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Properties

Label 16.8.21634605854...1424.4
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 223^{6}$
Root discriminant $51.10$
Ramified primes $2, 223$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3:S_4.C_2$ (as 16T764)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7121, -44556, -114020, -136060, 10452, 132400, 38240, -16092, 19499, 6808, -2588, 1644, -280, -36, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 36*x^13 - 280*x^12 + 1644*x^11 - 2588*x^10 + 6808*x^9 + 19499*x^8 - 16092*x^7 + 38240*x^6 + 132400*x^5 + 10452*x^4 - 136060*x^3 - 114020*x^2 - 44556*x - 7121)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 36*x^13 - 280*x^12 + 1644*x^11 - 2588*x^10 + 6808*x^9 + 19499*x^8 - 16092*x^7 + 38240*x^6 + 132400*x^5 + 10452*x^4 - 136060*x^3 - 114020*x^2 - 44556*x - 7121, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 36 x^{13} - 280 x^{12} + 1644 x^{11} - 2588 x^{10} + 6808 x^{9} + 19499 x^{8} - 16092 x^{7} + 38240 x^{6} + 132400 x^{5} + 10452 x^{4} - 136060 x^{3} - 114020 x^{2} - 44556 x - 7121 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2163460585448341410282471424=2^{44}\cdot 223^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.10$
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:  $2, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{1075} a^{14} + \frac{63}{1075} a^{13} - \frac{71}{1075} a^{12} + \frac{6}{1075} a^{11} + \frac{16}{43} a^{10} - \frac{14}{43} a^{9} + \frac{262}{1075} a^{8} + \frac{84}{215} a^{7} - \frac{69}{1075} a^{6} - \frac{291}{1075} a^{5} + \frac{21}{215} a^{4} - \frac{471}{1075} a^{3} + \frac{31}{1075} a^{2} + \frac{32}{215} a - \frac{119}{1075}$, $\frac{1}{66409274674941935372038723107077875} a^{15} - \frac{1965989363642522041470085413778}{13281854934988387074407744621415575} a^{14} + \frac{158923529119595833936410732424222}{2656370986997677414881548924283115} a^{13} + \frac{2016019135187453810355286095931364}{66409274674941935372038723107077875} a^{12} - \frac{5658240567934715083595137417658103}{66409274674941935372038723107077875} a^{11} - \frac{1879213751189212419285826952811447}{13281854934988387074407744621415575} a^{10} + \frac{202287636720739947217775713585139}{5108405744226302720926055623621375} a^{9} + \frac{13945419891612720759160746109599059}{66409274674941935372038723107077875} a^{8} + \frac{27479024956427972178772138235406261}{66409274674941935372038723107077875} a^{7} + \frac{13853405947927093966908893510484781}{66409274674941935372038723107077875} a^{6} - \frac{15510979120592088913933082280602902}{66409274674941935372038723107077875} a^{5} + \frac{7903572835137867020618379663055914}{66409274674941935372038723107077875} a^{4} + \frac{18999735818283789734085109652065904}{66409274674941935372038723107077875} a^{3} + \frac{32372024530272994584966959850505687}{66409274674941935372038723107077875} a^{2} + \frac{24884243536452703986261438477576096}{66409274674941935372038723107077875} a + \frac{583415387712669680125907527587679}{1544401736626556636559040072257625}$

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:  $11$
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:  \( 53511038.116 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:S_4.C_2$ (as 16T764):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $C_2^3:S_4.C_2$
Character table for $C_2^3:S_4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.14272.1, 8.8.3259039744.1

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
2Data not computed
223Data not computed