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Properties

Label 16.0.29145046131...8761.3
Degree $16$
Signature $[0, 8]$
Discriminant $37^{4}\cdot 41^{15}$
Root discriminant $80.17$
Ramified primes $37, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![387427, -1223664, 1779996, -1268099, 407030, 59049, -67861, 1428, 24109, -9661, 1379, 476, -92, 49, -15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 15*x^14 + 49*x^13 - 92*x^12 + 476*x^11 + 1379*x^10 - 9661*x^9 + 24109*x^8 + 1428*x^7 - 67861*x^6 + 59049*x^5 + 407030*x^4 - 1268099*x^3 + 1779996*x^2 - 1223664*x + 387427)
 
gp: K = bnfinit(x^16 - 3*x^15 - 15*x^14 + 49*x^13 - 92*x^12 + 476*x^11 + 1379*x^10 - 9661*x^9 + 24109*x^8 + 1428*x^7 - 67861*x^6 + 59049*x^5 + 407030*x^4 - 1268099*x^3 + 1779996*x^2 - 1223664*x + 387427, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 15 x^{14} + 49 x^{13} - 92 x^{12} + 476 x^{11} + 1379 x^{10} - 9661 x^{9} + 24109 x^{8} + 1428 x^{7} - 67861 x^{6} + 59049 x^{5} + 407030 x^{4} - 1268099 x^{3} + 1779996 x^{2} - 1223664 x + 387427 \)

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:  \(2914504613122855682608891338761=37^{4}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.17$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{296} a^{14} + \frac{5}{74} a^{13} - \frac{9}{74} a^{12} - \frac{39}{296} a^{11} + \frac{47}{296} a^{10} - \frac{27}{74} a^{9} - \frac{4}{37} a^{8} + \frac{3}{8} a^{7} + \frac{11}{148} a^{6} + \frac{47}{296} a^{5} - \frac{53}{148} a^{4} - \frac{9}{74} a^{3} - \frac{33}{74} a^{2} + \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{153973752264523792001209833057831643530728} a^{15} - \frac{42175920043839171990464262780429927767}{38493438066130948000302458264457910882682} a^{14} + \frac{3060618024644707142197342483111039543121}{76986876132261896000604916528915821765364} a^{13} - \frac{5158277398433682580687709251283403764649}{153973752264523792001209833057831643530728} a^{12} + \frac{33945202357263617457697689757846279649223}{153973752264523792001209833057831643530728} a^{11} + \frac{520782391934370205342958880580931676435}{19246719033065474000151229132228955441341} a^{10} - \frac{35770034650123815820387846178757249602631}{76986876132261896000604916528915821765364} a^{9} + \frac{37051363383169088678942787366952190413677}{153973752264523792001209833057831643530728} a^{8} - \frac{9141341005754267049566025747114297657383}{76986876132261896000604916528915821765364} a^{7} - \frac{827642489321759291796926241069161911753}{4161452763906048432465130623184639014344} a^{6} + \frac{6112896709087455526867482551866635553075}{76986876132261896000604916528915821765364} a^{5} - \frac{7257680534216906779374381566235303076203}{76986876132261896000604916528915821765364} a^{4} - \frac{28101720466959741185258199836508919428205}{76986876132261896000604916528915821765364} a^{3} - \frac{26638684009417937815726583628608045931367}{153973752264523792001209833057831643530728} a^{2} - \frac{586571145545997557551728296702898574531}{4161452763906048432465130623184639014344} a - \frac{439689827712999326749100107525031828857}{1040363190976512108116282655796159753586}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 110623817.988 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.0.194754273881.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
37Data not computed
41Data not computed