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Properties

Label 16.0.15998750436...8201.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{14}\cdot 59^{6}$
Root discriminant $118.92$
Ramified primes $41, 59$
Class number $16$ (GRH)
Class group $[4, 4]$ (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![724201, 0, -6175982, 0, 14327515, 0, -108359, 0, -221157, 0, 24280, 0, 342, 0, 55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 55*x^14 + 342*x^12 + 24280*x^10 - 221157*x^8 - 108359*x^6 + 14327515*x^4 - 6175982*x^2 + 724201)
 
gp: K = bnfinit(x^16 + 55*x^14 + 342*x^12 + 24280*x^10 - 221157*x^8 - 108359*x^6 + 14327515*x^4 - 6175982*x^2 + 724201, 1)
 

Normalized defining polynomial

\( x^{16} + 55 x^{14} + 342 x^{12} + 24280 x^{10} - 221157 x^{8} - 108359 x^{6} + 14327515 x^{4} - 6175982 x^{2} + 724201 \)

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:  \(1599875043672267792208865723998201=41^{14}\cdot 59^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.92$
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:  $41, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{10} + \frac{1}{10} a^{8} + \frac{2}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{11} + \frac{1}{10} a^{9} - \frac{1}{10} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{3}{10} a^{3} + \frac{3}{10} a - \frac{1}{2}$, $\frac{1}{354170138601021401889923180} a^{14} - \frac{129731509238477403477821}{35417013860102140188992318} a^{12} - \frac{3722621613722771417995101}{17708506930051070094496159} a^{10} + \frac{37281809394723746738349279}{177085069300510700944961590} a^{8} + \frac{119654464773683018465217621}{354170138601021401889923180} a^{6} - \frac{1}{2} a^{5} + \frac{62234289973873667013932909}{177085069300510700944961590} a^{4} - \frac{1}{2} a^{3} - \frac{21906688750717915249442625}{70834027720204280377984636} a^{2} - \frac{1}{2} a - \frac{65247767437004344060163593}{354170138601021401889923180}$, $\frac{1}{602797575898938426016649252360} a^{15} - \frac{1}{708340277202042803779846360} a^{14} - \frac{40006304979163004466963634}{75349696987367303252081156545} a^{13} - \frac{8529924691929341538553527}{177085069300510700944961590} a^{12} + \frac{36105836428097006348686709509}{301398787949469213008324626180} a^{11} + \frac{19517709207176644085454851}{354170138601021401889923180} a^{10} + \frac{51498202948123133441344187333}{301398787949469213008324626180} a^{9} + \frac{33552218325480533639635357}{354170138601021401889923180} a^{8} + \frac{125637551584975667848253992613}{602797575898938426016649252360} a^{7} + \frac{92847618386929822668736287}{708340277202042803779846360} a^{6} - \frac{48281989629065547690960581161}{301398787949469213008324626180} a^{5} + \frac{114850779326637033931028681}{354170138601021401889923180} a^{4} + \frac{298207974299886737235635081389}{602797575898938426016649252360} a^{3} + \frac{3282402173283155680236171}{708340277202042803779846360} a^{2} + \frac{175532306950949406712963749051}{602797575898938426016649252360} a - \frac{218088343443812777451774951}{708340277202042803779846360}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( 617060533.04 \) (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, 8.2.280256150219.1, 8.4.677939627379761.1, 8.2.39998438015405899.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ 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$$0$Trivial$[\ ]$
59.2.1.2$x^{2} + 177$$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$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$