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Properties

Label 16.8.16726428348...0000.4
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 761^{5}$
Root discriminant $50.29$
Ramified primes $2, 5, 761$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2.D_4^2.C_2$ (as 16T659)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![215591, 413814, -72267, -285350, 51682, 65798, -36949, -11584, 6872, 1066, -397, 182, 164, -34, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 30*x^14 - 34*x^13 + 164*x^12 + 182*x^11 - 397*x^10 + 1066*x^9 + 6872*x^8 - 11584*x^7 - 36949*x^6 + 65798*x^5 + 51682*x^4 - 285350*x^3 - 72267*x^2 + 413814*x + 215591)
 
gp: K = bnfinit(x^16 - 30*x^14 - 34*x^13 + 164*x^12 + 182*x^11 - 397*x^10 + 1066*x^9 + 6872*x^8 - 11584*x^7 - 36949*x^6 + 65798*x^5 + 51682*x^4 - 285350*x^3 - 72267*x^2 + 413814*x + 215591, 1)
 

Normalized defining polynomial

\( x^{16} - 30 x^{14} - 34 x^{13} + 164 x^{12} + 182 x^{11} - 397 x^{10} + 1066 x^{9} + 6872 x^{8} - 11584 x^{7} - 36949 x^{6} + 65798 x^{5} + 51682 x^{4} - 285350 x^{3} - 72267 x^{2} + 413814 x + 215591 \)

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:  \(1672642834856679833600000000=2^{24}\cdot 5^{8}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.29$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{13} + \frac{2}{25} a^{12} + \frac{3}{25} a^{11} - \frac{12}{25} a^{10} + \frac{2}{25} a^{9} - \frac{6}{25} a^{8} - \frac{1}{25} a^{7} + \frac{7}{25} a^{6} + \frac{3}{25} a^{5} + \frac{11}{25} a^{3} + \frac{6}{25} a^{2} - \frac{1}{5} a - \frac{11}{25}$, $\frac{1}{127270063529080666170275124388599440725} a^{15} + \frac{1399383482617035999771601819497590456}{127270063529080666170275124388599440725} a^{14} - \frac{679884455231720490407690307095642049}{25454012705816133234055024877719888145} a^{13} + \frac{1313861963705081098227737646390587122}{127270063529080666170275124388599440725} a^{12} + \frac{11183952945263030564195312340281137074}{127270063529080666170275124388599440725} a^{11} - \frac{41202687140037195888134660207002979667}{127270063529080666170275124388599440725} a^{10} - \frac{979532497289160579539222737130271677}{127270063529080666170275124388599440725} a^{9} - \frac{21914236997646774044733022416135697713}{127270063529080666170275124388599440725} a^{8} + \frac{5555763962170076088383458875628132127}{25454012705816133234055024877719888145} a^{7} - \frac{44002675761466602698204057387840865443}{127270063529080666170275124388599440725} a^{6} + \frac{5307061672450048603110613293892316136}{127270063529080666170275124388599440725} a^{5} + \frac{47893690482911304896591536869936927576}{127270063529080666170275124388599440725} a^{4} + \frac{18973398001032416055184151613379433143}{127270063529080666170275124388599440725} a^{3} - \frac{24088058314716883331510792564623035688}{127270063529080666170275124388599440725} a^{2} - \frac{19092424304291931585235186882249386761}{127270063529080666170275124388599440725} a - \frac{54920719852181677348225880110269018362}{127270063529080666170275124388599440725}$

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:  \( 18606574.5705 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.D_4^2.C_2$ (as 16T659):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 25 conjugacy class representatives for $C_2.D_4^2.C_2$
Character table for $C_2.D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$5.4.2.1$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}$
5.4.2.1$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}$
761Data not computed