↑

Properties

Label 16.4.18484762498...0000.3
Degree $16$
Signature $[4, 6]$
Discriminant $2^{32}\cdot 5^{10}\cdot 761^{3}$
Root discriminant $37.95$
Ramified primes $2, 5, 761$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1398

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79, 1804, 4972, 4952, 1874, 152, -566, -936, -296, -552, 70, -112, 67, -12, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 14*x^14 - 12*x^13 + 67*x^12 - 112*x^11 + 70*x^10 - 552*x^9 - 296*x^8 - 936*x^7 - 566*x^6 + 152*x^5 + 1874*x^4 + 4952*x^3 + 4972*x^2 + 1804*x + 79)
 
gp: K = bnfinit(x^16 + 14*x^14 - 12*x^13 + 67*x^12 - 112*x^11 + 70*x^10 - 552*x^9 - 296*x^8 - 936*x^7 - 566*x^6 + 152*x^5 + 1874*x^4 + 4952*x^3 + 4972*x^2 + 1804*x + 79, 1)
 

Normalized defining polynomial

\( x^{16} + 14 x^{14} - 12 x^{13} + 67 x^{12} - 112 x^{11} + 70 x^{10} - 552 x^{9} - 296 x^{8} - 936 x^{7} - 566 x^{6} + 152 x^{5} + 1874 x^{4} + 4952 x^{3} + 4972 x^{2} + 1804 x + 79 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18484762498826240000000000=2^{32}\cdot 5^{10}\cdot 761^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.95$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3511} a^{14} + \frac{559}{3511} a^{13} + \frac{38}{3511} a^{12} - \frac{1582}{3511} a^{11} + \frac{1337}{3511} a^{10} - \frac{266}{3511} a^{9} + \frac{164}{3511} a^{8} + \frac{1008}{3511} a^{7} + \frac{1513}{3511} a^{6} - \frac{210}{3511} a^{5} - \frac{406}{3511} a^{4} + \frac{304}{3511} a^{3} + \frac{1372}{3511} a^{2} - \frac{854}{3511} a + \frac{156}{3511}$, $\frac{1}{33381464503816934369871} a^{15} + \frac{4397373958226927206}{33381464503816934369871} a^{14} - \frac{1839410096986989338138}{11127154834605644789957} a^{13} - \frac{5282964993692382940483}{11127154834605644789957} a^{12} + \frac{14149707069209470872082}{33381464503816934369871} a^{11} + \frac{920253006836500436732}{11127154834605644789957} a^{10} + \frac{14884310830163971595464}{33381464503816934369871} a^{9} + \frac{11699528070903495618982}{33381464503816934369871} a^{8} + \frac{5142499095409231839278}{33381464503816934369871} a^{7} + \frac{6848166931332049027724}{33381464503816934369871} a^{6} + \frac{1843494270720752318531}{11127154834605644789957} a^{5} - \frac{11144326909576428789532}{33381464503816934369871} a^{4} - \frac{12624531310113915144551}{33381464503816934369871} a^{3} + \frac{5547100902445359947455}{11127154834605644789957} a^{2} - \frac{9857407732097068426796}{33381464503816934369871} a + \frac{767261290795924569290}{33381464503816934369871}$

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:  $9$
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:  \( 1819821.5615 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1398:

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 t16n1398 are not computed
Character table for t16n1398 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
761Data not computed