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Properties

Label 16.0.41816828273...3125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{11}\cdot 29^{3}\cdot 101^{6}\cdot 149^{3}$
Root discriminant $82.00$
Ramified primes $5, 29, 101, 149$
Class number $96$ (GRH)
Class group $[2, 4, 12]$ (GRH)
Galois group 16T1642

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5530880, 0, -5530880, 0, 2534880, 0, -691280, 0, 122309, 0, -14352, 0, 1090, 0, -49, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 49*x^14 + 1090*x^12 - 14352*x^10 + 122309*x^8 - 691280*x^6 + 2534880*x^4 - 5530880*x^2 + 5530880)
 
gp: K = bnfinit(x^16 - 49*x^14 + 1090*x^12 - 14352*x^10 + 122309*x^8 - 691280*x^6 + 2534880*x^4 - 5530880*x^2 + 5530880, 1)
 

Normalized defining polynomial

\( x^{16} - 49 x^{14} + 1090 x^{12} - 14352 x^{10} + 122309 x^{8} - 691280 x^{6} + 2534880 x^{4} - 5530880 x^{2} + 5530880 \)

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:  \(4181682827362654424548876953125=5^{11}\cdot 29^{3}\cdot 101^{6}\cdot 149^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.00$
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:  $5, 29, 101, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{10} + \frac{3}{16} a^{8} - \frac{1}{8} a^{6} - \frac{1}{2} a^{3} + \frac{5}{16} a^{2} - \frac{1}{4}$, $\frac{1}{64} a^{11} - \frac{1}{32} a^{10} - \frac{5}{64} a^{9} + \frac{5}{32} a^{8} - \frac{5}{32} a^{7} - \frac{3}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{5}{64} a^{3} + \frac{11}{32} a^{2} + \frac{3}{16} a - \frac{3}{8}$, $\frac{1}{256} a^{12} + \frac{7}{256} a^{10} - \frac{3}{128} a^{8} - \frac{1}{16} a^{6} + \frac{37}{256} a^{4} - \frac{7}{32} a^{2} - \frac{7}{16}$, $\frac{1}{1024} a^{13} - \frac{1}{512} a^{12} + \frac{7}{1024} a^{11} - \frac{7}{512} a^{10} - \frac{3}{512} a^{9} + \frac{3}{256} a^{8} - \frac{1}{64} a^{7} + \frac{1}{32} a^{6} - \frac{219}{1024} a^{5} + \frac{219}{512} a^{4} - \frac{39}{128} a^{3} + \frac{7}{64} a^{2} - \frac{23}{64} a - \frac{9}{32}$, $\frac{1}{4096} a^{14} + \frac{3}{4096} a^{12} - \frac{17}{2048} a^{10} - \frac{63}{512} a^{8} - \frac{667}{4096} a^{6} + \frac{269}{1024} a^{4} + \frac{119}{256} a^{2} - \frac{1}{2} a + \frac{23}{64}$, $\frac{1}{16384} a^{15} - \frac{1}{8192} a^{14} + \frac{3}{16384} a^{13} - \frac{3}{8192} a^{12} - \frac{17}{8192} a^{11} + \frac{17}{4096} a^{10} - \frac{63}{2048} a^{9} + \frac{63}{1024} a^{8} - \frac{2715}{16384} a^{7} - \frac{1381}{8192} a^{6} - \frac{243}{4096} a^{5} + \frac{243}{2048} a^{4} - \frac{265}{1024} a^{3} + \frac{9}{512} a^{2} - \frac{41}{256} a + \frac{41}{128}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{12}$, which has order $96$ (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:  \( 6710629.59661 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1642:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1642 are not computed
Character table for t16n1642 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.0.13912283190625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$149$149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.3.4$x^{4} + 1192$$4$$1$$3$$C_4$$[\ ]_{4}$
149.8.0.1$x^{8} - x + 11$$1$$8$$0$$C_8$$[\ ]^{8}$