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Properties

Label 16.8.35767658757...1056.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{34}\cdot 113^{6}$
Root discriminant $25.68$
Ramified primes $2, 113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.D_4$ (as 16T339)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 7, -6, -2, 34, -67, -194, 138, 194, -67, -34, -2, 6, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 7*x^14 + 6*x^13 - 2*x^12 - 34*x^11 - 67*x^10 + 194*x^9 + 138*x^8 - 194*x^7 - 67*x^6 + 34*x^5 - 2*x^4 - 6*x^3 + 7*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 7*x^14 + 6*x^13 - 2*x^12 - 34*x^11 - 67*x^10 + 194*x^9 + 138*x^8 - 194*x^7 - 67*x^6 + 34*x^5 - 2*x^4 - 6*x^3 + 7*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 7 x^{14} + 6 x^{13} - 2 x^{12} - 34 x^{11} - 67 x^{10} + 194 x^{9} + 138 x^{8} - 194 x^{7} - 67 x^{6} + 34 x^{5} - 2 x^{4} - 6 x^{3} + 7 x^{2} + 6 x + 1 \)

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:  \(35767658757222150701056=2^{34}\cdot 113^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{56} a^{12} - \frac{1}{28} a^{11} - \frac{5}{56} a^{10} - \frac{5}{28} a^{9} + \frac{13}{56} a^{8} + \frac{3}{7} a^{7} + \frac{11}{28} a^{6} - \frac{3}{7} a^{5} - \frac{15}{56} a^{4} + \frac{5}{28} a^{3} + \frac{23}{56} a^{2} + \frac{1}{28} a + \frac{1}{56}$, $\frac{1}{56} a^{13} + \frac{5}{56} a^{11} - \frac{3}{28} a^{10} - \frac{1}{8} a^{9} - \frac{3}{28} a^{8} + \frac{3}{28} a^{6} - \frac{3}{8} a^{5} - \frac{3}{28} a^{4} + \frac{15}{56} a^{3} + \frac{5}{14} a^{2} - \frac{9}{56} a + \frac{2}{7}$, $\frac{1}{4928} a^{14} + \frac{5}{1232} a^{13} + \frac{101}{2464} a^{11} - \frac{191}{2464} a^{10} - \frac{241}{1232} a^{9} + \frac{711}{4928} a^{8} - \frac{543}{1232} a^{7} - \frac{1943}{4928} a^{6} - \frac{241}{1232} a^{5} - \frac{337}{2464} a^{4} - \frac{75}{2464} a^{3} + \frac{3}{28} a^{2} + \frac{93}{1232} a + \frac{201}{704}$, $\frac{1}{167552} a^{15} + \frac{15}{167552} a^{14} + \frac{327}{41888} a^{13} - \frac{339}{83776} a^{12} + \frac{29}{1309} a^{11} + \frac{83}{7616} a^{10} - \frac{15237}{167552} a^{9} - \frac{25615}{167552} a^{8} - \frac{70107}{167552} a^{7} + \frac{303}{167552} a^{6} + \frac{41585}{83776} a^{5} + \frac{7537}{41888} a^{4} + \frac{14191}{83776} a^{3} + \frac{19277}{41888} a^{2} + \frac{3595}{167552} a - \frac{12491}{167552}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 436562.027036 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T339):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7232.1, 8.8.5910106112.1, 8.4.1673658368.1, 8.4.189123395584.6

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$113$113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$