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Properties

Label 16.0.58625838959...5696.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 17^{6}\cdot 47^{2}$
Root discriminant $26.49$
Ramified primes $2, 17, 47$
Class number $2$
Class group $[2]$
Galois group 16T1445

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14, 32, 68, -180, -99, 296, 158, -92, -3, 0, 56, -44, 23, -8, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 8*x^13 + 23*x^12 - 44*x^11 + 56*x^10 - 3*x^8 - 92*x^7 + 158*x^6 + 296*x^5 - 99*x^4 - 180*x^3 + 68*x^2 + 32*x + 14)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 - 8*x^13 + 23*x^12 - 44*x^11 + 56*x^10 - 3*x^8 - 92*x^7 + 158*x^6 + 296*x^5 - 99*x^4 - 180*x^3 + 68*x^2 + 32*x + 14, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} - 8 x^{13} + 23 x^{12} - 44 x^{11} + 56 x^{10} - 3 x^{8} - 92 x^{7} + 158 x^{6} + 296 x^{5} - 99 x^{4} - 180 x^{3} + 68 x^{2} + 32 x + 14 \)

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:  \(58625838959875846045696=2^{40}\cdot 17^{6}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.49$
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:  $2, 17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4}$, $\frac{1}{368} a^{13} + \frac{19}{368} a^{12} - \frac{15}{184} a^{11} - \frac{1}{8} a^{10} + \frac{15}{92} a^{9} - \frac{3}{92} a^{8} - \frac{17}{184} a^{7} - \frac{1}{8} a^{6} - \frac{183}{368} a^{5} - \frac{45}{368} a^{4} + \frac{21}{46} a^{3} - \frac{15}{92} a^{2} + \frac{21}{184} a - \frac{61}{184}$, $\frac{1}{368} a^{14} - \frac{1}{16} a^{12} - \frac{7}{92} a^{11} + \frac{7}{184} a^{10} - \frac{3}{23} a^{9} + \frac{5}{184} a^{8} + \frac{3}{23} a^{7} + \frac{139}{368} a^{6} + \frac{15}{46} a^{5} + \frac{103}{368} a^{4} + \frac{15}{92} a^{3} + \frac{39}{184} a^{2} - \frac{1}{2} a + \frac{55}{184}$, $\frac{1}{184029072736} a^{15} - \frac{222902473}{184029072736} a^{14} - \frac{8325209}{8001264032} a^{13} + \frac{2990237171}{184029072736} a^{12} - \frac{4923586207}{92014536368} a^{11} + \frac{3115300501}{92014536368} a^{10} - \frac{14916516115}{92014536368} a^{9} + \frac{8941281963}{92014536368} a^{8} - \frac{87033839901}{184029072736} a^{7} + \frac{40008279645}{184029072736} a^{6} - \frac{8544674521}{184029072736} a^{5} - \frac{53942200195}{184029072736} a^{4} - \frac{36978005151}{92014536368} a^{3} + \frac{11421661749}{92014536368} a^{2} - \frac{27981729749}{92014536368} a - \frac{34393083047}{92014536368}$

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:  \( 189647.391197 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1445:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1445 are not computed
Character table for t16n1445 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.1088.2, 8.0.890175488.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.10.3$x^{4} + 6 x^{2} - 9$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.10.3$x^{4} + 6 x^{2} - 9$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.8.20.1$x^{8} + 4 x^{7} + 14 x^{4} + 4$$4$$2$$20$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$