↑

Properties

Label 16.0.26436931289...625.24
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{8}$
Root discriminant $33.60$
Ramified primes $5, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1263

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15073775, 9449200, -10235150, -5525450, 4084295, 1467105, -1035255, -257335, 164376, 30231, -16213, -2082, 1035, 72, -43, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 43*x^14 + 72*x^13 + 1035*x^12 - 2082*x^11 - 16213*x^10 + 30231*x^9 + 164376*x^8 - 257335*x^7 - 1035255*x^6 + 1467105*x^5 + 4084295*x^4 - 5525450*x^3 - 10235150*x^2 + 9449200*x + 15073775)
 
gp: K = bnfinit(x^16 - x^15 - 43*x^14 + 72*x^13 + 1035*x^12 - 2082*x^11 - 16213*x^10 + 30231*x^9 + 164376*x^8 - 257335*x^7 - 1035255*x^6 + 1467105*x^5 + 4084295*x^4 - 5525450*x^3 - 10235150*x^2 + 9449200*x + 15073775, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 43 x^{14} + 72 x^{13} + 1035 x^{12} - 2082 x^{11} - 16213 x^{10} + 30231 x^{9} + 164376 x^{8} - 257335 x^{7} - 1035255 x^{6} + 1467105 x^{5} + 4084295 x^{4} - 5525450 x^{3} - 10235150 x^{2} + 9449200 x + 15073775 \)

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:  \(2643693128974804931640625=5^{12}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.60$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{4}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{6}$, $\frac{1}{15} a^{12} - \frac{1}{15} a^{9} - \frac{7}{15} a^{7} + \frac{1}{3} a^{6} - \frac{1}{5} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{15} a^{13} - \frac{1}{15} a^{10} - \frac{1}{15} a^{8} + \frac{2}{15} a^{7} - \frac{2}{5} a^{6} - \frac{4}{15} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{825} a^{14} - \frac{17}{825} a^{13} + \frac{1}{165} a^{11} + \frac{4}{165} a^{10} - \frac{82}{825} a^{9} - \frac{1}{75} a^{8} + \frac{5}{33} a^{7} + \frac{4}{55} a^{6} + \frac{58}{165} a^{5} + \frac{1}{5} a^{4} + \frac{7}{33} a^{3} - \frac{1}{3} a^{2} - \frac{14}{33} a - \frac{5}{11}$, $\frac{1}{13281210183334739443370737635860813302140075} a^{15} + \frac{522366966889414181779608883036541486922}{4427070061111579814456912545286937767380025} a^{14} + \frac{45357922490717140639329442269824405407813}{4427070061111579814456912545286937767380025} a^{13} + \frac{28811965098375603200457536707129510061374}{2656242036666947888674147527172162660428015} a^{12} + \frac{45659957851911360903553498964437299877473}{885414012222315962891382509057387553476005} a^{11} + \frac{337957481246309226461028614989218278836313}{13281210183334739443370737635860813302140075} a^{10} + \frac{261194498031324899280932804192770121520113}{13281210183334739443370737635860813302140075} a^{9} - \frac{378321107909000350168703655192955910617436}{4427070061111579814456912545286937767380025} a^{8} - \frac{17566758033581207483618152561289689543599}{885414012222315962891382509057387553476005} a^{7} - \frac{49663977952452817401878134371963895076086}{241476548787904353515831593379287514584365} a^{6} + \frac{326418332734040616917738653462383164603696}{2656242036666947888674147527172162660428015} a^{5} + \frac{182306840043454790433232432016760035295871}{885414012222315962891382509057387553476005} a^{4} - \frac{243171900484545171796099987201111022894093}{531248407333389577734829505434432532085603} a^{3} - \frac{38737967488485688417455373456289342659173}{177082802444463192578276501811477510695201} a^{2} - \frac{23480877875137414010985981621773986543360}{177082802444463192578276501811477510695201} a + \frac{804844330781815829886598905811055769060}{177082802444463192578276501811477510695201}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{65964514256728339351188718751160629}{885414012222315962891382509057387553476005} a^{15} - \frac{946641493926981138114668479660435641}{885414012222315962891382509057387553476005} a^{14} - \frac{782444092565008892690672748773316799}{177082802444463192578276501811477510695201} a^{13} + \frac{42867959730139207728319618652398894352}{885414012222315962891382509057387553476005} a^{12} + \frac{71156326882630013500807613111969954701}{885414012222315962891382509057387553476005} a^{11} - \frac{1093745119445633195002580397794370246284}{885414012222315962891382509057387553476005} a^{10} - \frac{684379800177655846870728468512232118218}{885414012222315962891382509057387553476005} a^{9} + \frac{3445629827318667223844386735607680911190}{177082802444463192578276501811477510695201} a^{8} + \frac{4580030997892105413195385471968555495921}{885414012222315962891382509057387553476005} a^{7} - \frac{14575392837867931433999210058146871283112}{80492182929301451171943864459762504861455} a^{6} - \frac{12774428979897304304641015235071550264903}{885414012222315962891382509057387553476005} a^{5} + \frac{172610023225095022760060545486020070522537}{177082802444463192578276501811477510695201} a^{4} - \frac{41876258166824759713240086121107483809747}{177082802444463192578276501811477510695201} a^{3} - \frac{556272071195442578100256569684882210896825}{177082802444463192578276501811477510695201} a^{2} + \frac{240812001592749226466712928854131815267313}{177082802444463192578276501811477510695201} a + \frac{875337467778966069110209289616691912542674}{177082802444463192578276501811477510695201} \) (order $10$)
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:  \( 414754.628528 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.1578125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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}$
101Data not computed