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Properties

Label 16.0.69019132572...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{6}\cdot 101^{6}\cdot 409^{4}$
Root discriminant $200.65$
Ramified primes $5, 29, 101, 409$
Class number $1280$ (GRH)
Class group $[2, 2, 2, 4, 40]$ (GRH)
Galois group 16T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63667451, 216524690, 314979348, 259099211, 158818931, 48345606, 14573429, -3344945, 947703, -658147, 53770, -16484, 4902, -153, 91, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 91*x^14 - 153*x^13 + 4902*x^12 - 16484*x^11 + 53770*x^10 - 658147*x^9 + 947703*x^8 - 3344945*x^7 + 14573429*x^6 + 48345606*x^5 + 158818931*x^4 + 259099211*x^3 + 314979348*x^2 + 216524690*x + 63667451)
 
gp: K = bnfinit(x^16 - 3*x^15 + 91*x^14 - 153*x^13 + 4902*x^12 - 16484*x^11 + 53770*x^10 - 658147*x^9 + 947703*x^8 - 3344945*x^7 + 14573429*x^6 + 48345606*x^5 + 158818931*x^4 + 259099211*x^3 + 314979348*x^2 + 216524690*x + 63667451, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 91 x^{14} - 153 x^{13} + 4902 x^{12} - 16484 x^{11} + 53770 x^{10} - 658147 x^{9} + 947703 x^{8} - 3344945 x^{7} + 14573429 x^{6} + 48345606 x^{5} + 158818931 x^{4} + 259099211 x^{3} + 314979348 x^{2} + 216524690 x + 63667451 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6901913257237170531380177586750390625=5^{8}\cdot 29^{6}\cdot 101^{6}\cdot 409^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.65$
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:  $5, 29, 101, 409$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{41} a^{14} + \frac{3}{41} a^{13} + \frac{18}{41} a^{12} + \frac{10}{41} a^{11} + \frac{3}{41} a^{10} + \frac{8}{41} a^{9} - \frac{1}{41} a^{8} - \frac{11}{41} a^{7} + \frac{13}{41} a^{6} + \frac{12}{41} a^{5} + \frac{16}{41} a^{4} - \frac{7}{41} a^{3} - \frac{3}{41} a^{2} + \frac{2}{41} a + \frac{3}{41}$, $\frac{1}{68524827073059121440284995798391289794610864643444417771349082163} a^{15} + \frac{90871260460588048512934890864219215611977399512639249876953}{40764323065472410136992858892558768467942215730781926098363523} a^{14} - \frac{32532561373309150615980597549700613984637375570253906373928364648}{68524827073059121440284995798391289794610864643444417771349082163} a^{13} + \frac{14312480586020885766428738891893260152490951867166548191382790805}{68524827073059121440284995798391289794610864643444417771349082163} a^{12} - \frac{4618441289041516647843094656890534651371245386703982275191036661}{68524827073059121440284995798391289794610864643444417771349082163} a^{11} - \frac{23172556301106541758886415489276521164362881555153344624336690338}{68524827073059121440284995798391289794610864643444417771349082163} a^{10} - \frac{28204606621294782173445960119395574255486371984793625903268222107}{68524827073059121440284995798391289794610864643444417771349082163} a^{9} - \frac{24165192386849337990654699640035018006766189221067493863027590712}{68524827073059121440284995798391289794610864643444417771349082163} a^{8} - \frac{22842402469074778077718899005908885306080774295312404973973088971}{68524827073059121440284995798391289794610864643444417771349082163} a^{7} + \frac{19890184770566245608184005967103896472119106604388115871866125522}{68524827073059121440284995798391289794610864643444417771349082163} a^{6} - \frac{7971724681871022373331665426432963777936442083006188913193223386}{68524827073059121440284995798391289794610864643444417771349082163} a^{5} + \frac{11004651694380372327143245764196680379742731192011496465826478007}{68524827073059121440284995798391289794610864643444417771349082163} a^{4} - \frac{12395864989778161633056654358420495755869124004399377083780557946}{68524827073059121440284995798391289794610864643444417771349082163} a^{3} - \frac{7922671755240607292020693459836652182704818928094346208667348226}{68524827073059121440284995798391289794610864643444417771349082163} a^{2} - \frac{31525891282437404283174860384725645078114568139753444047298444369}{68524827073059121440284995798391289794610864643444417771349082163} a + \frac{6020683962753987282860775491338552924060036937192859224763395604}{68524827073059121440284995798391289794610864643444417771349082163}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{40}$, which has order $1280$ (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:  $7$
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:  \( 357925647.873 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n839 are not computed
Character table for t16n839 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1032725.2, 4.4.73225.1, 4.0.296525.2, 8.0.896944098450625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.2$x^{4} - 101 x^{2} + 30603$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
409Data not computed