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Properties

Label 16.4.11509937633...1696.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{46}\cdot 3^{8}\cdot 4993^{2}$
Root discriminant $36.84$
Ramified primes $2, 3, 4993$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1385

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 10, -8, -83, 8, 210, -108, -102, 68, 14, -24, -3, 16, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 2*x^14 + 16*x^13 - 3*x^12 - 24*x^11 + 14*x^10 + 68*x^9 - 102*x^8 - 108*x^7 + 210*x^6 + 8*x^5 - 83*x^4 - 8*x^3 + 10*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 2*x^14 + 16*x^13 - 3*x^12 - 24*x^11 + 14*x^10 + 68*x^9 - 102*x^8 - 108*x^7 + 210*x^6 + 8*x^5 - 83*x^4 - 8*x^3 + 10*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 2 x^{14} + 16 x^{13} - 3 x^{12} - 24 x^{11} + 14 x^{10} + 68 x^{9} - 102 x^{8} - 108 x^{7} + 210 x^{6} + 8 x^{5} - 83 x^{4} - 8 x^{3} + 10 x^{2} + 4 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11509937633380058787741696=2^{46}\cdot 3^{8}\cdot 4993^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.84$
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:  $2, 3, 4993$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10749402289} a^{15} + \frac{4909855223}{10749402289} a^{14} + \frac{84497025}{228710687} a^{13} + \frac{4234157025}{10749402289} a^{12} - \frac{4220791985}{10749402289} a^{11} + \frac{1488909042}{10749402289} a^{10} + \frac{105325601}{228710687} a^{9} - \frac{3579625494}{10749402289} a^{8} - \frac{4616915503}{10749402289} a^{7} + \frac{398559843}{10749402289} a^{6} + \frac{4762657166}{10749402289} a^{5} - \frac{2863795886}{10749402289} a^{4} + \frac{2682835582}{10749402289} a^{3} + \frac{3704323133}{10749402289} a^{2} - \frac{4771598026}{10749402289} a - \frac{2268479434}{10749402289}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3351477.32722 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1385:

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 t16n1385 are not computed
Character table for t16n1385 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.18432.1, 8.4.1696314949632.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.24.5$x^{8} - 15$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
3Data not computed
4993Data not computed