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Properties

Label 16.0.31686294687...8096.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 3^{14}\cdot 7^{8}\cdot 67^{2}$
Root discriminant $16.55$
Ramified primes $2, 3, 7, 67$
Class number $1$
Class group Trivial
Galois group 16T984

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![127, -340, 161, 546, 98, 331, 196, 0, 169, 6, -113, -1, 47, 9, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 10*x^14 + 9*x^13 + 47*x^12 - x^11 - 113*x^10 + 6*x^9 + 169*x^8 + 196*x^6 + 331*x^5 + 98*x^4 + 546*x^3 + 161*x^2 - 340*x + 127)
 
gp: K = bnfinit(x^16 - 2*x^15 - 10*x^14 + 9*x^13 + 47*x^12 - x^11 - 113*x^10 + 6*x^9 + 169*x^8 + 196*x^6 + 331*x^5 + 98*x^4 + 546*x^3 + 161*x^2 - 340*x + 127, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 10 x^{14} + 9 x^{13} + 47 x^{12} - x^{11} - 113 x^{10} + 6 x^{9} + 169 x^{8} + 196 x^{6} + 331 x^{5} + 98 x^{4} + 546 x^{3} + 161 x^{2} - 340 x + 127 \)

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:  \(31686294687883428096=2^{8}\cdot 3^{14}\cdot 7^{8}\cdot 67^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.55$
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:  $2, 3, 7, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{11} - \frac{3}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{9} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{9} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{3871} a^{14} - \frac{200}{3871} a^{13} + \frac{106}{3871} a^{12} - \frac{5}{3871} a^{11} - \frac{269}{3871} a^{10} + \frac{1831}{3871} a^{9} + \frac{5}{49} a^{8} + \frac{1573}{3871} a^{7} - \frac{139}{553} a^{6} - \frac{139}{3871} a^{5} - \frac{16}{3871} a^{4} + \frac{1591}{3871} a^{3} + \frac{1614}{3871} a^{2} - \frac{1507}{3871} a - \frac{82}{3871}$, $\frac{1}{2202407288725} a^{15} - \frac{253626396}{2202407288725} a^{14} - \frac{8506189698}{314629612675} a^{13} - \frac{68744232707}{2202407288725} a^{12} + \frac{30207426686}{440481457745} a^{11} + \frac{102254695079}{2202407288725} a^{10} - \frac{1067344341689}{2202407288725} a^{9} - \frac{698272031453}{2202407288725} a^{8} + \frac{1011770086351}{2202407288725} a^{7} + \frac{740158254781}{2202407288725} a^{6} + \frac{70812637076}{314629612675} a^{5} - \frac{945918942502}{2202407288725} a^{4} + \frac{335327149536}{2202407288725} a^{3} + \frac{186003811762}{2202407288725} a^{2} - \frac{696872910792}{2202407288725} a + \frac{4509752279}{17341789675}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{2223021134}{44947087525} a^{15} - \frac{8704732339}{44947087525} a^{14} - \frac{6639859799}{44947087525} a^{13} + \frac{36626739062}{44947087525} a^{12} + \frac{7670769514}{8989417505} a^{11} - \frac{89995693364}{44947087525} a^{10} - \frac{102825951151}{44947087525} a^{9} + \frac{238987043773}{44947087525} a^{8} - \frac{25391313366}{44947087525} a^{7} - \frac{24826789996}{44947087525} a^{6} + \frac{478677649963}{44947087525} a^{5} - \frac{232473126743}{44947087525} a^{4} + \frac{456306702449}{44947087525} a^{3} + \frac{372484004258}{44947087525} a^{2} - \frac{595243506678}{44947087525} a + \frac{1726154361}{353914075} \) (order $6$)
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:  \( 3355.6312611 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T984:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n984
Character table for t16n984 is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), 4.2.1323.1 x2, 4.0.189.1 x2, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.1750329.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
3Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67Data not computed