421263029836661148262644966829126884172378129819} a + \frac{62093138286729203816055999450938047746430122446401927}{7059121836328890966405741970949940417497538583866319} \) (order $10$)
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:  \( 162503363.421 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-235}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{5}, \sqrt{-47})\), 4.4.276125.1, \(\Q(\zeta_{5})\), 8.0.76245015625.1, 8.4.842126197578125.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
47Data not computed