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Properties

Label 16.10.6002904701...1875.1
Degree $16$
Signature $[10, 3]$
Discriminant $-\,5^{12}\cdot 71^{3}\cdot 1901^{3}$
Root discriminant $30.63$
Ramified primes $5, 71, 1901$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1871

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-79, -887, -3857, -5404, 3508, 15544, 11368, -2292, -5965, -1816, 552, 516, 121, -36, -23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 23*x^14 - 36*x^13 + 121*x^12 + 516*x^11 + 552*x^10 - 1816*x^9 - 5965*x^8 - 2292*x^7 + 11368*x^6 + 15544*x^5 + 3508*x^4 - 5404*x^3 - 3857*x^2 - 887*x - 79)
 
gp: K = bnfinit(x^16 - 23*x^14 - 36*x^13 + 121*x^12 + 516*x^11 + 552*x^10 - 1816*x^9 - 5965*x^8 - 2292*x^7 + 11368*x^6 + 15544*x^5 + 3508*x^4 - 5404*x^3 - 3857*x^2 - 887*x - 79, 1)
 

Normalized defining polynomial

\( x^{16} - 23 x^{14} - 36 x^{13} + 121 x^{12} + 516 x^{11} + 552 x^{10} - 1816 x^{9} - 5965 x^{8} - 2292 x^{7} + 11368 x^{6} + 15544 x^{5} + 3508 x^{4} - 5404 x^{3} - 3857 x^{2} - 887 x - 79 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-600290470112453857421875=-\,5^{12}\cdot 71^{3}\cdot 1901^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.63$
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:  $5, 71, 1901$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13365108185234281135711} a^{15} + \frac{4179014274506093722145}{13365108185234281135711} a^{14} - \frac{4452504572479012670851}{13365108185234281135711} a^{13} - \frac{3524871553331242063336}{13365108185234281135711} a^{12} + \frac{825592584585449327202}{13365108185234281135711} a^{11} + \frac{6132143526294742909181}{13365108185234281135711} a^{10} - \frac{238641623610374104967}{13365108185234281135711} a^{9} + \frac{1874649739341016309223}{13365108185234281135711} a^{8} + \frac{1247891589202953601142}{13365108185234281135711} a^{7} + \frac{2634725544879507075776}{13365108185234281135711} a^{6} + \frac{5520992827592942119824}{13365108185234281135711} a^{5} + \frac{576272519791495779310}{13365108185234281135711} a^{4} + \frac{6365031237903382858784}{13365108185234281135711} a^{3} - \frac{2026082092644513534714}{13365108185234281135711} a^{2} + \frac{5881330569838280493018}{13365108185234281135711} a - \frac{1584530974516167580801}{13365108185234281135711}$

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:  $12$
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:  \( 1125327.83508 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1871:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 104 conjugacy class representatives for t16n1871 are not computed
Character table for t16n1871 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.6.84356875.1

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 $16$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ $16$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.3.1$x^{4} + 142$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
71.8.0.1$x^{8} - 7 x + 13$$1$$8$$0$$C_8$$[\ ]^{8}$
1901Data not computed