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Properties

Label 16.0.31347719877...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{9}\cdot 11^{5}\cdot 31^{5}\cdot 59^{2}$
Root discriminant $25.47$
Ramified primes $5, 11, 31, 59$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1782

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 6, -5, -10, -34, 45, 70, 131, 70, 45, -34, -10, -5, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 6*x^14 - 5*x^13 - 10*x^12 - 34*x^11 + 45*x^10 + 70*x^9 + 131*x^8 + 70*x^7 + 45*x^6 - 34*x^5 - 10*x^4 - 5*x^3 + 6*x^2 - x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 6*x^14 - 5*x^13 - 10*x^12 - 34*x^11 + 45*x^10 + 70*x^9 + 131*x^8 + 70*x^7 + 45*x^6 - 34*x^5 - 10*x^4 - 5*x^3 + 6*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 6 x^{14} - 5 x^{13} - 10 x^{12} - 34 x^{11} + 45 x^{10} + 70 x^{9} + 131 x^{8} + 70 x^{7} + 45 x^{6} - 34 x^{5} - 10 x^{4} - 5 x^{3} + 6 x^{2} - x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31347719877728869140625=5^{9}\cdot 11^{5}\cdot 31^{5}\cdot 59^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.47$
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:  $5, 11, 31, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5}$, $\frac{1}{15} a^{12} - \frac{1}{15} a^{11} - \frac{2}{15} a^{9} + \frac{7}{15} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{2}{15} a^{4} + \frac{7}{15} a^{3} - \frac{4}{15} a + \frac{4}{15}$, $\frac{1}{75} a^{13} - \frac{1}{75} a^{12} + \frac{2}{25} a^{11} + \frac{1}{75} a^{10} + \frac{4}{75} a^{9} + \frac{9}{25} a^{8} + \frac{2}{25} a^{7} + \frac{8}{25} a^{6} - \frac{2}{75} a^{5} - \frac{14}{75} a^{4} + \frac{8}{25} a^{3} - \frac{31}{75} a^{2} + \frac{31}{75} a + \frac{8}{25}$, $\frac{1}{21675} a^{14} - \frac{76}{21675} a^{13} - \frac{364}{21675} a^{12} + \frac{1361}{21675} a^{11} + \frac{874}{21675} a^{10} - \frac{8278}{21675} a^{9} - \frac{4219}{21675} a^{8} - \frac{1177}{7225} a^{7} - \frac{10577}{21675} a^{6} + \frac{7906}{21675} a^{5} + \frac{8099}{21675} a^{4} + \frac{1939}{21675} a^{3} + \frac{9751}{21675} a^{2} - \frac{2966}{21675} a - \frac{809}{4335}$, $\frac{1}{21675} a^{15} - \frac{71}{21675} a^{13} - \frac{194}{7225} a^{12} + \frac{559}{21675} a^{11} - \frac{54}{1445} a^{10} - \frac{242}{7225} a^{9} - \frac{2807}{21675} a^{8} - \frac{8429}{21675} a^{7} - \frac{8}{4335} a^{6} - \frac{1916}{7225} a^{5} - \frac{7933}{21675} a^{4} + \frac{5101}{21675} a^{3} + \frac{8096}{21675} a^{2} + \frac{4918}{21675} a + \frac{8746}{21675}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 40462.347657 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1782:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 130 conjugacy class representatives for t16n1782 are not computed
Character table for t16n1782 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.8525.1, 8.2.4287861875.4

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.8.4.1$x^{8} + 32674 x^{4} - 119164 x^{2} + 266897569$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$