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Properties

Label 16.0.31170560000...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 761$
Root discriminant $14.32$
Ramified primes $2, 5, 761$
Class number $1$
Class group Trivial
Galois group 16T1379

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 7, -4, -7, 4, 29, -132, 293, -384, 346, -248, 148, -68, 23, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 23*x^14 - 68*x^13 + 148*x^12 - 248*x^11 + 346*x^10 - 384*x^9 + 293*x^8 - 132*x^7 + 29*x^6 + 4*x^5 - 7*x^4 - 4*x^3 + 7*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 23*x^14 - 68*x^13 + 148*x^12 - 248*x^11 + 346*x^10 - 384*x^9 + 293*x^8 - 132*x^7 + 29*x^6 + 4*x^5 - 7*x^4 - 4*x^3 + 7*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 23 x^{14} - 68 x^{13} + 148 x^{12} - 248 x^{11} + 346 x^{10} - 384 x^{9} + 293 x^{8} - 132 x^{7} + 29 x^{6} + 4 x^{5} - 7 x^{4} - 4 x^{3} + 7 x^{2} - 2 x + 1 \)

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:  \(3117056000000000000=2^{24}\cdot 5^{12}\cdot 761\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.32$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} - \frac{2}{19} a^{13} - \frac{9}{19} a^{12} + \frac{1}{19} a^{11} + \frac{7}{19} a^{10} + \frac{3}{19} a^{9} + \frac{8}{19} a^{6} - \frac{5}{19} a^{5} + \frac{7}{19} a^{4} - \frac{4}{19} a^{2} - \frac{1}{19} a + \frac{4}{19}$, $\frac{1}{14988739} a^{15} + \frac{49890}{14988739} a^{14} + \frac{391908}{14988739} a^{13} - \frac{215007}{788881} a^{12} + \frac{174770}{365579} a^{11} + \frac{6100224}{14988739} a^{10} - \frac{5067990}{14988739} a^{9} + \frac{11558}{788881} a^{8} + \frac{12036}{365579} a^{7} - \frac{4019813}{14988739} a^{6} + \frac{404831}{14988739} a^{5} - \frac{1428187}{14988739} a^{4} + \frac{5113219}{14988739} a^{3} + \frac{5848701}{14988739} a^{2} - \frac{807703}{14988739} a + \frac{4359162}{14988739}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{2956608}{14988739} a^{15} - \frac{15573880}{14988739} a^{14} + \frac{61981410}{14988739} a^{13} - \frac{181834067}{14988739} a^{12} + \frac{9671498}{365579} a^{11} - \frac{694073575}{14988739} a^{10} + \frac{51961000}{788881} a^{9} - \frac{57860207}{788881} a^{8} + \frac{22208768}{365579} a^{7} - \frac{437754561}{14988739} a^{6} - \frac{6279526}{14988739} a^{5} + \frac{6964117}{788881} a^{4} - \frac{67796594}{14988739} a^{3} + \frac{9146563}{14988739} a^{2} + \frac{36256252}{14988739} a - \frac{531316}{788881} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1585.81715153 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1379:

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 t16n1379 are not computed
Character table for t16n1379 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.64000000.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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}$
761Data not computed