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Properties

Label 16.8.29630251663...0000.6
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $33.84$
Ramified primes $2, 5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-199, 480, 1140, -5972, 7687, 1660, -9896, 4432, 1584, -1088, -266, 456, -221, 36, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 + 36*x^13 - 221*x^12 + 456*x^11 - 266*x^10 - 1088*x^9 + 1584*x^8 + 4432*x^7 - 9896*x^6 + 1660*x^5 + 7687*x^4 - 5972*x^3 + 1140*x^2 + 480*x - 199)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 + 36*x^13 - 221*x^12 + 456*x^11 - 266*x^10 - 1088*x^9 + 1584*x^8 + 4432*x^7 - 9896*x^6 + 1660*x^5 + 7687*x^4 - 5972*x^3 + 1140*x^2 + 480*x - 199, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} + 36 x^{13} - 221 x^{12} + 456 x^{11} - 266 x^{10} - 1088 x^{9} + 1584 x^{8} + 4432 x^{7} - 9896 x^{6} + 1660 x^{5} + 7687 x^{4} - 5972 x^{3} + 1140 x^{2} + 480 x - 199 \)

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:  \(2963025166336000000000000=2^{32}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.84$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a + \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{410} a^{14} - \frac{9}{205} a^{13} + \frac{2}{205} a^{12} - \frac{61}{205} a^{11} - \frac{72}{205} a^{10} + \frac{84}{205} a^{9} - \frac{77}{205} a^{8} + \frac{26}{205} a^{7} - \frac{63}{205} a^{6} - \frac{21}{205} a^{5} - \frac{91}{205} a^{4} + \frac{58}{205} a^{3} + \frac{57}{410} a^{2} + \frac{87}{205} a - \frac{56}{205}$, $\frac{1}{8283850735494412539310} a^{15} + \frac{1153657331496316014}{4141925367747206269655} a^{14} - \frac{34973673100889340003}{8283850735494412539310} a^{13} + \frac{3020449906547265519}{101022569945053811455} a^{12} + \frac{1157517809083229017233}{4141925367747206269655} a^{11} - \frac{165939998722489279507}{4141925367747206269655} a^{10} - \frac{593039230225531106128}{4141925367747206269655} a^{9} - \frac{482463267053407018919}{4141925367747206269655} a^{8} - \frac{1700250265519514718427}{4141925367747206269655} a^{7} - \frac{1985779441605733174983}{4141925367747206269655} a^{6} - \frac{452374299107227874556}{4141925367747206269655} a^{5} - \frac{183537680600916570591}{4141925367747206269655} a^{4} + \frac{436165093733774098499}{8283850735494412539310} a^{3} + \frac{1460751770913378909004}{4141925367747206269655} a^{2} + \frac{921259568640321193797}{8283850735494412539310} a + \frac{357362712550556607931}{828385073549441253931}$

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:  \( 2606459.0867 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$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}$
41Data not computed