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Properties

Label 16.0.59488299467...2809.3
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 89^{11}$
Root discriminant $72.59$
Ramified primes $11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47872, 196032, 402960, 544952, 548736, 441734, 287185, 141137, 45676, 7317, -105, -130, 35, 23, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 23*x^13 + 35*x^12 - 130*x^11 - 105*x^10 + 7317*x^9 + 45676*x^8 + 141137*x^7 + 287185*x^6 + 441734*x^5 + 548736*x^4 + 544952*x^3 + 402960*x^2 + 196032*x + 47872)
 
gp: K = bnfinit(x^16 - x^15 + 23*x^13 + 35*x^12 - 130*x^11 - 105*x^10 + 7317*x^9 + 45676*x^8 + 141137*x^7 + 287185*x^6 + 441734*x^5 + 548736*x^4 + 544952*x^3 + 402960*x^2 + 196032*x + 47872, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 23 x^{13} + 35 x^{12} - 130 x^{11} - 105 x^{10} + 7317 x^{9} + 45676 x^{8} + 141137 x^{7} + 287185 x^{6} + 441734 x^{5} + 548736 x^{4} + 544952 x^{3} + 402960 x^{2} + 196032 x + 47872 \)

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:  \(594882994674022504125031032809=11^{8}\cdot 89^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.59$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{6} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{48} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{1}{8} a^{7} - \frac{7}{48} a^{6} + \frac{5}{48} a^{5} + \frac{1}{48} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{96} a^{13} - \frac{1}{16} a^{11} + \frac{1}{96} a^{10} + \frac{1}{24} a^{9} - \frac{1}{8} a^{8} + \frac{11}{96} a^{7} + \frac{1}{12} a^{6} + \frac{5}{48} a^{5} - \frac{5}{96} a^{4} + \frac{5}{24} a^{3} + \frac{1}{12} a^{2} - \frac{5}{12} a$, $\frac{1}{1152} a^{14} - \frac{1}{192} a^{13} - \frac{1}{288} a^{12} + \frac{7}{1152} a^{11} + \frac{5}{144} a^{10} + \frac{1}{144} a^{9} - \frac{17}{1152} a^{8} + \frac{13}{576} a^{7} + \frac{59}{288} a^{6} - \frac{79}{1152} a^{5} + \frac{25}{288} a^{4} - \frac{11}{72} a^{3} - \frac{7}{48} a^{2} + \frac{17}{36} a - \frac{4}{9}$, $\frac{1}{518997040169346161784576} a^{15} + \frac{49177989480629233207}{518997040169346161784576} a^{14} + \frac{732312949937085207595}{259498520084673080892288} a^{13} + \frac{151766739216917082473}{19222112598864672658688} a^{12} + \frac{11407494277179393739811}{518997040169346161784576} a^{11} + \frac{709215664419661089413}{21624876673722756741024} a^{10} + \frac{2411079205406299775071}{57666337796594017976064} a^{9} - \frac{2273906734049401478249}{19222112598864672658688} a^{8} + \frac{8452667209773804859067}{259498520084673080892288} a^{7} + \frac{38434880741852799858925}{518997040169346161784576} a^{6} - \frac{33133679725287554853493}{172999013389782053928192} a^{5} - \frac{13129228855779880748857}{129749260042336540446144} a^{4} - \frac{2188556907412500362371}{64874630021168270223072} a^{3} + \frac{30845314410387823316699}{64874630021168270223072} a^{2} - \frac{3053754039637869673607}{16218657505292067555768} a - \frac{1256658099541040407669}{4054664376323016888942}$

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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2341079059.75 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.2.87131.1, 8.0.675671193329.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ R $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$89$89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$