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Properties

Label 16.0.95908900025...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{7}$
Root discriminant $27.31$
Ramified primes $5, 61$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![38651, -2501, 118920, -53285, 145600, -77318, 84573, -40145, 27440, -10800, 5328, -1708, 620, -155, 40, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 40*x^14 - 155*x^13 + 620*x^12 - 1708*x^11 + 5328*x^10 - 10800*x^9 + 27440*x^8 - 40145*x^7 + 84573*x^6 - 77318*x^5 + 145600*x^4 - 53285*x^3 + 118920*x^2 - 2501*x + 38651)
 
gp: K = bnfinit(x^16 - 6*x^15 + 40*x^14 - 155*x^13 + 620*x^12 - 1708*x^11 + 5328*x^10 - 10800*x^9 + 27440*x^8 - 40145*x^7 + 84573*x^6 - 77318*x^5 + 145600*x^4 - 53285*x^3 + 118920*x^2 - 2501*x + 38651, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 40 x^{14} - 155 x^{13} + 620 x^{12} - 1708 x^{11} + 5328 x^{10} - 10800 x^{9} + 27440 x^{8} - 40145 x^{7} + 84573 x^{6} - 77318 x^{5} + 145600 x^{4} - 53285 x^{3} + 118920 x^{2} - 2501 x + 38651 \)

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:  \(95908900025054931640625=5^{15}\cdot 61^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.31$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13547372656667781176292264758323394971} a^{15} + \frac{204040532698108213322163469508552012}{13547372656667781176292264758323394971} a^{14} - \frac{56613206423337923363194851005675281}{134132402541265160161309552062607871} a^{13} - \frac{2035599438478142964997539598964872924}{13547372656667781176292264758323394971} a^{12} - \frac{957563932447051752074642813574743319}{13547372656667781176292264758323394971} a^{11} - \frac{5159144805658386451716204742924024660}{13547372656667781176292264758323394971} a^{10} + \frac{799351798565853896921078734043917243}{13547372656667781176292264758323394971} a^{9} - \frac{1943926401792122819496966251222357472}{13547372656667781176292264758323394971} a^{8} + \frac{4649314116483044873157401809718387764}{13547372656667781176292264758323394971} a^{7} - \frac{129624516169180791173970635301781837}{13547372656667781176292264758323394971} a^{6} + \frac{35673623220259108558839872832006270}{134132402541265160161309552062607871} a^{5} - \frac{4430231222470957602609366546684275237}{13547372656667781176292264758323394971} a^{4} + \frac{1069553076413018058889378780831370721}{13547372656667781176292264758323394971} a^{3} + \frac{6077961435425961514645922850496096977}{13547372656667781176292264758323394971} a^{2} - \frac{4058698453046804399008078898831121942}{13547372656667781176292264758323394971} a - \frac{1040592739671551836423377080658796654}{13547372656667781176292264758323394971}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{17982781986182203010418890355}{431293898846511769007426212419961} a^{15} - \frac{120921680303585526880837496655}{431293898846511769007426212419961} a^{14} + \frac{791771744363725970889351760946}{431293898846511769007426212419961} a^{13} - \frac{3304966327765618501337687500607}{431293898846511769007426212419961} a^{12} + \frac{13069857167757681698740717511656}{431293898846511769007426212419961} a^{11} - \frac{38543057474440799536149578825443}{431293898846511769007426212419961} a^{10} + \frac{115885761835393816694005975199318}{431293898846511769007426212419961} a^{9} - \frac{257414229823706293647471793195692}{431293898846511769007426212419961} a^{8} + \frac{602620554610488756567212900019487}{431293898846511769007426212419961} a^{7} - \frac{1019598187041216754343175187922004}{431293898846511769007426212419961} a^{6} + \frac{1818328235173812619051626060949346}{431293898846511769007426212419961} a^{5} - \frac{2138101961831543716992535561414320}{431293898846511769007426212419961} a^{4} + \frac{2717103167829994583299593841195734}{431293898846511769007426212419961} a^{3} - \frac{1855735658538018743590397102219997}{431293898846511769007426212419961} a^{2} + \frac{1044273792163187221173344146515831}{431293898846511769007426212419961} a - \frac{490430522787126080822323211357566}{431293898846511769007426212419961} \) (order $10$)
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:  \( 50484.8966192 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.4765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
61Data not computed