↑

Properties

Label 16.0.15244671129...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 29^{6}$
Root discriminant $13.69$
Ramified primes $3, 5, 29$
Class number $1$
Class group Trivial
Galois group $C_2\times C_4\wr C_2$ (as 16T111)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 11, 6, 19, -3, 27, -16, 14, -22, 8, -9, 9, -3, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 4*x^14 - 3*x^13 + 9*x^12 - 9*x^11 + 8*x^10 - 22*x^9 + 14*x^8 - 16*x^7 + 27*x^6 - 3*x^5 + 19*x^4 + 6*x^3 + 11*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 4*x^14 - 3*x^13 + 9*x^12 - 9*x^11 + 8*x^10 - 22*x^9 + 14*x^8 - 16*x^7 + 27*x^6 - 3*x^5 + 19*x^4 + 6*x^3 + 11*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 4 x^{14} - 3 x^{13} + 9 x^{12} - 9 x^{11} + 8 x^{10} - 22 x^{9} + 14 x^{8} - 16 x^{7} + 27 x^{6} - 3 x^{5} + 19 x^{4} + 6 x^{3} + 11 x^{2} + 3 x + 1 \)

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:  \(1524467112922265625=3^{8}\cdot 5^{8}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.69$
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:  $3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{13} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{2}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{10115567} a^{15} - \frac{175046}{10115567} a^{14} + \frac{2319712}{10115567} a^{13} - \frac{2121934}{10115567} a^{12} + \frac{170677}{919597} a^{11} + \frac{166545}{1445081} a^{10} + \frac{4252153}{10115567} a^{9} + \frac{2638925}{10115567} a^{8} + \frac{3965849}{10115567} a^{7} + \frac{4758612}{10115567} a^{6} - \frac{283248}{1445081} a^{5} - \frac{408711}{1445081} a^{4} + \frac{4008272}{10115567} a^{3} + \frac{2205696}{10115567} a^{2} + \frac{3575433}{10115567} a + \frac{4356699}{10115567}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{488287}{1445081} a^{15} - \frac{480295}{1445081} a^{14} + \frac{1823924}{1445081} a^{13} - \frac{1270706}{1445081} a^{12} + \frac{356690}{131371} a^{11} - \frac{4114182}{1445081} a^{10} + \frac{3217488}{1445081} a^{9} - \frac{10060334}{1445081} a^{8} + \frac{6167423}{1445081} a^{7} - \frac{6263876}{1445081} a^{6} + \frac{12369095}{1445081} a^{5} - \frac{732727}{1445081} a^{4} + \frac{8420851}{1445081} a^{3} + \frac{1038857}{1445081} a^{2} + \frac{4750470}{1445081} a + \frac{1293703}{1445081} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 651.844970784 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4\wr C_2$ (as 16T111):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $C_2\times C_4\wr C_2$
Character table for $C_2\times C_4\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.4.725.1, 4.0.6525.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.4.15243125.1, 8.4.1234693125.1, 8.0.42575625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data 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}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$