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Properties

Label 16.16.3001873343...9744.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 17^{4}\cdot 113^{2}$
Root discriminant $29.33$
Ramified primes $2, 17, 113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T547)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-31, 168, 1316, -152, -4624, 80, 6484, -736, -4447, 1056, 1496, -536, -214, 112, 4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 4*x^14 + 112*x^13 - 214*x^12 - 536*x^11 + 1496*x^10 + 1056*x^9 - 4447*x^8 - 736*x^7 + 6484*x^6 + 80*x^5 - 4624*x^4 - 152*x^3 + 1316*x^2 + 168*x - 31)
 
gp: K = bnfinit(x^16 - 8*x^15 + 4*x^14 + 112*x^13 - 214*x^12 - 536*x^11 + 1496*x^10 + 1056*x^9 - 4447*x^8 - 736*x^7 + 6484*x^6 + 80*x^5 - 4624*x^4 - 152*x^3 + 1316*x^2 + 168*x - 31, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 4 x^{14} + 112 x^{13} - 214 x^{12} - 536 x^{11} + 1496 x^{10} + 1056 x^{9} - 4447 x^{8} - 736 x^{7} + 6484 x^{6} + 80 x^{5} - 4624 x^{4} - 152 x^{3} + 1316 x^{2} + 168 x - 31 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(300187334364663585439744=2^{48}\cdot 17^{4}\cdot 113^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.33$
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:  $2, 17, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{4}{9} a^{6} - \frac{2}{9} a^{5} - \frac{1}{3} a^{4} - \frac{2}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{3} a^{5} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2} + \frac{4}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} - \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{28953} a^{15} + \frac{1601}{28953} a^{14} - \frac{268}{9651} a^{13} - \frac{290}{28953} a^{12} - \frac{359}{28953} a^{11} + \frac{893}{28953} a^{10} - \frac{961}{9651} a^{9} + \frac{1480}{9651} a^{8} + \frac{4207}{28953} a^{7} + \frac{6193}{28953} a^{6} - \frac{11330}{28953} a^{5} + \frac{4066}{28953} a^{4} + \frac{4498}{9651} a^{3} + \frac{6595}{28953} a^{2} + \frac{9439}{28953} a + \frac{4310}{9651}$

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:  $15$
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:  \( 3415967.60395 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T547):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.34816.1, \(\Q(\zeta_{16})^+\), 4.4.4352.1, 8.8.547893542912.1, 8.8.136973385728.1, 8.8.4848615424.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$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$$0$Trivial$[\ ]$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{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$$[\ ]^{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$