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Properties

Label 16.0.25916019301...625.12
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{6}$
Root discriminant $18.87$
Ramified primes $5, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5041, -3479, 12119, -5424, 11209, -3986, 5648, -2087, 1891, -860, 493, -237, 104, -35, 15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 15*x^14 - 35*x^13 + 104*x^12 - 237*x^11 + 493*x^10 - 860*x^9 + 1891*x^8 - 2087*x^7 + 5648*x^6 - 3986*x^5 + 11209*x^4 - 5424*x^3 + 12119*x^2 - 3479*x + 5041)
 
gp: K = bnfinit(x^16 - 2*x^15 + 15*x^14 - 35*x^13 + 104*x^12 - 237*x^11 + 493*x^10 - 860*x^9 + 1891*x^8 - 2087*x^7 + 5648*x^6 - 3986*x^5 + 11209*x^4 - 5424*x^3 + 12119*x^2 - 3479*x + 5041, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 15 x^{14} - 35 x^{13} + 104 x^{12} - 237 x^{11} + 493 x^{10} - 860 x^{9} + 1891 x^{8} - 2087 x^{7} + 5648 x^{6} - 3986 x^{5} + 11209 x^{4} - 5424 x^{3} + 12119 x^{2} - 3479 x + 5041 \)

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:  \(259160193017822265625=5^{12}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.87$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{32365512640164631935031} a^{15} - \frac{4118617289331707306916}{32365512640164631935031} a^{14} - \frac{14746950173556684925296}{32365512640164631935031} a^{13} - \frac{6231357823106687423951}{32365512640164631935031} a^{12} + \frac{9932998351115014242142}{32365512640164631935031} a^{11} + \frac{7141070998906811069898}{32365512640164631935031} a^{10} - \frac{10345471289736257844161}{32365512640164631935031} a^{9} + \frac{4045403028262294257793}{32365512640164631935031} a^{8} + \frac{4778491452210181285765}{32365512640164631935031} a^{7} + \frac{12647430672520091676978}{32365512640164631935031} a^{6} - \frac{9925821505477975716884}{32365512640164631935031} a^{5} - \frac{985656012436600308973}{32365512640164631935031} a^{4} + \frac{8542455133203817727780}{32365512640164631935031} a^{3} + \frac{1580618657369376391740}{32365512640164631935031} a^{2} + \frac{263346415697451793759}{32365512640164631935031} a + \frac{159679525482700053966}{455852290706544111761}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{4951145675706059159}{1044048794844020385001} a^{15} - \frac{13596892400226106520}{1044048794844020385001} a^{14} + \frac{70542323483369498203}{1044048794844020385001} a^{13} - \frac{193594546831675306891}{1044048794844020385001} a^{12} + \frac{478627303913287523966}{1044048794844020385001} a^{11} - \frac{1068736034649502349771}{1044048794844020385001} a^{10} + \frac{2129150690789142976419}{1044048794844020385001} a^{9} - \frac{3402730291693752251596}{1044048794844020385001} a^{8} + \frac{7152712820625952516195}{1044048794844020385001} a^{7} - \frac{8585416532099784422206}{1044048794844020385001} a^{6} + \frac{18114939293307323692961}{1044048794844020385001} a^{5} - \frac{17030594990475097047943}{1044048794844020385001} a^{4} + \frac{27390982535147587060109}{1044048794844020385001} a^{3} - \frac{18318745451584444307628}{1044048794844020385001} a^{2} + \frac{16263518454063609516759}{1044048794844020385001} a - \frac{71309558864557845978}{14704912603436906831} \) (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:  \( 8532.43184084 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.1578125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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}$
101Data not computed