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Properties

Label 16.8.18484762498...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{10}\cdot 761^{3}$
Root discriminant $37.95$
Ramified primes $2, 5, 761$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1398

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2169, -9768, 12878, 5744, -20563, -2372, 15362, 1704, -6843, -948, 1324, 320, 29, -24, -22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 22*x^14 - 24*x^13 + 29*x^12 + 320*x^11 + 1324*x^10 - 948*x^9 - 6843*x^8 + 1704*x^7 + 15362*x^6 - 2372*x^5 - 20563*x^4 + 5744*x^3 + 12878*x^2 - 9768*x + 2169)
 
gp: K = bnfinit(x^16 - 22*x^14 - 24*x^13 + 29*x^12 + 320*x^11 + 1324*x^10 - 948*x^9 - 6843*x^8 + 1704*x^7 + 15362*x^6 - 2372*x^5 - 20563*x^4 + 5744*x^3 + 12878*x^2 - 9768*x + 2169, 1)
 

Normalized defining polynomial

\( x^{16} - 22 x^{14} - 24 x^{13} + 29 x^{12} + 320 x^{11} + 1324 x^{10} - 948 x^{9} - 6843 x^{8} + 1704 x^{7} + 15362 x^{6} - 2372 x^{5} - 20563 x^{4} + 5744 x^{3} + 12878 x^{2} - 9768 x + 2169 \)

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:  \(18484762498826240000000000=2^{32}\cdot 5^{10}\cdot 761^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.95$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{191} a^{14} + \frac{50}{191} a^{13} + \frac{24}{191} a^{12} - \frac{48}{191} a^{11} + \frac{44}{191} a^{10} - \frac{18}{191} a^{9} - \frac{19}{191} a^{8} + \frac{63}{191} a^{7} - \frac{42}{191} a^{6} + \frac{94}{191} a^{5} - \frac{65}{191} a^{4} - \frac{31}{191} a^{3} + \frac{68}{191} a^{2} + \frac{32}{191} a + \frac{24}{191}$, $\frac{1}{2201689835295033427801999500003} a^{15} + \frac{107497743813562137370729777}{733896611765011142600666500001} a^{14} - \frac{472447323320494546471633363267}{2201689835295033427801999500003} a^{13} + \frac{68319059813919941262548554422}{733896611765011142600666500001} a^{12} - \frac{123131704832458565727302187097}{2201689835295033427801999500003} a^{11} + \frac{286696128958289514254814997997}{2201689835295033427801999500003} a^{10} - \frac{126376086169731766013261109584}{2201689835295033427801999500003} a^{9} + \frac{307838386932168416652382542403}{733896611765011142600666500001} a^{8} + \frac{137551912453527774388961543350}{733896611765011142600666500001} a^{7} + \frac{243062533809610309902705732573}{733896611765011142600666500001} a^{6} - \frac{136137340138715786676907855006}{2201689835295033427801999500003} a^{5} + \frac{996337890810636364384052969215}{2201689835295033427801999500003} a^{4} + \frac{514068991822807090561413237128}{2201689835295033427801999500003} a^{3} + \frac{105916882409478127353538943669}{2201689835295033427801999500003} a^{2} + \frac{372025287736854577432580716877}{2201689835295033427801999500003} a + \frac{158558524325088366208630568857}{733896611765011142600666500001}$

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:  \( 4993446.62306 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1398:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 71 conjugacy class representatives for t16n1398 are not computed
Character table for t16n1398 is not computed

Intermediate fields

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
761Data not computed