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Properties

Label 16.8.50374030414...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 119851^{4}$
Root discriminant $62.21$
Ramified primes $5, 119851$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![130562021, 37541221, -24665835, 4705425, -2500217, -3629051, 2297256, 417900, -477001, 10218, 35742, -5997, -208, 377, -68, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 68*x^14 + 377*x^13 - 208*x^12 - 5997*x^11 + 35742*x^10 + 10218*x^9 - 477001*x^8 + 417900*x^7 + 2297256*x^6 - 3629051*x^5 - 2500217*x^4 + 4705425*x^3 - 24665835*x^2 + 37541221*x + 130562021)
 
gp: K = bnfinit(x^16 - 3*x^15 - 68*x^14 + 377*x^13 - 208*x^12 - 5997*x^11 + 35742*x^10 + 10218*x^9 - 477001*x^8 + 417900*x^7 + 2297256*x^6 - 3629051*x^5 - 2500217*x^4 + 4705425*x^3 - 24665835*x^2 + 37541221*x + 130562021, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 68 x^{14} + 377 x^{13} - 208 x^{12} - 5997 x^{11} + 35742 x^{10} + 10218 x^{9} - 477001 x^{8} + 417900 x^{7} + 2297256 x^{6} - 3629051 x^{5} - 2500217 x^{4} + 4705425 x^{3} - 24665835 x^{2} + 37541221 x + 130562021 \)

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:  \(50374030414813809668212890625=5^{12}\cdot 119851^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.21$
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:  $5, 119851$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{9759053975980220729325212480447748748254307885239373967242} a^{15} - \frac{76772441323395685986952194878043865607659405219178908136}{4879526987990110364662606240223874374127153942619686983621} a^{14} + \frac{622606533565967092884718481132425215965469692307612983930}{4879526987990110364662606240223874374127153942619686983621} a^{13} + \frac{504220713020294979348538617279771614117174703979238302585}{9759053975980220729325212480447748748254307885239373967242} a^{12} - \frac{1372320031498500531755531372237401015926511209094704813935}{4879526987990110364662606240223874374127153942619686983621} a^{11} - \frac{2598482549032288167651517682929824497098266640499603703869}{9759053975980220729325212480447748748254307885239373967242} a^{10} + \frac{1486884394550425294087338371495451681355680373388542326901}{4879526987990110364662606240223874374127153942619686983621} a^{9} + \frac{1013835031558564166138547019013701093652740769399242142779}{9759053975980220729325212480447748748254307885239373967242} a^{8} + \frac{3641699523554249769876935948861340699764042419058147341945}{9759053975980220729325212480447748748254307885239373967242} a^{7} - \frac{3656418613963151894019147264522512281648684774953803641623}{9759053975980220729325212480447748748254307885239373967242} a^{6} - \frac{435576591432359512323656238621367615728596925185235003138}{4879526987990110364662606240223874374127153942619686983621} a^{5} + \frac{2133941008656786947085523505919507297258558788549742625791}{4879526987990110364662606240223874374127153942619686983621} a^{4} + \frac{1255061039804399295903346408349987290591082723336623162317}{9759053975980220729325212480447748748254307885239373967242} a^{3} + \frac{735067617019358715781664445579728749945197781541384111784}{4879526987990110364662606240223874374127153942619686983621} a^{2} + \frac{4229598801244396497513173294073489697279822627637627052055}{9759053975980220729325212480447748748254307885239373967242} a + \frac{19758235535256205849299023211058984911103941638651047322}{4879526987990110364662606240223874374127153942619686983621}$

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:  \( 196167511.55 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1869 are not computed
Character table for t16n1869 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.6.74906875.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
119851Data not computed