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Properties

Label 16.8.36484848775...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 29^{10}\cdot 149^{2}$
Root discriminant $34.29$
Ramified primes $5, 29, 149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2:C_2^2.C_2$ (as 16T406)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5849, -10131, 18383, 9864, -5774, -13202, 6445, 579, -714, 364, -140, -52, 51, 24, -17, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 17*x^14 + 24*x^13 + 51*x^12 - 52*x^11 - 140*x^10 + 364*x^9 - 714*x^8 + 579*x^7 + 6445*x^6 - 13202*x^5 - 5774*x^4 + 9864*x^3 + 18383*x^2 - 10131*x - 5849)
 
gp: K = bnfinit(x^16 - x^15 - 17*x^14 + 24*x^13 + 51*x^12 - 52*x^11 - 140*x^10 + 364*x^9 - 714*x^8 + 579*x^7 + 6445*x^6 - 13202*x^5 - 5774*x^4 + 9864*x^3 + 18383*x^2 - 10131*x - 5849, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 17 x^{14} + 24 x^{13} + 51 x^{12} - 52 x^{11} - 140 x^{10} + 364 x^{9} - 714 x^{8} + 579 x^{7} + 6445 x^{6} - 13202 x^{5} - 5774 x^{4} + 9864 x^{3} + 18383 x^{2} - 10131 x - 5849 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3648484877538188437890625=5^{8}\cdot 29^{10}\cdot 149^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.29$
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:  $5, 29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{109} a^{14} - \frac{51}{109} a^{13} - \frac{38}{109} a^{12} - \frac{44}{109} a^{11} - \frac{4}{109} a^{10} + \frac{21}{109} a^{9} + \frac{47}{109} a^{8} + \frac{49}{109} a^{7} + \frac{41}{109} a^{6} - \frac{29}{109} a^{5} + \frac{39}{109} a^{4} + \frac{2}{109} a^{3} + \frac{23}{109} a^{2} - \frac{25}{109} a - \frac{42}{109}$, $\frac{1}{118050893520700263247498731989177} a^{15} + \frac{8833978098182461966475802889}{9080837963130789480576825537629} a^{14} - \frac{24924265447833579080574515585252}{118050893520700263247498731989177} a^{13} + \frac{15512084213847292209621767809389}{118050893520700263247498731989177} a^{12} + \frac{51459428798275779357035657072485}{118050893520700263247498731989177} a^{11} + \frac{27668527860467220814315634429029}{118050893520700263247498731989177} a^{10} + \frac{35800522986292384138151059755323}{118050893520700263247498731989177} a^{9} + \frac{26708392984425071944981038951189}{118050893520700263247498731989177} a^{8} - \frac{54768487985896233623240277634942}{118050893520700263247498731989177} a^{7} - \frac{6303869569479858510367690854029}{118050893520700263247498731989177} a^{6} + \frac{17875700593083233565022101902523}{118050893520700263247498731989177} a^{5} - \frac{27161401623497336383296717634551}{118050893520700263247498731989177} a^{4} - \frac{12194221096556425257587773032108}{118050893520700263247498731989177} a^{3} - \frac{10051165023987819915678643689500}{118050893520700263247498731989177} a^{2} + \frac{36808872659165784149022354087436}{118050893520700263247498731989177} a + \frac{22841900970352403824722913836253}{118050893520700263247498731989177}$

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:  $11$
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:  \( 1537140.32332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2.C_2$ (as 16T406):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_4^2:C_2^2.C_2$
Character table for $C_4^2:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$149$$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$