↑

Properties

Label 16.0.40886114501953125.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{14}\cdot 1021$
Root discriminant $10.92$
Ramified primes $3, 5, 1021$
Class number $1$
Class group Trivial
Galois group 16T1379

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 4, -10, 2, -5, 32, -25, -5, -5, 18, 10, -28, 10, 6, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 6*x^14 + 10*x^13 - 28*x^12 + 10*x^11 + 18*x^10 - 5*x^9 - 5*x^8 - 25*x^7 + 32*x^6 - 5*x^5 + 2*x^4 - 10*x^3 + 4*x^2 + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 6*x^14 + 10*x^13 - 28*x^12 + 10*x^11 + 18*x^10 - 5*x^9 - 5*x^8 - 25*x^7 + 32*x^6 - 5*x^5 + 2*x^4 - 10*x^3 + 4*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 6 x^{14} + 10 x^{13} - 28 x^{12} + 10 x^{11} + 18 x^{10} - 5 x^{9} - 5 x^{8} - 25 x^{7} + 32 x^{6} - 5 x^{5} + 2 x^{4} - 10 x^{3} + 4 x^{2} + 1 \)

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:  \(40886114501953125=3^{8}\cdot 5^{14}\cdot 1021\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.92$
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:  $3, 5, 1021$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{21421} a^{15} + \frac{4773}{21421} a^{14} - \frac{7965}{21421} a^{13} + \frac{8357}{21421} a^{12} + \frac{974}{21421} a^{11} + \frac{175}{691} a^{10} + \frac{1258}{21421} a^{9} - \frac{8582}{21421} a^{8} - \frac{5007}{21421} a^{7} + \frac{3786}{21421} a^{6} + \frac{10216}{21421} a^{5} - \frac{6416}{21421} a^{4} - \frac{2195}{21421} a^{3} + \frac{8570}{21421} a^{2} - \frac{9488}{21421} a - \frac{6828}{21421}$

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:  \( \frac{3681}{21421} a^{15} - \frac{17228}{21421} a^{14} + \frac{27605}{21421} a^{13} + \frac{1561}{21421} a^{12} - \frac{77697}{21421} a^{11} + \frac{4309}{691} a^{10} - \frac{60501}{21421} a^{9} - \frac{80051}{21421} a^{8} + \frac{55556}{21421} a^{7} - \frac{8805}{21421} a^{6} + \frac{161188}{21421} a^{5} - \frac{204143}{21421} a^{4} + \frac{38764}{21421} a^{3} - \frac{6963}{21421} a^{2} + \frac{55165}{21421} a - \frac{7035}{21421} \) (order $30$)
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:  \( 487.904905457 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1379:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 71 conjugacy class representatives for t16n1379 are not computed
Character table for t16n1379 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
1021Data not computed