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Properties

Label 16.6.65203029999...8687.6
Degree $16$
Signature $[6, 5]$
Discriminant $-\,17^{12}\cdot 47^{9}$
Root discriminant $73.01$
Ramified primes $17, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![184301, 372274, 78893, -920155, -505143, 782415, 216515, -224360, -31887, 19111, 3699, -953, 137, -3, -26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 26*x^14 - 3*x^13 + 137*x^12 - 953*x^11 + 3699*x^10 + 19111*x^9 - 31887*x^8 - 224360*x^7 + 216515*x^6 + 782415*x^5 - 505143*x^4 - 920155*x^3 + 78893*x^2 + 372274*x + 184301)
 
gp: K = bnfinit(x^16 - 26*x^14 - 3*x^13 + 137*x^12 - 953*x^11 + 3699*x^10 + 19111*x^9 - 31887*x^8 - 224360*x^7 + 216515*x^6 + 782415*x^5 - 505143*x^4 - 920155*x^3 + 78893*x^2 + 372274*x + 184301, 1)
 

Normalized defining polynomial

\( x^{16} - 26 x^{14} - 3 x^{13} + 137 x^{12} - 953 x^{11} + 3699 x^{10} + 19111 x^{9} - 31887 x^{8} - 224360 x^{7} + 216515 x^{6} + 782415 x^{5} - 505143 x^{4} - 920155 x^{3} + 78893 x^{2} + 372274 x + 184301 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-652030299991134977060343848687=-\,17^{12}\cdot 47^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.01$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13242334464612654631510201099958975591024952661} a^{15} - \frac{4682136142275001038725140894270692915623903520}{13242334464612654631510201099958975591024952661} a^{14} - \frac{1221553440231200405293363240048298572613316638}{13242334464612654631510201099958975591024952661} a^{13} + \frac{4787416700139762751371322177135151218752895608}{13242334464612654631510201099958975591024952661} a^{12} + \frac{1949519609194177897201867282986630034310348420}{13242334464612654631510201099958975591024952661} a^{11} + \frac{4107138392985286216238873773017306019920340416}{13242334464612654631510201099958975591024952661} a^{10} + \frac{2710827303741263542441286501211691894651730317}{13242334464612654631510201099958975591024952661} a^{9} - \frac{944080553268911864113143851371410361244126245}{13242334464612654631510201099958975591024952661} a^{8} + \frac{1596905009829108864738016905737292059713595162}{13242334464612654631510201099958975591024952661} a^{7} + \frac{4047374694570275366790372383420562630788863104}{13242334464612654631510201099958975591024952661} a^{6} + \frac{73242627793261637674227698256792123382489058}{13242334464612654631510201099958975591024952661} a^{5} + \frac{398264349162704133409408279726463297097498350}{13242334464612654631510201099958975591024952661} a^{4} - \frac{169427383531866125915895423392925743934248101}{1018641112662511894731553930766075045463457897} a^{3} - \frac{2396340083088355888119710844583482568050526091}{13242334464612654631510201099958975591024952661} a^{2} + \frac{133590760839509402633140473014302393519256651}{13242334464612654631510201099958975591024952661} a - \frac{454848079425219388213218586166442687528174132}{1018641112662511894731553930766075045463457897}$

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:  $10$
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:  \( 470864643.569 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.6.2506034826287.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
47Data not computed