↑

Properties

Label 16.0.30232191979...5625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{4}\cdot 149^{6}$
Root discriminant $33.89$
Ramified primes $5, 29, 149$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T392)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2239, -12891, 26588, -30527, 31832, -21532, 16911, -9468, 5318, -2441, 1278, -390, 161, -60, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 60*x^13 + 161*x^12 - 390*x^11 + 1278*x^10 - 2441*x^9 + 5318*x^8 - 9468*x^7 + 16911*x^6 - 21532*x^5 + 31832*x^4 - 30527*x^3 + 26588*x^2 - 12891*x + 2239)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 60*x^13 + 161*x^12 - 390*x^11 + 1278*x^10 - 2441*x^9 + 5318*x^8 - 9468*x^7 + 16911*x^6 - 21532*x^5 + 31832*x^4 - 30527*x^3 + 26588*x^2 - 12891*x + 2239, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 60 x^{13} + 161 x^{12} - 390 x^{11} + 1278 x^{10} - 2441 x^{9} + 5318 x^{8} - 9468 x^{7} + 16911 x^{6} - 21532 x^{5} + 31832 x^{4} - 30527 x^{3} + 26588 x^{2} - 12891 x + 2239 \)

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:  \(3023219197928805422265625=5^{8}\cdot 29^{4}\cdot 149^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.89$
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:  $5, 29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{30} a^{12} - \frac{1}{15} a^{11} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{7}{15} a^{7} + \frac{1}{6} a^{6} + \frac{7}{15} a^{5} - \frac{1}{5} a^{4} + \frac{13}{30} a^{3} + \frac{1}{3} a^{2} + \frac{2}{5} a - \frac{1}{30}$, $\frac{1}{870} a^{13} - \frac{7}{870} a^{12} - \frac{43}{435} a^{11} + \frac{17}{290} a^{10} - \frac{83}{290} a^{9} + \frac{134}{435} a^{8} + \frac{1}{290} a^{7} + \frac{277}{870} a^{6} - \frac{8}{435} a^{5} + \frac{397}{870} a^{4} - \frac{271}{870} a^{3} + \frac{128}{435} a^{2} + \frac{299}{870} a + \frac{107}{870}$, $\frac{1}{870} a^{14} + \frac{1}{87} a^{12} + \frac{1}{30} a^{11} - \frac{11}{145} a^{10} + \frac{89}{435} a^{9} - \frac{7}{174} a^{8} + \frac{4}{435} a^{7} - \frac{31}{87} a^{6} + \frac{401}{870} a^{5} + \frac{14}{29} a^{4} + \frac{122}{435} a^{3} - \frac{113}{870} a^{2} - \frac{31}{435} a + \frac{43}{87}$, $\frac{1}{33059499369481100288850} a^{15} - \frac{2278803429708219808}{5509916561580183381475} a^{14} + \frac{7168514904105727187}{33059499369481100288850} a^{13} + \frac{67000351995343774526}{16529749684740550144425} a^{12} + \frac{736813330391503488164}{16529749684740550144425} a^{11} + \frac{572726801710348275043}{33059499369481100288850} a^{10} + \frac{353235754053783066764}{2361392812105792877775} a^{9} + \frac{109176698743064421927}{220396662463207335259} a^{8} - \frac{8230898293776391383637}{33059499369481100288850} a^{7} + \frac{14330040655045111314}{157426187473719525185} a^{6} - \frac{1712285400929938865762}{16529749684740550144425} a^{5} - \frac{4993417717115722590541}{33059499369481100288850} a^{4} - \frac{1425618554600246734744}{5509916561580183381475} a^{3} - \frac{3433133473826107352498}{16529749684740550144425} a^{2} + \frac{65846199448307874699}{1001803011196396978450} a + \frac{14818978518942636548621}{33059499369481100288850}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 75079.232878 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T392):

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 $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.108025.2, 4.4.725.1, 4.0.3725.1, 8.0.11669400625.1, 8.4.11669400625.1, 8.4.11669400625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$149$149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$