↑

Properties

Label 16.8.22866941668...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{10}\cdot 29^{6}\cdot 89^{8}$
Root discriminant $91.19$
Ramified primes $5, 29, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T516)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-30234725, -94897975, -78387230, 811415, 7128834, -396868, 7514944, 554051, 64529, 239203, -75553, 4526, 341, -442, 59, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 59*x^14 - 442*x^13 + 341*x^12 + 4526*x^11 - 75553*x^10 + 239203*x^9 + 64529*x^8 + 554051*x^7 + 7514944*x^6 - 396868*x^5 + 7128834*x^4 + 811415*x^3 - 78387230*x^2 - 94897975*x - 30234725)
 
gp: K = bnfinit(x^16 - 6*x^15 + 59*x^14 - 442*x^13 + 341*x^12 + 4526*x^11 - 75553*x^10 + 239203*x^9 + 64529*x^8 + 554051*x^7 + 7514944*x^6 - 396868*x^5 + 7128834*x^4 + 811415*x^3 - 78387230*x^2 - 94897975*x - 30234725, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 59 x^{14} - 442 x^{13} + 341 x^{12} + 4526 x^{11} - 75553 x^{10} + 239203 x^{9} + 64529 x^{8} + 554051 x^{7} + 7514944 x^{6} - 396868 x^{5} + 7128834 x^{4} + 811415 x^{3} - 78387230 x^{2} - 94897975 x - 30234725 \)

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:  \(22866941668155620674130869140625=5^{10}\cdot 29^{6}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.19$
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:  $5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{60} a^{14} + \frac{7}{30} a^{13} + \frac{1}{15} a^{12} - \frac{1}{5} a^{11} + \frac{7}{20} a^{10} + \frac{13}{30} a^{9} + \frac{1}{5} a^{8} - \frac{9}{20} a^{7} + \frac{29}{60} a^{6} - \frac{2}{5} a^{5} + \frac{3}{20} a^{4} - \frac{7}{15} a^{3} + \frac{19}{60} a^{2} + \frac{1}{4} a + \frac{1}{12}$, $\frac{1}{374227521131359410558937565545135830560809257506351851204751400} a^{15} + \frac{118844580903764211606428281008330547060075519650921483702571}{19696185322703126871523029765533464766358381974018518484460600} a^{14} + \frac{4782610575812063381367516400287520651969573862163872252219857}{187113760565679705279468782772567915280404628753175925602375700} a^{13} - \frac{17267247176519328210379021694770810144796007408524267768066983}{93556880282839852639734391386283957640202314376587962801187850} a^{12} + \frac{20268070477818863341575753194219608299151240837777669109384607}{124742507043786470186312521848378610186936419168783950401583800} a^{11} - \frac{88744582176442770116903543891057540995725824155385700998421739}{374227521131359410558937565545135830560809257506351851204751400} a^{10} - \frac{46148864019217930177825844315864513542075659777260937761967019}{187113760565679705279468782772567915280404628753175925602375700} a^{9} + \frac{33478579040407540797460142432482673728423705214322336392931591}{124742507043786470186312521848378610186936419168783950401583800} a^{8} - \frac{9731070049550375055865268804270738740475680633060912866675559}{93556880282839852639734391386283957640202314376587962801187850} a^{7} - \frac{151090486691307452237130436405356244139892752509027471782316349}{374227521131359410558937565545135830560809257506351851204751400} a^{6} - \frac{7119946418535873732064282030694191117236924252740578989969437}{124742507043786470186312521848378610186936419168783950401583800} a^{5} + \frac{1476592921583450731693409492881219392355246976364866911206773}{19696185322703126871523029765533464766358381974018518484460600} a^{4} - \frac{77383300563926321719750342373652396532885686828980957540749441}{374227521131359410558937565545135830560809257506351851204751400} a^{3} - \frac{2656872837445307035530366150969295553917777002133631157135088}{9355688028283985263973439138628395764020231437658796280118785} a^{2} + \frac{10124182242086682048948803973080135233116974094196792102575041}{37422752113135941055893756554513583056080925750635185120475140} a - \frac{5592515621573863185653678655762222222384780672441395401889849}{14969100845254376422357502621805433222432370300254074048190056}$

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:  $11$
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:  \( 2889847525.14 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T516):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.2, 4.4.2225.1, 4.4.725.1, 8.8.4163475625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$