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Properties

Label 16.8.36582036480...0000.4
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{10}\cdot 5^{15}\cdot 11^{2}$
Root discriminant $34.29$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1584

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![961, -7936, 21800, -22930, 1250, 15502, -11592, 4000, -1365, -430, 1328, -668, 30, 50, 0, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 50*x^13 + 30*x^12 - 668*x^11 + 1328*x^10 - 430*x^9 - 1365*x^8 + 4000*x^7 - 11592*x^6 + 15502*x^5 + 1250*x^4 - 22930*x^3 + 21800*x^2 - 7936*x + 961)
 
gp: K = bnfinit(x^16 - 6*x^15 + 50*x^13 + 30*x^12 - 668*x^11 + 1328*x^10 - 430*x^9 - 1365*x^8 + 4000*x^7 - 11592*x^6 + 15502*x^5 + 1250*x^4 - 22930*x^3 + 21800*x^2 - 7936*x + 961, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 50 x^{13} + 30 x^{12} - 668 x^{11} + 1328 x^{10} - 430 x^{9} - 1365 x^{8} + 4000 x^{7} - 11592 x^{6} + 15502 x^{5} + 1250 x^{4} - 22930 x^{3} + 21800 x^{2} - 7936 x + 961 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3658203648000000000000000=2^{24}\cdot 3^{10}\cdot 5^{15}\cdot 11^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.29$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{124} a^{13} + \frac{9}{124} a^{12} + \frac{21}{124} a^{11} + \frac{21}{124} a^{10} + \frac{7}{31} a^{9} + \frac{6}{31} a^{8} - \frac{47}{124} a^{7} - \frac{27}{124} a^{6} + \frac{27}{62} a^{5} + \frac{7}{62} a^{4} + \frac{39}{124} a^{3} - \frac{17}{124} a^{2} + \frac{21}{124} a - \frac{1}{4}$, $\frac{1}{372} a^{14} + \frac{1}{372} a^{13} + \frac{7}{62} a^{12} - \frac{23}{372} a^{11} - \frac{47}{372} a^{10} - \frac{7}{186} a^{9} + \frac{71}{372} a^{8} - \frac{49}{124} a^{7} + \frac{115}{372} a^{6} + \frac{35}{93} a^{5} + \frac{175}{372} a^{4} - \frac{27}{124} a^{3} + \frac{16}{93} a^{2} + \frac{37}{124} a + \frac{1}{12}$, $\frac{1}{74441029108992377028} a^{15} + \frac{1941010494981170}{18610257277248094257} a^{14} - \frac{14025147750622628}{18610257277248094257} a^{13} + \frac{1405749545375513683}{18610257277248094257} a^{12} + \frac{3814485203416615781}{74441029108992377028} a^{11} + \frac{3336704651580655829}{18610257277248094257} a^{10} - \frac{1379602958258581877}{24813676369664125676} a^{9} + \frac{9236190272332152595}{37220514554496188514} a^{8} - \frac{27694537095012092105}{74441029108992377028} a^{7} - \frac{1060727064954575253}{6203419092416031419} a^{6} - \frac{11997322587252622613}{24813676369664125676} a^{5} + \frac{2097325949778679139}{37220514554496188514} a^{4} + \frac{4223089674231538381}{18610257277248094257} a^{3} + \frac{8935341678406216633}{18610257277248094257} a^{2} + \frac{13033234248956194531}{74441029108992377028} a + \frac{193609506618421289}{1200661759822457694}$

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:  $11$
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:  \( 3301037.26463 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1584:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1584 are not computed
Character table for t16n1584 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.1620000000.2

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$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}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$