↑

Properties

Label 16.0.42786095104...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 5^{14}\cdot 101^{2}$
Root discriminant $34.63$
Ramified primes $2, 5, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T554)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3061, 1824, -6962, -4552, 5716, 1524, -960, 1204, -171, -832, 600, -28, -74, -4, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 22*x^14 - 4*x^13 - 74*x^12 - 28*x^11 + 600*x^10 - 832*x^9 - 171*x^8 + 1204*x^7 - 960*x^6 + 1524*x^5 + 5716*x^4 - 4552*x^3 - 6962*x^2 + 1824*x + 3061)
 
gp: K = bnfinit(x^16 - 8*x^15 + 22*x^14 - 4*x^13 - 74*x^12 - 28*x^11 + 600*x^10 - 832*x^9 - 171*x^8 + 1204*x^7 - 960*x^6 + 1524*x^5 + 5716*x^4 - 4552*x^3 - 6962*x^2 + 1824*x + 3061, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 22 x^{14} - 4 x^{13} - 74 x^{12} - 28 x^{11} + 600 x^{10} - 832 x^{9} - 171 x^{8} + 1204 x^{7} - 960 x^{6} + 1524 x^{5} + 5716 x^{4} - 4552 x^{3} - 6962 x^{2} + 1824 x + 3061 \)

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:  \(4278609510400000000000000=2^{36}\cdot 5^{14}\cdot 101^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.63$
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:  $2, 5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{2}{11} a^{13} - \frac{3}{11} a^{12} + \frac{4}{11} a^{11} - \frac{3}{11} a^{9} - \frac{1}{11} a^{8} + \frac{4}{11} a^{7} - \frac{2}{11} a^{6} - \frac{4}{11} a^{5} - \frac{1}{11} a^{4} - \frac{3}{11} a^{3} + \frac{2}{11} a^{2} - \frac{2}{11} a - \frac{4}{11}$, $\frac{1}{105199267105433252489509572889} a^{15} - \frac{2022113183389418304987268385}{105199267105433252489509572889} a^{14} - \frac{18906332793381999195793093801}{105199267105433252489509572889} a^{13} + \frac{25993759279959180905315304002}{105199267105433252489509572889} a^{12} + \frac{18345308812304643916257127854}{105199267105433252489509572889} a^{11} + \frac{47029843116797450357863456269}{105199267105433252489509572889} a^{10} + \frac{2035804293712884330624334833}{9563569736857568408137233899} a^{9} + \frac{822370112526784121262102625}{5536803531864908025763661731} a^{8} - \frac{2748839148953275343963265284}{5536803531864908025763661731} a^{7} - \frac{25474389321772347012563179944}{105199267105433252489509572889} a^{6} - \frac{34986377866609306787437364034}{105199267105433252489509572889} a^{5} - \frac{3940335773948176570784375910}{105199267105433252489509572889} a^{4} - \frac{16283049841359293096301963653}{105199267105433252489509572889} a^{3} + \frac{27571969242796622312892701083}{105199267105433252489509572889} a^{2} - \frac{12951473756982563334443647479}{105199267105433252489509572889} a + \frac{39159082708845717547420456751}{105199267105433252489509572889}$

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:  $7$
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:  \( 152649.730269 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T554):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.0.5120000000.1, 8.4.6464000000.2, 8.4.517120000000.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$