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Properties

Label 16.4.45960213837...7121.9
Degree $16$
Signature $[4, 6]$
Discriminant $41^{14}\cdot 59^{4}$
Root discriminant $71.43$
Ramified primes $41, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![652721, -695194, 226816, -366712, 221773, -57772, 51954, -20076, 9414, -2228, 2016, -366, 86, -74, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 74*x^13 + 86*x^12 - 366*x^11 + 2016*x^10 - 2228*x^9 + 9414*x^8 - 20076*x^7 + 51954*x^6 - 57772*x^5 + 221773*x^4 - 366712*x^3 + 226816*x^2 - 695194*x + 652721)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 - 74*x^13 + 86*x^12 - 366*x^11 + 2016*x^10 - 2228*x^9 + 9414*x^8 - 20076*x^7 + 51954*x^6 - 57772*x^5 + 221773*x^4 - 366712*x^3 + 226816*x^2 - 695194*x + 652721, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} - 74 x^{13} + 86 x^{12} - 366 x^{11} + 2016 x^{10} - 2228 x^{9} + 9414 x^{8} - 20076 x^{7} + 51954 x^{6} - 57772 x^{5} + 221773 x^{4} - 366712 x^{3} + 226816 x^{2} - 695194 x + 652721 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(459602138371809190522512417121=41^{14}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.43$
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:  $41, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{8} a$, $\frac{1}{1016} a^{14} - \frac{27}{508} a^{13} + \frac{37}{1016} a^{12} - \frac{103}{1016} a^{11} + \frac{49}{1016} a^{10} - \frac{45}{1016} a^{9} + \frac{10}{127} a^{8} - \frac{1}{254} a^{7} - \frac{17}{508} a^{6} + \frac{31}{1016} a^{5} - \frac{11}{127} a^{4} + \frac{49}{508} a^{3} - \frac{65}{254} a^{2} + \frac{79}{1016} a + \frac{471}{1016}$, $\frac{1}{433062904741260602796593089631032} a^{15} + \frac{31588341302750142539411883367}{433062904741260602796593089631032} a^{14} + \frac{1789639581967496397799829746577}{433062904741260602796593089631032} a^{13} - \frac{10265584464846392695707174748989}{216531452370630301398296544815516} a^{12} + \frac{2281674018782145625521886348655}{54132863092657575349574136203879} a^{11} - \frac{6888746113140315374748210137675}{108265726185315150699148272407758} a^{10} + \frac{42959644680400503164486451271741}{433062904741260602796593089631032} a^{9} - \frac{727892238006756112912002927883}{108265726185315150699148272407758} a^{8} - \frac{9830552560613601828040864286647}{108265726185315150699148272407758} a^{7} + \frac{48367411511122851975834686499213}{433062904741260602796593089631032} a^{6} + \frac{28746373007014819527220589352435}{433062904741260602796593089631032} a^{5} - \frac{16160828286866843557948480196855}{216531452370630301398296544815516} a^{4} + \frac{38955092305965668523013100064033}{216531452370630301398296544815516} a^{3} - \frac{61679748030775630564799259609337}{433062904741260602796593089631032} a^{2} + \frac{16716925230805823508025706611630}{54132863092657575349574136203879} a + \frac{36822842439728600166903235936835}{433062904741260602796593089631032}$

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:  $9$
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:  \( 471160744.049 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

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 $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.4.677939627379761.1, 8.2.280256150219.1, 8.6.11490502158979.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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
41Data not computed
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$