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Properties

Label 16.4.12584463237...5625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{8}\cdot 29^{6}\cdot 41\cdot 1321$
Root discriminant $15.62$
Ramified primes $5, 29, 41, 1321$
Class number $1$
Class group Trivial
Galois group 16T1722

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 15, -52, 95, -75, -81, 376, -710, 927, -936, 757, -498, 265, -112, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 265*x^12 - 498*x^11 + 757*x^10 - 936*x^9 + 927*x^8 - 710*x^7 + 376*x^6 - 81*x^5 - 75*x^4 + 95*x^3 - 52*x^2 + 15*x - 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 265*x^12 - 498*x^11 + 757*x^10 - 936*x^9 + 927*x^8 - 710*x^7 + 376*x^6 - 81*x^5 - 75*x^4 + 95*x^3 - 52*x^2 + 15*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 112 x^{13} + 265 x^{12} - 498 x^{11} + 757 x^{10} - 936 x^{9} + 927 x^{8} - 710 x^{7} + 376 x^{6} - 81 x^{5} - 75 x^{4} + 95 x^{3} - 52 x^{2} + 15 x - 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12584463237766015625=5^{8}\cdot 29^{6}\cdot 41\cdot 1321\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.62$
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:  $5, 29, 41, 1321$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{139} a^{14} - \frac{7}{139} a^{13} - \frac{31}{139} a^{12} - \frac{1}{139} a^{11} + \frac{39}{139} a^{10} + \frac{18}{139} a^{9} - \frac{36}{139} a^{8} + \frac{33}{139} a^{7} + \frac{62}{139} a^{6} + \frac{13}{139} a^{5} + \frac{5}{139} a^{4} - \frac{22}{139} a^{3} + \frac{20}{139} a^{2} + \frac{45}{139} a + \frac{44}{139}$, $\frac{1}{139} a^{15} + \frac{59}{139} a^{13} + \frac{60}{139} a^{12} + \frac{32}{139} a^{11} + \frac{13}{139} a^{10} - \frac{49}{139} a^{9} + \frac{59}{139} a^{8} + \frac{15}{139} a^{7} + \frac{30}{139} a^{6} - \frac{43}{139} a^{5} + \frac{13}{139} a^{4} + \frac{5}{139} a^{3} + \frac{46}{139} a^{2} - \frac{58}{139} a + \frac{30}{139}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1346.61973519 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1722:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 104 conjugacy class representatives for t16n1722 are not computed
Character table for t16n1722 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.15243125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed
1321Data not computed