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Properties

Label 16.4.70883963839...8125.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{8}\cdot 31^{5}\cdot 859^{3}$
Root discriminant $23.21$
Ramified primes $5, 31, 859$
Class number $1$
Class group Trivial
Galois group 16T1871

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1121, 3028, -4267, -4844, 1865, 3124, -1090, -1427, 563, 573, -174, -160, 45, 33, -9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 9*x^14 + 33*x^13 + 45*x^12 - 160*x^11 - 174*x^10 + 573*x^9 + 563*x^8 - 1427*x^7 - 1090*x^6 + 3124*x^5 + 1865*x^4 - 4844*x^3 - 4267*x^2 + 3028*x + 1121)
 
gp: K = bnfinit(x^16 - 3*x^15 - 9*x^14 + 33*x^13 + 45*x^12 - 160*x^11 - 174*x^10 + 573*x^9 + 563*x^8 - 1427*x^7 - 1090*x^6 + 3124*x^5 + 1865*x^4 - 4844*x^3 - 4267*x^2 + 3028*x + 1121, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 9 x^{14} + 33 x^{13} + 45 x^{12} - 160 x^{11} - 174 x^{10} + 573 x^{9} + 563 x^{8} - 1427 x^{7} - 1090 x^{6} + 3124 x^{5} + 1865 x^{4} - 4844 x^{3} - 4267 x^{2} + 3028 x + 1121 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7088396383905323828125=5^{8}\cdot 31^{5}\cdot 859^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.21$
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:  $5, 31, 859$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4840216811791613410205939417} a^{15} + \frac{1858443485590239057216605620}{4840216811791613410205939417} a^{14} + \frac{541229816024530813147472488}{4840216811791613410205939417} a^{13} - \frac{1280739257127212461315519931}{4840216811791613410205939417} a^{12} + \frac{402864041525452831389555791}{4840216811791613410205939417} a^{11} + \frac{367816992794979005139853629}{4840216811791613410205939417} a^{10} + \frac{272923368916664426748489643}{4840216811791613410205939417} a^{9} - \frac{1576286878358679262403249599}{4840216811791613410205939417} a^{8} + \frac{1644960848065983237569337718}{4840216811791613410205939417} a^{7} + \frac{810882329545667470262216477}{4840216811791613410205939417} a^{6} + \frac{2381684052822741541108532237}{4840216811791613410205939417} a^{5} - \frac{373512779274501932949661513}{4840216811791613410205939417} a^{4} - \frac{1347464425044489999073359681}{4840216811791613410205939417} a^{3} + \frac{2310076422758111739518396358}{4840216811791613410205939417} a^{2} + \frac{1441673383716321180936793125}{4840216811791613410205939417} a + \frac{958958847392116677666339575}{4840216811791613410205939417}$

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:  $9$
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:  \( 32258.961955 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1871:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 104 conjugacy class representatives for t16n1871 are not computed
Character table for t16n1871 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.4.16643125.1

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ $16$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
31Data not computed
859Data not computed