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Properties

Label 16.8.24126669511...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{8}\cdot 5^{12}\cdot 3881^{2}$
Root discriminant $16.27$
Ramified primes $3, 5, 3881$
Class number $1$
Class group Trivial
Galois group 16T1497

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -14, 43, -57, 69, -20, -102, 126, -60, 11, 28, -28, -9, 16, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^13 - 9*x^12 - 28*x^11 + 28*x^10 + 11*x^9 - 60*x^8 + 126*x^7 - 102*x^6 - 20*x^5 + 69*x^4 - 57*x^3 + 43*x^2 - 14*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 16*x^13 - 9*x^12 - 28*x^11 + 28*x^10 + 11*x^9 - 60*x^8 + 126*x^7 - 102*x^6 - 20*x^5 + 69*x^4 - 57*x^3 + 43*x^2 - 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 16 x^{13} - 9 x^{12} - 28 x^{11} + 28 x^{10} + 11 x^{9} - 60 x^{8} + 126 x^{7} - 102 x^{6} - 20 x^{5} + 69 x^{4} - 57 x^{3} + 43 x^{2} - 14 x + 1 \)

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:  \(24126669511962890625=3^{8}\cdot 5^{12}\cdot 3881^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.27$
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:  $3, 5, 3881$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{121826695} a^{15} - \frac{789993}{24365339} a^{14} - \frac{39665929}{121826695} a^{13} - \frac{665363}{24365339} a^{12} + \frac{50509387}{121826695} a^{11} - \frac{2516742}{24365339} a^{10} - \frac{10945729}{24365339} a^{9} + \frac{25256631}{121826695} a^{8} - \frac{45172901}{121826695} a^{7} + \frac{18717103}{121826695} a^{6} - \frac{54390046}{121826695} a^{5} + \frac{34600349}{121826695} a^{4} + \frac{54747819}{121826695} a^{3} + \frac{38760738}{121826695} a^{2} + \frac{5168209}{121826695} a - \frac{8291624}{24365339}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4697.01767862 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1497:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1497
Character table for t16n1497 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.4.196475625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
3881Data not computed