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Properties

Label 16.8.22176987219...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{28}\cdot 5^{8}\cdot 59^{2}\cdot 157^{4}$
Root discriminant $44.32$
Ramified primes $2, 5, 59, 157$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-171, -1986, 8360, -2786, -4234, -2278, -1932, 3614, 2834, -138, -808, -506, -50, 42, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 4*x^14 + 42*x^13 - 50*x^12 - 506*x^11 - 808*x^10 - 138*x^9 + 2834*x^8 + 3614*x^7 - 1932*x^6 - 2278*x^5 - 4234*x^4 - 2786*x^3 + 8360*x^2 - 1986*x - 171)
 
gp: K = bnfinit(x^16 - 2*x^15 - 4*x^14 + 42*x^13 - 50*x^12 - 506*x^11 - 808*x^10 - 138*x^9 + 2834*x^8 + 3614*x^7 - 1932*x^6 - 2278*x^5 - 4234*x^4 - 2786*x^3 + 8360*x^2 - 1986*x - 171, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 4 x^{14} + 42 x^{13} - 50 x^{12} - 506 x^{11} - 808 x^{10} - 138 x^{9} + 2834 x^{8} + 3614 x^{7} - 1932 x^{6} - 2278 x^{5} - 4234 x^{4} - 2786 x^{3} + 8360 x^{2} - 1986 x - 171 \)

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:  \(221769872198179225600000000=2^{28}\cdot 5^{8}\cdot 59^{2}\cdot 157^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.32$
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:  $2, 5, 59, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{66} a^{14} + \frac{1}{22} a^{13} - \frac{3}{22} a^{12} - \frac{4}{33} a^{11} + \frac{13}{66} a^{10} + \frac{5}{66} a^{9} - \frac{3}{22} a^{8} + \frac{7}{33} a^{7} - \frac{7}{66} a^{6} - \frac{23}{66} a^{5} + \frac{29}{66} a^{4} + \frac{5}{33} a^{3} - \frac{23}{66} a^{2} + \frac{1}{66} a + \frac{1}{22}$, $\frac{1}{74593942671123088535322} a^{15} + \frac{220286315676798178481}{37296971335561544267661} a^{14} + \frac{7959428467229315473}{308239432525302018741} a^{13} - \frac{1921697571279121113103}{8288215852347009837258} a^{12} - \frac{988202838269216612347}{6781267515556644412302} a^{11} - \frac{16003781626223335937}{88068409292943433926} a^{10} - \frac{7712724577353073197265}{74593942671123088535322} a^{9} + \frac{683277544952134417553}{2762738617449003279086} a^{8} - \frac{22993916788803126449611}{74593942671123088535322} a^{7} - \frac{794526246590172002395}{3390633757778322206151} a^{6} - \frac{2812938826560254494556}{12432323778520514755887} a^{5} + \frac{33129332620558725576023}{74593942671123088535322} a^{4} + \frac{7340839094460329313929}{74593942671123088535322} a^{3} - \frac{15704582168056551288851}{74593942671123088535322} a^{2} + \frac{34528688098026098886509}{74593942671123088535322} a + \frac{212690480077909235471}{24864647557041029511774}$

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:  $11$
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:  \( 45541377.1395 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12288
The 74 conjugacy class representatives for t16n1765 are not computed
Character table for t16n1765 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.4.63101440000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ R

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.2$x^{4} - 59 x^{2} + 6962$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
$157$157.4.0.1$x^{4} - x + 15$$1$$4$$0$$C_4$$[\ ]^{4}$
157.4.0.1$x^{4} - x + 15$$1$$4$$0$$C_4$$[\ ]^{4}$
157.8.4.1$x^{8} + 739470 x^{4} - 3869893 x^{2} + 136703970225$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$