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Properties

Label 16.8.72976987184...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{12}\cdot 61^{6}\cdot 97^{4}$
Root discriminant $98.05$
Ramified primes $2, 5, 61, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T864

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6298321, 23497154, -97669675, 3934772, 7386366, -7838166, 6809527, -470392, -545932, 213152, -86878, 21866, -3104, 218, 15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 15*x^14 + 218*x^13 - 3104*x^12 + 21866*x^11 - 86878*x^10 + 213152*x^9 - 545932*x^8 - 470392*x^7 + 6809527*x^6 - 7838166*x^5 + 7386366*x^4 + 3934772*x^3 - 97669675*x^2 + 23497154*x + 6298321)
 
gp: K = bnfinit(x^16 - 4*x^15 + 15*x^14 + 218*x^13 - 3104*x^12 + 21866*x^11 - 86878*x^10 + 213152*x^9 - 545932*x^8 - 470392*x^7 + 6809527*x^6 - 7838166*x^5 + 7386366*x^4 + 3934772*x^3 - 97669675*x^2 + 23497154*x + 6298321, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 15 x^{14} + 218 x^{13} - 3104 x^{12} + 21866 x^{11} - 86878 x^{10} + 213152 x^{9} - 545932 x^{8} - 470392 x^{7} + 6809527 x^{6} - 7838166 x^{5} + 7386366 x^{4} + 3934772 x^{3} - 97669675 x^{2} + 23497154 x + 6298321 \)

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:  \(72976987184482631056000000000000=2^{16}\cdot 5^{12}\cdot 61^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.05$
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:  $2, 5, 61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{15} a^{12} - \frac{1}{15} a^{11} - \frac{2}{15} a^{10} - \frac{2}{15} a^{9} - \frac{4}{15} a^{7} - \frac{7}{15} a^{6} - \frac{1}{15} a^{5} + \frac{1}{3} a^{4} + \frac{2}{15} a^{3} - \frac{7}{15} a^{2} - \frac{4}{15} a - \frac{4}{15}$, $\frac{1}{15} a^{13} + \frac{2}{15} a^{11} + \frac{1}{15} a^{10} - \frac{2}{15} a^{9} - \frac{4}{15} a^{8} + \frac{4}{15} a^{7} + \frac{7}{15} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{3} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{15}$, $\frac{1}{8955} a^{14} - \frac{62}{8955} a^{13} + \frac{277}{8955} a^{12} + \frac{139}{2985} a^{11} - \frac{128}{2985} a^{10} + \frac{19}{1791} a^{9} - \frac{217}{995} a^{8} - \frac{947}{2985} a^{7} - \frac{89}{1791} a^{6} - \frac{572}{2985} a^{5} - \frac{2329}{8955} a^{4} + \frac{4279}{8955} a^{3} - \frac{314}{995} a^{2} + \frac{1792}{8955} a + \frac{4273}{8955}$, $\frac{1}{1548828614886864488364770931893260234324522522635952112565} a^{15} + \frac{4824528079320691429055859856524685454858005333104656}{172092068320762720929418992432584470480502502515105790285} a^{14} - \frac{3394069191133936025857574845765019857043121981364564574}{103255240992457632557651395459550682288301501509063474171} a^{13} + \frac{4369510357648015600065107350812584041577044681267437074}{1548828614886864488364770931893260234324522522635952112565} a^{12} + \frac{21158732789528766848866578983999954265670465290240614636}{516276204962288162788256977297753411441507507545317370855} a^{11} - \frac{197949074146465595121354819479174612821647127522965643459}{1548828614886864488364770931893260234324522522635952112565} a^{10} - \frac{107223106561294811961412360447842359142558880865597295803}{1548828614886864488364770931893260234324522522635952112565} a^{9} + \frac{82520833443502464369988310218588150483998235867464791627}{516276204962288162788256977297753411441507507545317370855} a^{8} + \frac{294926111097797900368248224671378304066771731446255120254}{1548828614886864488364770931893260234324522522635952112565} a^{7} + \frac{387646465502482626390650094233384544064927741914982827289}{1548828614886864488364770931893260234324522522635952112565} a^{6} - \frac{112578227179026792918647089491612110447131651627131298523}{309765722977372897672954186378652046864904504527190422513} a^{5} - \frac{75058177807075401630795310654709210082316004952732236450}{309765722977372897672954186378652046864904504527190422513} a^{4} + \frac{10868445701814411574785551356812249962081111974461643328}{1548828614886864488364770931893260234324522522635952112565} a^{3} + \frac{562795663520812769568966555140732441715825476732172592726}{1548828614886864488364770931893260234324522522635952112565} a^{2} - \frac{9020711609461355653264590633341800049796351541124772592}{34418413664152544185883798486516894096100500503021158057} a - \frac{693859547286183006561817280605513675253423444465149186687}{1548828614886864488364770931893260234324522522635952112565}$

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:  \( 8748995701.53 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T864:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n864
Character table for t16n864 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.14884000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$