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Properties

Label 16.8.21451549715...0144.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{48}\cdot 17^{2}\cdot 271^{4}\cdot 1487^{4}$
Root discriminant $287.22$
Ramified primes $2, 17, 271, 1487$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![73984, 0, 517888, 0, -55520, 0, -922800, 0, 456561, 0, 5804, 0, -1930, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 1930*x^12 + 5804*x^10 + 456561*x^8 - 922800*x^6 - 55520*x^4 + 517888*x^2 + 73984)
 
gp: K = bnfinit(x^16 - 4*x^14 - 1930*x^12 + 5804*x^10 + 456561*x^8 - 922800*x^6 - 55520*x^4 + 517888*x^2 + 73984, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 1930 x^{12} + 5804 x^{10} + 456561 x^{8} - 922800 x^{6} - 55520 x^{4} + 517888 x^{2} + 73984 \)

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:  \(2145154971555059070700713179815969030144=2^{48}\cdot 17^{2}\cdot 271^{4}\cdot 1487^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $287.22$
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:  $2, 17, 271, 1487$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{3}$, $\frac{1}{128} a^{8} - \frac{1}{64} a^{6} - \frac{7}{128} a^{4} + \frac{1}{16} a^{2} - \frac{1}{8}$, $\frac{1}{128} a^{9} - \frac{1}{64} a^{7} - \frac{7}{128} a^{5} + \frac{1}{16} a^{3} - \frac{1}{8} a$, $\frac{1}{768} a^{10} - \frac{1}{256} a^{9} + \frac{1}{384} a^{8} + \frac{1}{128} a^{7} + \frac{17}{768} a^{6} - \frac{25}{256} a^{5} + \frac{19}{192} a^{4} + \frac{3}{32} a^{3} + \frac{1}{48} a^{2} - \frac{7}{16} a - \frac{5}{12}$, $\frac{1}{768} a^{11} - \frac{1}{768} a^{9} + \frac{23}{768} a^{7} + \frac{1}{768} a^{5} + \frac{11}{96} a^{3} + \frac{7}{48} a$, $\frac{1}{15588864} a^{12} - \frac{1}{5196288} a^{10} - \frac{51845}{15588864} a^{8} + \frac{34565}{5196288} a^{6} + \frac{1165}{1948608} a^{4} - \frac{3823}{974304} a^{2} + \frac{664}{1791}$, $\frac{1}{31177728} a^{13} - \frac{1}{31177728} a^{12} - \frac{1}{10392576} a^{11} + \frac{1}{10392576} a^{10} + \frac{69943}{31177728} a^{9} - \frac{69943}{31177728} a^{8} + \frac{602909}{10392576} a^{7} + \frac{46627}{10392576} a^{6} - \frac{210799}{7794432} a^{5} - \frac{763505}{7794432} a^{4} + \frac{422435}{1948608} a^{3} + \frac{186505}{1948608} a^{2} + \frac{10685}{28656} a + \frac{10807}{28656}$, $\frac{1}{80157938688} a^{14} + \frac{641}{40078969344} a^{12} - \frac{988229}{20039484672} a^{10} - \frac{1}{256} a^{9} + \frac{128171429}{40078969344} a^{8} - \frac{7}{128} a^{7} + \frac{2503982603}{80157938688} a^{6} - \frac{25}{256} a^{5} + \frac{2086132051}{20039484672} a^{4} - \frac{3}{32} a^{3} - \frac{6084163}{1669957056} a^{2} + \frac{5}{16} a + \frac{15675251}{73674576}$, $\frac{1}{160315877376} a^{15} + \frac{641}{80157938688} a^{13} - \frac{1}{31177728} a^{12} + \frac{12552425}{20039484672} a^{11} + \frac{1}{10392576} a^{10} - \frac{80573203}{80157938688} a^{9} - \frac{69943}{31177728} a^{8} + \frac{5530779767}{160315877376} a^{7} + \frac{46627}{10392576} a^{6} + \frac{2660179789}{40078969344} a^{5} + \frac{210799}{7794432} a^{4} + \frac{133078925}{3339914112} a^{3} - \frac{57071}{1948608} a^{2} + \frac{31024121}{147349152} a - \frac{3521}{28656}$

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:  $11$
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:  \( 105250391887000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.825296896.1, \(\Q(\zeta_{16})^+\), 4.4.25790528.1, 8.8.681114966547234816.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.26.4$x^{8} + 8 x^{7} + 12 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
271Data not computed
1487Data not computed