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Properties

Label 16.0.14923692200...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{15}\cdot 421^{3}$
Root discriminant $28.08$
Ramified primes $2, 5, 421$
Class number $4$
Class group $[2, 2]$
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![205, 200, -165, -340, 95, -250, 65, -120, 195, -20, 125, -10, 50, 0, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 10*x^14 + 50*x^12 - 10*x^11 + 125*x^10 - 20*x^9 + 195*x^8 - 120*x^7 + 65*x^6 - 250*x^5 + 95*x^4 - 340*x^3 - 165*x^2 + 200*x + 205)
 
gp: K = bnfinit(x^16 + 10*x^14 + 50*x^12 - 10*x^11 + 125*x^10 - 20*x^9 + 195*x^8 - 120*x^7 + 65*x^6 - 250*x^5 + 95*x^4 - 340*x^3 - 165*x^2 + 200*x + 205, 1)
 

Normalized defining polynomial

\( x^{16} + 10 x^{14} + 50 x^{12} - 10 x^{11} + 125 x^{10} - 20 x^{9} + 195 x^{8} - 120 x^{7} + 65 x^{6} - 250 x^{5} + 95 x^{4} - 340 x^{3} - 165 x^{2} + 200 x + 205 \)

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:  \(149236922000000000000000=2^{16}\cdot 5^{15}\cdot 421^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.08$
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:  $2, 5, 421$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{295262548309507995791} a^{15} + \frac{57801934829956219992}{295262548309507995791} a^{14} - \frac{86133362556940374667}{295262548309507995791} a^{13} + \frac{145823446075581695872}{295262548309507995791} a^{12} - \frac{55670726996106048930}{295262548309507995791} a^{11} - \frac{122401770688206297053}{295262548309507995791} a^{10} + \frac{114252198172875888400}{295262548309507995791} a^{9} - \frac{11410092177380666568}{295262548309507995791} a^{8} - \frac{55529013514372902075}{295262548309507995791} a^{7} + \frac{90732639546785696819}{295262548309507995791} a^{6} - \frac{5718520785166330691}{295262548309507995791} a^{5} - \frac{5770662437648759090}{295262548309507995791} a^{4} + \frac{139977477725637116829}{295262548309507995791} a^{3} - \frac{82667512837665020633}{295262548309507995791} a^{2} + \frac{29172858549842822628}{295262548309507995791} a - \frac{102888154474201547075}{295262548309507995791}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 22225.0626018 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.8420000000.1

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 $16$ R $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
2Data not computed
5Data not computed
421Data not computed