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Properties

Label 16.0.83344647990...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{8}\cdot 11^{6}\cdot 19^{6}$
Root discriminant $23.45$
Ramified primes $2, 5, 11, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T511)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 0, 832, 0, 1424, 0, 1456, 0, 936, 0, 364, 0, 89, 0, 13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 13*x^14 + 89*x^12 + 364*x^10 + 936*x^8 + 1456*x^6 + 1424*x^4 + 832*x^2 + 256)
 
gp: K = bnfinit(x^16 + 13*x^14 + 89*x^12 + 364*x^10 + 936*x^8 + 1456*x^6 + 1424*x^4 + 832*x^2 + 256, 1)
 

Normalized defining polynomial

\( x^{16} + 13 x^{14} + 89 x^{12} + 364 x^{10} + 936 x^{8} + 1456 x^{6} + 1424 x^{4} + 832 x^{2} + 256 \)

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:  \(8334464799024100000000=2^{8}\cdot 5^{8}\cdot 11^{6}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.45$
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:  $2, 5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{10} + \frac{1}{32} a^{8} - \frac{1}{8} a^{7} - \frac{7}{32} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{64} a^{11} + \frac{1}{64} a^{9} - \frac{7}{64} a^{7} - \frac{1}{4} a^{6} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{128} a^{12} + \frac{1}{128} a^{10} - \frac{7}{128} a^{8} - \frac{1}{8} a^{7} + \frac{7}{32} a^{6} + \frac{1}{8} a^{5} + \frac{5}{32} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{11} - \frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{5}{32} a^{5} - \frac{3}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{512} a^{14} - \frac{1}{256} a^{13} - \frac{1}{512} a^{12} - \frac{1}{256} a^{11} - \frac{1}{512} a^{10} + \frac{7}{256} a^{9} - \frac{7}{256} a^{8} - \frac{7}{64} a^{7} - \frac{23}{128} a^{6} + \frac{11}{64} a^{5} - \frac{9}{64} a^{4} + \frac{1}{4} a^{3} + \frac{3}{16} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{1024} a^{15} + \frac{3}{1024} a^{13} - \frac{1}{256} a^{12} + \frac{3}{1024} a^{11} + \frac{3}{256} a^{10} + \frac{11}{512} a^{9} - \frac{5}{256} a^{8} - \frac{11}{256} a^{7} - \frac{5}{32} a^{6} - \frac{7}{128} a^{5} - \frac{5}{64} a^{4} + \frac{7}{32} a^{3} - \frac{5}{16} a^{2} - \frac{1}{8} a + \frac{1}{4}$

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:  $7$
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:  \( 19499.8389662 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T511):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.5225.1, 4.2.475.1, 4.2.275.1, 8.0.91293290000.2, 8.0.91293290000.1, 8.0.27300625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$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$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.8.6.2$x^{8} - 19 x^{4} + 5776$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$