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Properties

Label 16.0.29145046131...8761.2
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![50153, 142491, 231785, 121382, -49199, -74851, -14665, 12419, 6045, -1417, 7, 842, 59, -19, 28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 28*x^14 - 19*x^13 + 59*x^12 + 842*x^11 + 7*x^10 - 1417*x^9 + 6045*x^8 + 12419*x^7 - 14665*x^6 - 74851*x^5 - 49199*x^4 + 121382*x^3 + 231785*x^2 + 142491*x + 50153)
 
gp: K = bnfinit(x^16 - 4*x^15 + 28*x^14 - 19*x^13 + 59*x^12 + 842*x^11 + 7*x^10 - 1417*x^9 + 6045*x^8 + 12419*x^7 - 14665*x^6 - 74851*x^5 - 49199*x^4 + 121382*x^3 + 231785*x^2 + 142491*x + 50153, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 28 x^{14} - 19 x^{13} + 59 x^{12} + 842 x^{11} + 7 x^{10} - 1417 x^{9} + 6045 x^{8} + 12419 x^{7} - 14665 x^{6} - 74851 x^{5} - 49199 x^{4} + 121382 x^{3} + 231785 x^{2} + 142491 x + 50153 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{82581295077622710192897854139649786842637} a^{15} + \frac{6649513075478823461911721527036069845223}{82581295077622710192897854139649786842637} a^{14} + \frac{592209320725178043848836398673623947832}{2663912744439442264287027552891928607827} a^{13} - \frac{36457497904290984782421268821146914069032}{82581295077622710192897854139649786842637} a^{12} + \frac{28955233459769165317304362630908759944506}{82581295077622710192897854139649786842637} a^{11} - \frac{2194077700229875039382665085486901288010}{82581295077622710192897854139649786842637} a^{10} - \frac{25028610733945078096112731290301867304702}{82581295077622710192897854139649786842637} a^{9} + \frac{30012276739258732937683733580133729260698}{82581295077622710192897854139649786842637} a^{8} + \frac{9723754074951809063121172870428875982508}{82581295077622710192897854139649786842637} a^{7} - \frac{2695850621193405401205564766405060412135}{82581295077622710192897854139649786842637} a^{6} + \frac{18357335811377829829915900295819304397051}{82581295077622710192897854139649786842637} a^{5} + \frac{14941249341689638908032066451553258075088}{82581295077622710192897854139649786842637} a^{4} - \frac{9638285061837929292911271202685871360537}{82581295077622710192897854139649786842637} a^{3} + \frac{23424569025335727666840435743565657508081}{82581295077622710192897854139649786842637} a^{2} - \frac{927567873445053509927019972798916546916}{2231926893989802978186428490260805049801} a - \frac{8779670528652243005543667165208433237963}{82581295077622710192897854139649786842637}$

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:  \( 130743479.743 \) (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 ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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