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Properties

Label 16.0.48825707677...5649.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{15}\cdot 157^{5}$
Root discriminant $53.77$
Ramified primes $13, 157$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16913, 85111, 254696, 167440, 74919, 38246, 19968, 13052, 5603, 1703, 1157, -182, 208, 0, 26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 26*x^14 + 208*x^12 - 182*x^11 + 1157*x^10 + 1703*x^9 + 5603*x^8 + 13052*x^7 + 19968*x^6 + 38246*x^5 + 74919*x^4 + 167440*x^3 + 254696*x^2 + 85111*x + 16913)
 
gp: K = bnfinit(x^16 + 26*x^14 + 208*x^12 - 182*x^11 + 1157*x^10 + 1703*x^9 + 5603*x^8 + 13052*x^7 + 19968*x^6 + 38246*x^5 + 74919*x^4 + 167440*x^3 + 254696*x^2 + 85111*x + 16913, 1)
 

Normalized defining polynomial

\( x^{16} + 26 x^{14} + 208 x^{12} - 182 x^{11} + 1157 x^{10} + 1703 x^{9} + 5603 x^{8} + 13052 x^{7} + 19968 x^{6} + 38246 x^{5} + 74919 x^{4} + 167440 x^{3} + 254696 x^{2} + 85111 x + 16913 \)

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:  \(4882570767744501515595495649=13^{15}\cdot 157^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.77$
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:  $13, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{261} a^{14} + \frac{4}{29} a^{13} + \frac{2}{29} a^{12} - \frac{11}{29} a^{11} + \frac{55}{261} a^{10} - \frac{128}{261} a^{9} - \frac{1}{87} a^{8} - \frac{11}{29} a^{7} - \frac{52}{261} a^{6} + \frac{119}{261} a^{5} - \frac{73}{261} a^{4} - \frac{20}{261} a^{3} + \frac{2}{261} a^{2} - \frac{70}{261} a + \frac{52}{261}$, $\frac{1}{890735647483629962122326903430520924007} a^{15} - \frac{333810755198879501112522998394609539}{890735647483629962122326903430520924007} a^{14} + \frac{8354581900681977511121366305377113782}{296911882494543320707442301143506974669} a^{13} - \frac{20508997830455181199153902257518583963}{296911882494543320707442301143506974669} a^{12} + \frac{2719742714040765804101416231373950157}{30715022327021722831804375980362790483} a^{11} - \frac{261425911246226701776942578368568337643}{890735647483629962122326903430520924007} a^{10} + \frac{6664438096673946294255009266974077077}{30715022327021722831804375980362790483} a^{9} - \frac{96366704577681549095923214455498932385}{296911882494543320707442301143506974669} a^{8} + \frac{442545757683397138764744795115761833948}{890735647483629962122326903430520924007} a^{7} - \frac{213472018689737263062789763029943190174}{890735647483629962122326903430520924007} a^{6} - \frac{6230935582418041872113923572596870362}{38727636847114346179231604496979170609} a^{5} - \frac{6706798286148560029399689827120884290}{98970627498181106902480767047835658223} a^{4} - \frac{27959115227651478400521866867395735688}{98970627498181106902480767047835658223} a^{3} + \frac{283493333960793832310647414189572456727}{890735647483629962122326903430520924007} a^{2} - \frac{36688584602579825989921465843433942544}{98970627498181106902480767047835658223} a + \frac{275316841910579449378704273158516651020}{890735647483629962122326903430520924007}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 38532285.5451 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.9851517169.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ $16$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
13Data not computed
157Data not computed