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Properties

Label 16.8.20152322170...3616.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{22}\cdot 13^{2}\cdot 17^{12}\cdot 47^{4}$
Root discriminant $78.35$
Ramified primes $2, 13, 17, 47$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1086

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![195364, 0, 37570, 0, -83249, 0, 3349, 0, 6291, 0, -256, 0, -159, 0, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + x^14 - 159*x^12 - 256*x^10 + 6291*x^8 + 3349*x^6 - 83249*x^4 + 37570*x^2 + 195364)
 
gp: K = bnfinit(x^16 + x^14 - 159*x^12 - 256*x^10 + 6291*x^8 + 3349*x^6 - 83249*x^4 + 37570*x^2 + 195364, 1)
 

Normalized defining polynomial

\( x^{16} + x^{14} - 159 x^{12} - 256 x^{10} + 6291 x^{8} + 3349 x^{6} - 83249 x^{4} + 37570 x^{2} + 195364 \)

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:  \(2015232217016213221512488943616=2^{22}\cdot 13^{2}\cdot 17^{12}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.35$
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:  $2, 13, 17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{816} a^{12} - \frac{169}{816} a^{10} + \frac{33}{136} a^{8} + \frac{169}{816} a^{6} + \frac{307}{816} a^{4} - \frac{1}{8} a^{2} + \frac{1}{12}$, $\frac{1}{1632} a^{13} - \frac{1}{1632} a^{12} - \frac{169}{1632} a^{11} + \frac{169}{1632} a^{10} - \frac{35}{272} a^{9} + \frac{35}{272} a^{8} + \frac{169}{1632} a^{7} - \frac{169}{1632} a^{6} - \frac{509}{1632} a^{5} + \frac{509}{1632} a^{4} - \frac{5}{16} a^{3} + \frac{5}{16} a^{2} - \frac{11}{24} a + \frac{11}{24}$, $\frac{1}{829185819072} a^{14} + \frac{42735713}{414592909536} a^{12} + \frac{150121295659}{829185819072} a^{10} + \frac{6851046115}{829185819072} a^{8} + \frac{41010627153}{138197636512} a^{6} - \frac{408212580157}{829185819072} a^{4} + \frac{10328093765}{24387818208} a^{2} + \frac{175897547}{937993008}$, $\frac{1}{14096158924224} a^{15} + \frac{49462581}{1174679910352} a^{13} - \frac{1}{1632} a^{12} + \frac{64255852385}{14096158924224} a^{11} + \frac{169}{1632} a^{10} + \frac{2387711798671}{14096158924224} a^{9} + \frac{35}{272} a^{8} - \frac{342023220457}{881009932764} a^{7} - \frac{169}{1632} a^{6} - \frac{234327549727}{829185819072} a^{5} + \frac{509}{1632} a^{4} + \frac{27094718783}{414592909536} a^{3} + \frac{5}{16} a^{2} - \frac{456350587}{937993008} a + \frac{11}{24}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1123195360.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1086:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 97 conjugacy class representatives for t16n1086 are not computed
Character table for t16n1086 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.54332.1, 4.4.4913.1, 4.4.923644.1, 8.8.853118238736.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
47Data not computed