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Properties

Label 16.8.10961832082...8096.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 761^{5}$
Root discriminant $31.80$
Ramified primes $2, 761$
Class number $1$
Class group Trivial
Galois group 16T1870

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![127, -748, 1192, 904, -4713, 4268, 1626, -4656, 1241, 1512, -710, -184, 154, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 + 154*x^12 - 184*x^11 - 710*x^10 + 1512*x^9 + 1241*x^8 - 4656*x^7 + 1626*x^6 + 4268*x^5 - 4713*x^4 + 904*x^3 + 1192*x^2 - 748*x + 127)
 
gp: K = bnfinit(x^16 - 16*x^14 + 154*x^12 - 184*x^11 - 710*x^10 + 1512*x^9 + 1241*x^8 - 4656*x^7 + 1626*x^6 + 4268*x^5 - 4713*x^4 + 904*x^3 + 1192*x^2 - 748*x + 127, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} + 154 x^{12} - 184 x^{11} - 710 x^{10} + 1512 x^{9} + 1241 x^{8} - 4656 x^{7} + 1626 x^{6} + 4268 x^{5} - 4713 x^{4} + 904 x^{3} + 1192 x^{2} - 748 x + 127 \)

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:  \(1096183208251673695748096=2^{32}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.80$
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:  $2, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1607} a^{14} + \frac{150}{1607} a^{13} - \frac{125}{1607} a^{12} - \frac{46}{1607} a^{11} + \frac{701}{1607} a^{10} + \frac{796}{1607} a^{9} + \frac{704}{1607} a^{8} - \frac{528}{1607} a^{7} - \frac{224}{1607} a^{6} - \frac{539}{1607} a^{5} + \frac{278}{1607} a^{4} - \frac{265}{1607} a^{3} + \frac{208}{1607} a^{2} + \frac{453}{1607} a - \frac{549}{1607}$, $\frac{1}{672489603467809231661} a^{15} - \frac{160075173212932332}{672489603467809231661} a^{14} + \frac{1290771491389904277}{23189296671303766609} a^{13} - \frac{781580954524264481}{23189296671303766609} a^{12} - \frac{128378565844043671403}{672489603467809231661} a^{11} - \frac{129637019044739402606}{672489603467809231661} a^{10} + \frac{253647149057724886239}{672489603467809231661} a^{9} + \frac{13796463935362818196}{672489603467809231661} a^{8} - \frac{189549361139181045418}{672489603467809231661} a^{7} + \frac{122838455634866047559}{672489603467809231661} a^{6} - \frac{87500035832448113859}{672489603467809231661} a^{5} - \frac{328145407092877382524}{672489603467809231661} a^{4} + \frac{238261286786837832878}{672489603467809231661} a^{3} + \frac{44672996965921113292}{672489603467809231661} a^{2} + \frac{38223132086608312059}{672489603467809231661} a - \frac{50454301414554611255}{672489603467809231661}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2363912.21481 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1870:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 83 conjugacy class representatives for t16n1870 are not computed
Character table for t16n1870 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 8.4.2372079616.2

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$2$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.12.24.314$x^{12} + 14 x^{10} - 4 x^{9} + 4 x^{8} + 16 x^{7} - 4 x^{6} + 16 x^{5} - 12 x^{4} - 8 x^{2} + 16 x - 8$$4$$3$$24$$C_2^2 \times A_4$$[2, 2, 2, 3]^{3}$
761Data not computed