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Properties

Label 16.0.42819656572...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{8}\cdot 761^{5}$
Root discriminant $71.12$
Ramified primes $2, 5, 761$
Class number $9912$ (GRH)
Class group $[2, 2, 2478]$ (GRH)
Galois group 16T1163

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15982929, 957816, 14554588, 1006896, 5379139, 771132, 1018278, 252364, 135322, 27576, 18902, 220, 1724, -44, 68, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 68*x^14 - 44*x^13 + 1724*x^12 + 220*x^11 + 18902*x^10 + 27576*x^9 + 135322*x^8 + 252364*x^7 + 1018278*x^6 + 771132*x^5 + 5379139*x^4 + 1006896*x^3 + 14554588*x^2 + 957816*x + 15982929)
 
gp: K = bnfinit(x^16 + 68*x^14 - 44*x^13 + 1724*x^12 + 220*x^11 + 18902*x^10 + 27576*x^9 + 135322*x^8 + 252364*x^7 + 1018278*x^6 + 771132*x^5 + 5379139*x^4 + 1006896*x^3 + 14554588*x^2 + 957816*x + 15982929, 1)
 

Normalized defining polynomial

\( x^{16} + 68 x^{14} - 44 x^{13} + 1724 x^{12} + 220 x^{11} + 18902 x^{10} + 27576 x^{9} + 135322 x^{8} + 252364 x^{7} + 1018278 x^{6} + 771132 x^{5} + 5379139 x^{4} + 1006896 x^{3} + 14554588 x^{2} + 957816 x + 15982929 \)

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:  \(428196565723310037401600000000=2^{32}\cdot 5^{8}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.12$
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:  $2, 5, 761$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{368234072365984768549076701355703422596579099965303} a^{15} - \frac{38149243347625969288449099041468154144376718836780}{122744690788661589516358900451901140865526366655101} a^{14} + \frac{45336063488449441327928818748387538901951914787238}{368234072365984768549076701355703422596579099965303} a^{13} - \frac{150983569127317778667082925967296782284124580580662}{368234072365984768549076701355703422596579099965303} a^{12} - \frac{83089897946291665585414371348138802384610442897598}{368234072365984768549076701355703422596579099965303} a^{11} + \frac{2273540313728802916473572113927672997904205195489}{368234072365984768549076701355703422596579099965303} a^{10} + \frac{29215423353641192906973850577576114548312134271934}{368234072365984768549076701355703422596579099965303} a^{9} + \frac{11741726675214697043864649348209554705194443521986}{122744690788661589516358900451901140865526366655101} a^{8} + \frac{72041512887951696051385790077275959486674831150669}{368234072365984768549076701355703422596579099965303} a^{7} - \frac{151555432389682934475157206555034138143548446667897}{368234072365984768549076701355703422596579099965303} a^{6} - \frac{42354536438611609386020234013636728636496008326338}{122744690788661589516358900451901140865526366655101} a^{5} - \frac{2015087730137302202728536441376475248261171319496}{5336725686463547370276473932691353950675059419787} a^{4} + \frac{112366918201622398744241206430439541502065888756693}{368234072365984768549076701355703422596579099965303} a^{3} - \frac{35889140662414425948000488383458006370958893958760}{122744690788661589516358900451901140865526366655101} a^{2} + \frac{163652996902227000814000799699318846954174926961698}{368234072365984768549076701355703422596579099965303} a - \frac{29472071568732941904647397660608914205358198332444}{122744690788661589516358900451901140865526366655101}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2478}$, which has order $9912$ (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:  \( 9562.41394678 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1163:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 52 conjugacy class representatives for t16n1163 are not computed
Character table for t16n1163 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
761Data not computed