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Properties

Label 16.0.36086878424...2857.1
Degree $16$
Signature $[0, 8]$
Discriminant $29^{15}\cdot 53^{5}$
Root discriminant $81.25$
Ramified primes $29, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1693277, -5019549, 3509953, 1077282, -1276890, 184957, 330810, -201758, 65483, -33518, 14350, -5013, 1454, -304, 45, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 45*x^14 - 304*x^13 + 1454*x^12 - 5013*x^11 + 14350*x^10 - 33518*x^9 + 65483*x^8 - 201758*x^7 + 330810*x^6 + 184957*x^5 - 1276890*x^4 + 1077282*x^3 + 3509953*x^2 - 5019549*x + 1693277)
 
gp: K = bnfinit(x^16 - 3*x^15 + 45*x^14 - 304*x^13 + 1454*x^12 - 5013*x^11 + 14350*x^10 - 33518*x^9 + 65483*x^8 - 201758*x^7 + 330810*x^6 + 184957*x^5 - 1276890*x^4 + 1077282*x^3 + 3509953*x^2 - 5019549*x + 1693277, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 45 x^{14} - 304 x^{13} + 1454 x^{12} - 5013 x^{11} + 14350 x^{10} - 33518 x^{9} + 65483 x^{8} - 201758 x^{7} + 330810 x^{6} + 184957 x^{5} - 1276890 x^{4} + 1077282 x^{3} + 3509953 x^{2} - 5019549 x + 1693277 \)

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:  \(3608687842491875308187596442857=29^{15}\cdot 53^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.25$
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:  $29, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{546} a^{14} - \frac{127}{546} a^{13} + \frac{34}{273} a^{12} - \frac{193}{546} a^{11} - \frac{235}{546} a^{10} - \frac{31}{91} a^{9} - \frac{71}{182} a^{8} + \frac{193}{546} a^{7} - \frac{115}{273} a^{6} + \frac{19}{78} a^{5} + \frac{47}{182} a^{4} + \frac{76}{273} a^{3} + \frac{1}{546} a^{2} - \frac{185}{546} a - \frac{82}{273}$, $\frac{1}{3440292942139331454307186243922088714389106208314} a^{15} - \frac{1539196961528737853235933944933402904714133333}{1720146471069665727153593121961044357194553104157} a^{14} - \frac{43530460403027133653971362610944641520233079654}{573382157023221909051197707320348119064851034719} a^{13} - \frac{203042422576654759288738173188368562759651501917}{1146764314046443818102395414640696238129702069438} a^{12} - \frac{106654321694082062171957695502178285050225932477}{573382157023221909051197707320348119064851034719} a^{11} + \frac{49299729271633500115195528384605932667353560652}{132318959313051209781045624766234181322657931089} a^{10} - \frac{444771797054786658302285378851027469178149718671}{1146764314046443818102395414640696238129702069438} a^{9} + \frac{437600220119337111697165947950182521607954595624}{1720146471069665727153593121961044357194553104157} a^{8} + \frac{276494631090530440914229125262700930200210952141}{573382157023221909051197707320348119064851034719} a^{7} - \frac{14005481528171165999553658439570610058937080621}{88212639542034139854030416510822787548438620726} a^{6} + \frac{646256907883436834204429753533776987895522603444}{1720146471069665727153593121961044357194553104157} a^{5} + \frac{531014526097815439346827094677479711840287793256}{1720146471069665727153593121961044357194553104157} a^{4} - \frac{898348897480786174715271894371093203719481451911}{3440292942139331454307186243922088714389106208314} a^{3} - \frac{27898751934537676212479411592651942271676387206}{81911736717603129864456815331478302723550147817} a^{2} + \frac{152448162515724235281037647962248842376993593885}{573382157023221909051197707320348119064851034719} a - \frac{563475866570293097839215282575308986926635148435}{1720146471069665727153593121961044357194553104157}$

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:  \( 211738394.027 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.24389.1, 8.0.914243444377.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ $16$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
29Data not computed
53Data not computed