↑

Properties

Label 16.16.1917707186...8496.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 13^{8}\cdot 17^{4}$
Root discriminant $58.57$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1028, 1392, -9384, -7888, 32828, 11984, -49944, -4256, 32762, -480, -9332, 192, 1112, -8, -56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 56*x^14 - 8*x^13 + 1112*x^12 + 192*x^11 - 9332*x^10 - 480*x^9 + 32762*x^8 - 4256*x^7 - 49944*x^6 + 11984*x^5 + 32828*x^4 - 7888*x^3 - 9384*x^2 + 1392*x + 1028)
 
gp: K = bnfinit(x^16 - 56*x^14 - 8*x^13 + 1112*x^12 + 192*x^11 - 9332*x^10 - 480*x^9 + 32762*x^8 - 4256*x^7 - 49944*x^6 + 11984*x^5 + 32828*x^4 - 7888*x^3 - 9384*x^2 + 1392*x + 1028, 1)
 

Normalized defining polynomial

\( x^{16} - 56 x^{14} - 8 x^{13} + 1112 x^{12} + 192 x^{11} - 9332 x^{10} - 480 x^{9} + 32762 x^{8} - 4256 x^{7} - 49944 x^{6} + 11984 x^{5} + 32828 x^{4} - 7888 x^{3} - 9384 x^{2} + 1392 x + 1028 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(19177071869085684350631018496=2^{48}\cdot 13^{8}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.57$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{102} a^{13} + \frac{2}{51} a^{12} + \frac{1}{6} a^{11} - \frac{5}{34} a^{10} + \frac{5}{51} a^{9} + \frac{23}{102} a^{8} - \frac{14}{51} a^{7} - \frac{25}{51} a^{6} + \frac{5}{51} a^{5} - \frac{22}{51} a^{4} - \frac{13}{51} a^{3} - \frac{8}{51} a^{2} - \frac{14}{51} a - \frac{7}{51}$, $\frac{1}{102} a^{14} + \frac{1}{102} a^{12} + \frac{19}{102} a^{11} + \frac{19}{102} a^{10} - \frac{1}{6} a^{9} - \frac{3}{17} a^{8} - \frac{20}{51} a^{7} + \frac{1}{17} a^{6} + \frac{3}{17} a^{5} + \frac{8}{17} a^{4} - \frac{7}{51} a^{3} + \frac{6}{17} a^{2} - \frac{2}{51} a - \frac{23}{51}$, $\frac{1}{19587458546277627558} a^{15} + \frac{12494961993763569}{6529152848759209186} a^{14} - \frac{20262385325306827}{9793729273138813779} a^{13} - \frac{28610618663462215}{399744051964849542} a^{12} + \frac{882432878155640205}{6529152848759209186} a^{11} - \frac{1637708180159243114}{9793729273138813779} a^{10} - \frac{1722852430700774404}{9793729273138813779} a^{9} - \frac{1806063664662073192}{9793729273138813779} a^{8} - \frac{1377641119119436021}{9793729273138813779} a^{7} + \frac{1249519937426085991}{3264576424379604593} a^{6} + \frac{98359070933254700}{1399104181876973397} a^{5} + \frac{189606944589071134}{466368060625657799} a^{4} + \frac{555887188434330013}{3264576424379604593} a^{3} + \frac{4690467197231306}{25705326176217359} a^{2} - \frac{664971360249878429}{9793729273138813779} a - \frac{940335302992596206}{9793729273138813779}$

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:  $15$
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:  \( 1348718265.85 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{26}) \), 4.4.346112.1, \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{13})\), 8.8.119793516544.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$