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Properties

Label 16.0.87187048560...1504.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 3^{4}\cdot 17^{2}\cdot 193^{4}$
Root discriminant $23.51$
Ramified primes $2, 3, 17, 193$
Class number $5$
Class group $[5]$
Galois group 16T1116

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28, -104, 210, -248, 219, -222, 267, -274, 192, -150, 99, 2, -24, -2, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 13*x^14 - 2*x^13 - 24*x^12 + 2*x^11 + 99*x^10 - 150*x^9 + 192*x^8 - 274*x^7 + 267*x^6 - 222*x^5 + 219*x^4 - 248*x^3 + 210*x^2 - 104*x + 28)
 
gp: K = bnfinit(x^16 - 6*x^15 + 13*x^14 - 2*x^13 - 24*x^12 + 2*x^11 + 99*x^10 - 150*x^9 + 192*x^8 - 274*x^7 + 267*x^6 - 222*x^5 + 219*x^4 - 248*x^3 + 210*x^2 - 104*x + 28, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 13 x^{14} - 2 x^{13} - 24 x^{12} + 2 x^{11} + 99 x^{10} - 150 x^{9} + 192 x^{8} - 274 x^{7} + 267 x^{6} - 222 x^{5} + 219 x^{4} - 248 x^{3} + 210 x^{2} - 104 x + 28 \)

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:  \(8718704856053531541504=2^{28}\cdot 3^{4}\cdot 17^{2}\cdot 193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.51$
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:  $2, 3, 17, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{28} a^{12} - \frac{3}{14} a^{11} - \frac{1}{14} a^{10} - \frac{1}{4} a^{8} - \frac{3}{14} a^{7} - \frac{1}{14} a^{6} - \frac{1}{7} a^{5} + \frac{1}{28} a^{4} + \frac{1}{14} a^{3} + \frac{3}{14} a^{2} + \frac{1}{7} a$, $\frac{1}{28} a^{13} + \frac{1}{7} a^{11} + \frac{1}{14} a^{10} - \frac{1}{4} a^{9} - \frac{3}{14} a^{8} + \frac{1}{7} a^{7} - \frac{1}{14} a^{6} + \frac{5}{28} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{168} a^{14} - \frac{1}{168} a^{13} - \frac{1}{56} a^{12} - \frac{5}{28} a^{11} + \frac{11}{56} a^{10} + \frac{29}{168} a^{9} - \frac{11}{168} a^{8} - \frac{5}{42} a^{7} - \frac{1}{8} a^{6} - \frac{13}{56} a^{5} - \frac{27}{56} a^{4} + \frac{13}{28} a^{3} + \frac{9}{28} a^{2} + \frac{1}{42} a - \frac{1}{6}$, $\frac{1}{485222136} a^{15} + \frac{348353}{161740712} a^{14} - \frac{8238613}{485222136} a^{13} - \frac{570953}{80870356} a^{12} + \frac{4994481}{23105816} a^{11} - \frac{59054647}{485222136} a^{10} + \frac{24667745}{161740712} a^{9} + \frac{11776}{8664681} a^{8} + \frac{23205043}{485222136} a^{7} + \frac{15324635}{161740712} a^{6} - \frac{33914949}{161740712} a^{5} - \frac{35094261}{80870356} a^{4} + \frac{33097353}{80870356} a^{3} + \frac{28139029}{121305534} a^{2} + \frac{179855}{685342} a + \frac{3073384}{8664681}$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( 11917.5605387 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1116:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 76 conjugacy class representatives for t16n1116 are not computed
Character table for t16n1116 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.12352.2, 8.0.5492588544.3, 8.0.10374889472.1, 8.0.23343501312.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
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}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$193$193.4.2.1$x^{4} + 1737 x^{2} + 931225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
193.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
193.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
193.4.2.1$x^{4} + 1737 x^{2} + 931225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$