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Properties

Label 16.10.6950941496...2256.1
Degree $16$
Signature $[10, 3]$
Discriminant $-\,2^{24}\cdot 113^{6}\cdot 199$
Root discriminant $23.18$
Ramified primes $2, 113, 199$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1638

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, -23, -56, 110, 166, -190, -186, 129, 74, -50, -22, 22, 12, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 9*x^14 + 12*x^13 + 22*x^12 - 22*x^11 - 50*x^10 + 74*x^9 + 129*x^8 - 186*x^7 - 190*x^6 + 166*x^5 + 110*x^4 - 56*x^3 - 23*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 9*x^14 + 12*x^13 + 22*x^12 - 22*x^11 - 50*x^10 + 74*x^9 + 129*x^8 - 186*x^7 - 190*x^6 + 166*x^5 + 110*x^4 - 56*x^3 - 23*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 9 x^{14} + 12 x^{13} + 22 x^{12} - 22 x^{11} - 50 x^{10} + 74 x^{9} + 129 x^{8} - 186 x^{7} - 190 x^{6} + 166 x^{5} + 110 x^{4} - 56 x^{3} - 23 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-6950941496764851552256=-\,2^{24}\cdot 113^{6}\cdot 199\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.18$
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:  $2, 113, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{8} a^{12} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{8} a$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{1509989576} a^{15} + \frac{43638359}{1509989576} a^{14} - \frac{91648777}{1509989576} a^{13} - \frac{11623621}{377497394} a^{12} + \frac{282425069}{1509989576} a^{11} - \frac{645113391}{1509989576} a^{10} + \frac{79390763}{188748697} a^{9} + \frac{330667769}{1509989576} a^{8} + \frac{46885463}{754994788} a^{7} + \frac{622448201}{1509989576} a^{6} + \frac{49848101}{1509989576} a^{5} + \frac{35538825}{377497394} a^{4} + \frac{752672651}{1509989576} a^{3} - \frac{739739123}{1509989576} a^{2} - \frac{481393203}{1509989576} a + \frac{365235835}{754994788}$

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:  $12$
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:  \( 201416.713638 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1638:

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 t16n1638 are not computed
Character table for t16n1638 is not computed

Intermediate fields

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

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 $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $16$

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}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$113$113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$199$199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$