↑

Properties

Label 16.4.32959821542...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{16}\cdot 5^{12}\cdot 11^{8}\cdot 31^{2}$
Root discriminant $34.07$
Ramified primes $2, 5, 11, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1086

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29, 146, -74, -600, 930, 34, -935, 1050, -964, 720, -315, 114, -20, -10, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 11*x^14 - 10*x^13 - 20*x^12 + 114*x^11 - 315*x^10 + 720*x^9 - 964*x^8 + 1050*x^7 - 935*x^6 + 34*x^5 + 930*x^4 - 600*x^3 - 74*x^2 + 146*x - 29)
 
gp: K = bnfinit(x^16 - 4*x^15 + 11*x^14 - 10*x^13 - 20*x^12 + 114*x^11 - 315*x^10 + 720*x^9 - 964*x^8 + 1050*x^7 - 935*x^6 + 34*x^5 + 930*x^4 - 600*x^3 - 74*x^2 + 146*x - 29, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 11 x^{14} - 10 x^{13} - 20 x^{12} + 114 x^{11} - 315 x^{10} + 720 x^{9} - 964 x^{8} + 1050 x^{7} - 935 x^{6} + 34 x^{5} + 930 x^{4} - 600 x^{3} - 74 x^{2} + 146 x - 29 \)

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:  \(3295982154256000000000000=2^{16}\cdot 5^{12}\cdot 11^{8}\cdot 31^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.07$
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:  $2, 5, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{9} a^{14} + \frac{4}{9} a^{13} - \frac{1}{9} a^{12} + \frac{4}{9} a^{11} + \frac{2}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{4}{9} a^{7} - \frac{1}{3} a^{6} - \frac{4}{9} a^{5} + \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a^{2} + \frac{2}{9}$, $\frac{1}{2940870706097488209} a^{15} - \frac{127259851078950655}{2940870706097488209} a^{14} - \frac{287843975526033197}{980290235365829403} a^{13} + \frac{290245197737463361}{980290235365829403} a^{12} + \frac{98881631059339808}{980290235365829403} a^{11} - \frac{1131435794074736843}{2940870706097488209} a^{10} + \frac{42458729026736981}{2940870706097488209} a^{9} + \frac{238760441819069210}{2940870706097488209} a^{8} - \frac{1051182593508317659}{2940870706097488209} a^{7} - \frac{769422350346581050}{2940870706097488209} a^{6} + \frac{686230758610256851}{2940870706097488209} a^{5} - \frac{453821205796249415}{980290235365829403} a^{4} + \frac{713553826347649058}{2940870706097488209} a^{3} + \frac{1328070861476163058}{2940870706097488209} a^{2} - \frac{16067697163673353}{2940870706097488209} a + \frac{1305202385148259475}{2940870706097488209}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 667744.969667 \) (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

\(\Q(\sqrt{5}) \), 4.4.15125.1, 4.4.4400.1, 4.4.22000.1, 8.8.58564000000.2

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
31Data not computed