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Properties

Label 16.16.8820836049...5216.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{12}\cdot 19^{10}\cdot 37^{8}$
Root discriminant $64.43$
Ramified primes $2, 19, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.Q_8.C_6$ (as 16T732)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61009, 0, -219488, 0, 282663, 0, -166706, 0, 52478, 0, -9367, 0, 942, 0, -49, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 49*x^14 + 942*x^12 - 9367*x^10 + 52478*x^8 - 166706*x^6 + 282663*x^4 - 219488*x^2 + 61009)
 
gp: K = bnfinit(x^16 - 49*x^14 + 942*x^12 - 9367*x^10 + 52478*x^8 - 166706*x^6 + 282663*x^4 - 219488*x^2 + 61009, 1)
 

Normalized defining polynomial

\( x^{16} - 49 x^{14} + 942 x^{12} - 9367 x^{10} + 52478 x^{8} - 166706 x^{6} + 282663 x^{4} - 219488 x^{2} + 61009 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(88208360493688117191834505216=2^{12}\cdot 19^{10}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.43$
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:  $2, 19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} + \frac{8}{19} a^{8} - \frac{8}{19} a^{6}$, $\frac{1}{19} a^{11} + \frac{8}{19} a^{9} - \frac{8}{19} a^{7}$, $\frac{1}{19} a^{12} + \frac{4}{19} a^{8} + \frac{7}{19} a^{6}$, $\frac{1}{19} a^{13} + \frac{4}{19} a^{9} + \frac{7}{19} a^{7}$, $\frac{1}{233514902} a^{14} - \frac{1}{38} a^{13} - \frac{5885121}{233514902} a^{12} + \frac{2047011}{233514902} a^{10} + \frac{15}{38} a^{9} - \frac{56010913}{233514902} a^{8} + \frac{6}{19} a^{7} - \frac{70907509}{233514902} a^{6} - \frac{1}{2} a^{5} + \frac{1821275}{6145129} a^{4} - \frac{1}{2} a^{3} + \frac{2575003}{6145129} a^{2} - \frac{1}{2} a - \frac{5041115}{12290258}$, $\frac{1}{3035693726} a^{15} - \frac{18305383}{1517846863} a^{13} - \frac{1}{38} a^{12} - \frac{22533505}{3035693726} a^{11} + \frac{454387170}{1517846863} a^{9} + \frac{15}{38} a^{8} + \frac{1194989065}{3035693726} a^{7} + \frac{6}{19} a^{6} - \frac{51663611}{159773354} a^{5} - \frac{1}{2} a^{4} + \frac{60456167}{159773354} a^{3} - \frac{1}{2} a^{2} - \frac{36318767}{79886677} a - \frac{1}{2}$

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:  $15$
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:  \( 2881793519.03 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.Q_8.C_6$ (as 16T732):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 30 conjugacy class representatives for $C_2^3.Q_8.C_6$
Character table for $C_2^3.Q_8.C_6$ is not computed

Intermediate fields

4.4.494209.1, 8.8.244242535681.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.12.8.1$x^{12} - 111 x^{9} + 4107 x^{6} - 50653 x^{3} + 14993288$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$