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Properties

Label 16.0.10176066191...2064.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 41^{6}\cdot 113^{2}$
Root discriminant $20.56$
Ramified primes $2, 41, 113$
Class number $2$
Class group $[2]$
Galois group 16T876

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -32, 104, -176, 288, -440, 656, -592, 329, -148, 192, -214, 160, -76, 27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 27*x^14 - 76*x^13 + 160*x^12 - 214*x^11 + 192*x^10 - 148*x^9 + 329*x^8 - 592*x^7 + 656*x^6 - 440*x^5 + 288*x^4 - 176*x^3 + 104*x^2 - 32*x + 16)
 
gp: K = bnfinit(x^16 - 6*x^15 + 27*x^14 - 76*x^13 + 160*x^12 - 214*x^11 + 192*x^10 - 148*x^9 + 329*x^8 - 592*x^7 + 656*x^6 - 440*x^5 + 288*x^4 - 176*x^3 + 104*x^2 - 32*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 27 x^{14} - 76 x^{13} + 160 x^{12} - 214 x^{11} + 192 x^{10} - 148 x^{9} + 329 x^{8} - 592 x^{7} + 656 x^{6} - 440 x^{5} + 288 x^{4} - 176 x^{3} + 104 x^{2} - 32 x + 16 \)

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:  \(1017606619113208152064=2^{24}\cdot 41^{6}\cdot 113^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.56$
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:  $2, 41, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{376} a^{14} - \frac{3}{94} a^{13} + \frac{45}{376} a^{12} - \frac{37}{188} a^{11} + \frac{27}{188} a^{10} + \frac{37}{188} a^{9} - \frac{20}{47} a^{8} - \frac{22}{47} a^{7} - \frac{127}{376} a^{6} - \frac{51}{188} a^{5} - \frac{73}{188} a^{4} - \frac{9}{47} a^{3} - \frac{11}{94} a^{2} - \frac{20}{47} a + \frac{7}{47}$, $\frac{1}{35883456758296} a^{15} - \frac{24582952983}{35883456758296} a^{14} - \frac{352217717067}{35883456758296} a^{13} + \frac{2985176336701}{35883456758296} a^{12} - \frac{984034069751}{8970864189574} a^{11} - \frac{924188836301}{17941728379148} a^{10} + \frac{3057695120673}{17941728379148} a^{9} - \frac{3458712150949}{8970864189574} a^{8} + \frac{12411330733885}{35883456758296} a^{7} + \frac{2068570696127}{35883456758296} a^{6} - \frac{1076399485423}{8970864189574} a^{5} - \frac{887184080453}{8970864189574} a^{4} - \frac{518524694623}{1281552027082} a^{3} + \frac{1133778893988}{4485432094787} a^{2} + \frac{1660301471088}{4485432094787} a - \frac{1350819342291}{4485432094787}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2873.19936942 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T876:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 65 conjugacy class representatives for t16n876 are not computed
Character table for t16n876 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.282300416.1, 8.0.778047488.1, 8.0.31899947008.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$41$41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$