↑

Properties

Label 16.6.73589398538...6875.1
Degree $16$
Signature $[6, 5]$
Discriminant $-\,3^{8}\cdot 5^{14}\cdot 179^{5}$
Root discriminant $35.82$
Ramified primes $3, 5, 179$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1354

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![39511, -69144, 64680, -43515, 1025, 19148, -14637, 4880, 1755, -1840, 433, 83, -125, 20, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 + 20*x^13 - 125*x^12 + 83*x^11 + 433*x^10 - 1840*x^9 + 1755*x^8 + 4880*x^7 - 14637*x^6 + 19148*x^5 + 1025*x^4 - 43515*x^3 + 64680*x^2 - 69144*x + 39511)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 + 20*x^13 - 125*x^12 + 83*x^11 + 433*x^10 - 1840*x^9 + 1755*x^8 + 4880*x^7 - 14637*x^6 + 19148*x^5 + 1025*x^4 - 43515*x^3 + 64680*x^2 - 69144*x + 39511, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} + 20 x^{13} - 125 x^{12} + 83 x^{11} + 433 x^{10} - 1840 x^{9} + 1755 x^{8} + 4880 x^{7} - 14637 x^{6} + 19148 x^{5} + 1025 x^{4} - 43515 x^{3} + 64680 x^{2} - 69144 x + 39511 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7358939853847283935546875=-\,3^{8}\cdot 5^{14}\cdot 179^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.82$
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:  $3, 5, 179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{7} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{55} a^{14} + \frac{2}{55} a^{12} - \frac{1}{11} a^{11} + \frac{4}{55} a^{10} - \frac{4}{55} a^{9} - \frac{3}{11} a^{7} + \frac{23}{55} a^{6} + \frac{21}{55} a^{5} + \frac{5}{11} a^{4} - \frac{1}{11} a^{3} + \frac{13}{55} a^{2} - \frac{17}{55} a - \frac{18}{55}$, $\frac{1}{4630670495812328557295812205} a^{15} - \frac{25606142419832543721947694}{4630670495812328557295812205} a^{14} + \frac{73723967992908309588876745}{926134099162465711459162441} a^{13} + \frac{58582773910620772152472714}{4630670495812328557295812205} a^{12} + \frac{187474953286018404876299116}{4630670495812328557295812205} a^{11} + \frac{88564713432099283318662232}{4630670495812328557295812205} a^{10} - \frac{447248821790497960023722104}{4630670495812328557295812205} a^{9} - \frac{13687658296118973325476727}{926134099162465711459162441} a^{8} + \frac{689243388609625993693948919}{4630670495812328557295812205} a^{7} - \frac{49061552980007848646498354}{926134099162465711459162441} a^{6} + \frac{2268670097493280872701608574}{4630670495812328557295812205} a^{5} + \frac{41965449916708902004436750}{926134099162465711459162441} a^{4} - \frac{293547405887482117157673244}{926134099162465711459162441} a^{3} - \frac{254856509488229331781683240}{926134099162465711459162441} a^{2} + \frac{335121368880981438063635264}{4630670495812328557295812205} a + \frac{983316447287367722607036703}{4630670495812328557295812205}$

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:  $10$
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:  \( 1805527.1718 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1354:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 59 conjugacy class representatives for t16n1354 are not computed
Character table for t16n1354 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.226546875.1

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 $16$ R R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
179Data not computed