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Properties

Label 16.6.20423375579...9375.3
Degree $16$
Signature $[6, 5]$
Discriminant $-\,5^{12}\cdot 101^{6}\cdot 199^{3}$
Root discriminant $50.92$
Ramified primes $5, 101, 199$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1643

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -640, -4928, 14400, -23520, 29560, -27628, 15490, -7461, 3500, -1077, 475, -105, 10, 3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 3*x^14 + 10*x^13 - 105*x^12 + 475*x^11 - 1077*x^10 + 3500*x^9 - 7461*x^8 + 15490*x^7 - 27628*x^6 + 29560*x^5 - 23520*x^4 + 14400*x^3 - 4928*x^2 - 640*x + 256)
 
gp: K = bnfinit(x^16 - 5*x^15 + 3*x^14 + 10*x^13 - 105*x^12 + 475*x^11 - 1077*x^10 + 3500*x^9 - 7461*x^8 + 15490*x^7 - 27628*x^6 + 29560*x^5 - 23520*x^4 + 14400*x^3 - 4928*x^2 - 640*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 3 x^{14} + 10 x^{13} - 105 x^{12} + 475 x^{11} - 1077 x^{10} + 3500 x^{9} - 7461 x^{8} + 15490 x^{7} - 27628 x^{6} + 29560 x^{5} - 23520 x^{4} + 14400 x^{3} - 4928 x^{2} - 640 x + 256 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2042337557936057128662109375=-\,5^{12}\cdot 101^{6}\cdot 199^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.92$
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:  $5, 101, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{80} a^{12} + \frac{1}{80} a^{11} + \frac{9}{80} a^{10} - \frac{1}{10} a^{9} + \frac{31}{80} a^{8} + \frac{29}{80} a^{7} - \frac{23}{80} a^{6} - \frac{19}{40} a^{5} - \frac{1}{80} a^{4} + \frac{1}{20} a^{3} - \frac{9}{20} a^{2} + \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{160} a^{13} - \frac{1}{160} a^{12} + \frac{7}{160} a^{11} + \frac{7}{80} a^{10} + \frac{7}{160} a^{9} + \frac{7}{160} a^{8} + \frac{79}{160} a^{7} + \frac{3}{10} a^{6} - \frac{9}{32} a^{5} + \frac{23}{80} a^{4} - \frac{11}{40} a^{3} + \frac{1}{4} a^{2} + \frac{3}{10} a + \frac{1}{5}$, $\frac{1}{63680} a^{14} + \frac{47}{63680} a^{13} + \frac{179}{63680} a^{12} - \frac{335}{6368} a^{11} - \frac{2381}{63680} a^{10} - \frac{15897}{63680} a^{9} - \frac{1241}{12736} a^{8} - \frac{1293}{3184} a^{7} + \frac{8639}{63680} a^{6} + \frac{9283}{31840} a^{5} - \frac{2437}{7960} a^{4} - \frac{3739}{7960} a^{3} - \frac{133}{995} a^{2} + \frac{813}{1990} a - \frac{471}{995}$, $\frac{1}{527539854481108850560} a^{15} + \frac{1930446371589147}{527539854481108850560} a^{14} + \frac{245574975875111739}{105507970896221770112} a^{13} - \frac{170647523822296809}{52753985448110885056} a^{12} + \frac{16134204210451185987}{527539854481108850560} a^{11} + \frac{64441682457984930211}{527539854481108850560} a^{10} - \frac{80786774866866912201}{527539854481108850560} a^{9} + \frac{28549006492081739521}{65942481810138606320} a^{8} + \frac{101635681451429944967}{527539854481108850560} a^{7} + \frac{4504865764168496233}{263769927240554425280} a^{6} - \frac{7311676527245355999}{32971240905069303160} a^{5} + \frac{13857980837784197743}{32971240905069303160} a^{4} + \frac{4589348098571286949}{16485620452534651580} a^{3} + \frac{148547105309210318}{824281022626732579} a^{2} + \frac{132779865111076204}{4121405113133662895} a - \frac{496584947427370628}{4121405113133662895}$

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:  $10$
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:  \( 28054315.475 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1643:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 64 conjugacy class representatives for t16n1643 are not computed
Character table for t16n1643 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.16098453125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$