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Properties

Label 16.0.66560828909...0176.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 17^{4}\cdot 41^{6}$
Root discriminant $23.12$
Ramified primes $2, 17, 41$
Class number $4$
Class group $[4]$
Galois group 16T876

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 43, 114, 328, 520, 669, 608, 603, 292, 307, 68, 64, -10, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 5*x^14 - 10*x^13 + 64*x^12 + 68*x^11 + 307*x^10 + 292*x^9 + 603*x^8 + 608*x^7 + 669*x^6 + 520*x^5 + 328*x^4 + 114*x^3 + 43*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 5*x^14 - 10*x^13 + 64*x^12 + 68*x^11 + 307*x^10 + 292*x^9 + 603*x^8 + 608*x^7 + 669*x^6 + 520*x^5 + 328*x^4 + 114*x^3 + 43*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 5 x^{14} - 10 x^{13} + 64 x^{12} + 68 x^{11} + 307 x^{10} + 292 x^{9} + 603 x^{8} + 608 x^{7} + 669 x^{6} + 520 x^{5} + 328 x^{4} + 114 x^{3} + 43 x^{2} + 6 x + 1 \)

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:  \(6656082890982399410176=2^{24}\cdot 17^{4}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.12$
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:  $2, 17, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{47} a^{14} - \frac{10}{47} a^{13} - \frac{15}{47} a^{12} - \frac{18}{47} a^{11} + \frac{16}{47} a^{10} + \frac{1}{47} a^{9} + \frac{15}{47} a^{8} - \frac{22}{47} a^{7} - \frac{16}{47} a^{6} + \frac{22}{47} a^{5} - \frac{22}{47} a^{4} - \frac{10}{47} a^{2} + \frac{6}{47} a + \frac{8}{47}$, $\frac{1}{1467446019077585} a^{15} + \frac{3706443213672}{1467446019077585} a^{14} + \frac{712528559390778}{1467446019077585} a^{13} - \frac{515024288121928}{1467446019077585} a^{12} - \frac{111372923506448}{1467446019077585} a^{11} + \frac{390891100845786}{1467446019077585} a^{10} - \frac{537660204014284}{1467446019077585} a^{9} - \frac{22248569061519}{1467446019077585} a^{8} + \frac{148265969563787}{1467446019077585} a^{7} - \frac{602236216184629}{1467446019077585} a^{6} + \frac{500755005396008}{1467446019077585} a^{5} + \frac{263185517131382}{1467446019077585} a^{4} - \frac{193529922142109}{1467446019077585} a^{3} + \frac{119416410009578}{1467446019077585} a^{2} + \frac{55706377652689}{293489203815517} a - \frac{468133788970214}{1467446019077585}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -1 \) (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:  \( 3104.86729508 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T876:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 65 conjugacy class representatives for t16n876 are not computed
Character table for t16n876 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.0.81584820224.6, 8.4.4799107072.1, 8.4.117051392.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$