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Properties

Label 16.6.15855575040...0000.1
Degree $16$
Signature $[6, 5]$
Discriminant $-\,2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 59$
Root discriminant $21.14$
Ramified primes $2, 3, 5, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1726

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 4, 38, 34, -206, 166, -224, 230, -124, 92, 8, -45, 24, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 2*x^14 + 24*x^13 - 45*x^12 + 8*x^11 + 92*x^10 - 124*x^9 + 230*x^8 - 224*x^7 + 166*x^6 - 206*x^5 + 34*x^4 + 38*x^3 + 4*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 2*x^14 + 24*x^13 - 45*x^12 + 8*x^11 + 92*x^10 - 124*x^9 + 230*x^8 - 224*x^7 + 166*x^6 - 206*x^5 + 34*x^4 + 38*x^3 + 4*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 2 x^{14} + 24 x^{13} - 45 x^{12} + 8 x^{11} + 92 x^{10} - 124 x^{9} + 230 x^{8} - 224 x^{7} + 166 x^{6} - 206 x^{5} + 34 x^{4} + 38 x^{3} + 4 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1585557504000000000000=-\,2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 59\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.14$
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:  $2, 3, 5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{229} a^{14} + \frac{113}{229} a^{13} - \frac{6}{229} a^{12} - \frac{60}{229} a^{11} + \frac{59}{229} a^{10} - \frac{111}{229} a^{9} - \frac{98}{229} a^{8} + \frac{16}{229} a^{7} - \frac{16}{229} a^{6} - \frac{55}{229} a^{5} + \frac{76}{229} a^{4} + \frac{68}{229} a^{3} + \frac{50}{229} a^{2} + \frac{71}{229} a + \frac{106}{229}$, $\frac{1}{1233456499561421} a^{15} - \frac{2146764512529}{1233456499561421} a^{14} + \frac{43614087067641}{1233456499561421} a^{13} + \frac{375911792320480}{1233456499561421} a^{12} + \frac{8263661143883}{1233456499561421} a^{11} + \frac{612189758787463}{1233456499561421} a^{10} + \frac{260568078720743}{1233456499561421} a^{9} + \frac{333135860680058}{1233456499561421} a^{8} + \frac{310832908658655}{1233456499561421} a^{7} - \frac{408597497386425}{1233456499561421} a^{6} + \frac{163627131794729}{1233456499561421} a^{5} - \frac{161235733926429}{1233456499561421} a^{4} + \frac{247314462202539}{1233456499561421} a^{3} - \frac{560815252594731}{1233456499561421} a^{2} + \frac{251744874436647}{1233456499561421} a + \frac{204141201321159}{1233456499561421}$

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:  $10$
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:  \( 33584.1175065 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1726:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 116 conjugacy class representatives for t16n1726 are not computed
Character table for t16n1726 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.4.324000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
59Data not computed