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Properties

Label 16.0.24956872719...3125.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{11}\cdot 59^{10}$
Root discriminant $38.67$
Ramified primes $5, 59$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2549, -20, 17043, -911, 10677, -2708, 7588, -1599, 2063, 134, 141, 121, 36, -2, 14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 14*x^14 - 2*x^13 + 36*x^12 + 121*x^11 + 141*x^10 + 134*x^9 + 2063*x^8 - 1599*x^7 + 7588*x^6 - 2708*x^5 + 10677*x^4 - 911*x^3 + 17043*x^2 - 20*x + 2549)
 
gp: K = bnfinit(x^16 - 2*x^15 + 14*x^14 - 2*x^13 + 36*x^12 + 121*x^11 + 141*x^10 + 134*x^9 + 2063*x^8 - 1599*x^7 + 7588*x^6 - 2708*x^5 + 10677*x^4 - 911*x^3 + 17043*x^2 - 20*x + 2549, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 14 x^{14} - 2 x^{13} + 36 x^{12} + 121 x^{11} + 141 x^{10} + 134 x^{9} + 2063 x^{8} - 1599 x^{7} + 7588 x^{6} - 2708 x^{5} + 10677 x^{4} - 911 x^{3} + 17043 x^{2} - 20 x + 2549 \)

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:  \(24956872719757880908203125=5^{11}\cdot 59^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.67$
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:  $5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{14} - \frac{2}{25} a^{12} - \frac{1}{25} a^{11} + \frac{1}{25} a^{10} - \frac{6}{25} a^{9} - \frac{12}{25} a^{8} + \frac{11}{25} a^{7} + \frac{7}{25} a^{6} + \frac{9}{25} a^{5} - \frac{11}{25} a^{4} + \frac{6}{25} a^{3} - \frac{1}{5} a^{2} + \frac{8}{25} a - \frac{6}{25}$, $\frac{1}{187018867331727496874893425} a^{15} - \frac{914844907460533104707241}{62339622443909165624964475} a^{14} + \frac{11796920064506271962657198}{187018867331727496874893425} a^{13} - \frac{2154028573321760626874426}{37403773466345499374978685} a^{12} - \frac{18484262423342608520769266}{187018867331727496874893425} a^{11} - \frac{2278559059169568263271253}{62339622443909165624964475} a^{10} + \frac{26606287646885462636051857}{62339622443909165624964475} a^{9} + \frac{73301389268631187677112757}{187018867331727496874893425} a^{8} + \frac{432202933389165274698623}{5667238403991742329542225} a^{7} - \frac{12708477810020662647832849}{62339622443909165624964475} a^{6} + \frac{44774799838783996904710477}{187018867331727496874893425} a^{5} + \frac{1976339241171853342554788}{5667238403991742329542225} a^{4} + \frac{27387498076555784028183504}{62339622443909165624964475} a^{3} - \frac{5658898982703393309902447}{187018867331727496874893425} a^{2} - \frac{18065613560551561274639372}{37403773466345499374978685} a - \frac{57211737295000007690791897}{187018867331727496874893425}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -1 \) (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:  \( 1486421.09204 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1475.1, 8.0.37866753125.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ R

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
59Data not computed