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Properties

Label 16.0.15586275156...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{6}\cdot 181^{4}$
Root discriminant $43.36$
Ramified primes $5, 29, 181$
Class number $68$ (GRH)
Class group $[2, 34]$ (GRH)
Galois group $C_4^2:C_2^2.C_2$ (as 16T382)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44521, 67520, 68991, 62875, -693, -55, 353, -3390, 10430, -495, 3462, -70, 482, -5, 34, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 34*x^14 - 5*x^13 + 482*x^12 - 70*x^11 + 3462*x^10 - 495*x^9 + 10430*x^8 - 3390*x^7 + 353*x^6 - 55*x^5 - 693*x^4 + 62875*x^3 + 68991*x^2 + 67520*x + 44521)
 
gp: K = bnfinit(x^16 + 34*x^14 - 5*x^13 + 482*x^12 - 70*x^11 + 3462*x^10 - 495*x^9 + 10430*x^8 - 3390*x^7 + 353*x^6 - 55*x^5 - 693*x^4 + 62875*x^3 + 68991*x^2 + 67520*x + 44521, 1)
 

Normalized defining polynomial

\( x^{16} + 34 x^{14} - 5 x^{13} + 482 x^{12} - 70 x^{11} + 3462 x^{10} - 495 x^{9} + 10430 x^{8} - 3390 x^{7} + 353 x^{6} - 55 x^{5} - 693 x^{4} + 62875 x^{3} + 68991 x^{2} + 67520 x + 44521 \)

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:  \(155862751564078330322265625=5^{12}\cdot 29^{6}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.36$
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:  $5, 29, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{6} - \frac{1}{5} a^{4} - \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{5} a^{10} - \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{3}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{4}$, $\frac{1}{14332315389826155717983149177205} a^{15} - \frac{437971302793884644293533818}{13585133070925266083396349931} a^{14} + \frac{1411867070310134330727268077761}{14332315389826155717983149177205} a^{13} + \frac{569674099089249270992735948997}{14332315389826155717983149177205} a^{12} + \frac{45613322729030540544654992001}{2866463077965231143596629835441} a^{11} + \frac{948696245078706495730392170716}{14332315389826155717983149177205} a^{10} + \frac{291622304699563280888851560522}{14332315389826155717983149177205} a^{9} - \frac{177578362674895986677521209926}{2866463077965231143596629835441} a^{8} - \frac{6074043476177239718502561998641}{14332315389826155717983149177205} a^{7} - \frac{337251811000401661924460395377}{2866463077965231143596629835441} a^{6} + \frac{29092462976753623455880998876}{14332315389826155717983149177205} a^{5} + \frac{814051545940761089822807244406}{2866463077965231143596629835441} a^{4} - \frac{5842008363023640961626384529736}{14332315389826155717983149177205} a^{3} + \frac{5929028029671477188839479953591}{14332315389826155717983149177205} a^{2} + \frac{1839080528290276344988876154047}{14332315389826155717983149177205} a + \frac{14071644689556585559127213928}{67925665354626330416981749655}$

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

Class group and class number

$C_{2}\times C_{34}$, which has order $68$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 29196.3261178 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2.C_2$ (as 16T382):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_4^2:C_2^2.C_2$
Character table for $C_4^2:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.131225.1, 4.4.725.1, 4.4.4525.1, 8.8.17220000625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$181$181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$
181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$
181.8.4.1$x^{8} + 3538188 x^{4} - 5929741 x^{2} + 3129693580836$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$