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Properties

Label 16.0.99346860908...0249.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{7}\cdot 47^{15}$
Root discriminant $86.56$
Ramified primes $7, 47$
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4781, -5929, 26472, -9440, 25264, 30336, 7913, 13145, 13887, 9152, 3515, -489, -603, 24, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 + 24*x^13 - 603*x^12 - 489*x^11 + 3515*x^10 + 9152*x^9 + 13887*x^8 + 13145*x^7 + 7913*x^6 + 30336*x^5 + 25264*x^4 - 9440*x^3 + 26472*x^2 - 5929*x + 4781)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 + 24*x^13 - 603*x^12 - 489*x^11 + 3515*x^10 + 9152*x^9 + 13887*x^8 + 13145*x^7 + 7913*x^6 + 30336*x^5 + 25264*x^4 - 9440*x^3 + 26472*x^2 - 5929*x + 4781, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} + 24 x^{13} - 603 x^{12} - 489 x^{11} + 3515 x^{10} + 9152 x^{9} + 13887 x^{8} + 13145 x^{7} + 7913 x^{6} + 30336 x^{5} + 25264 x^{4} - 9440 x^{3} + 26472 x^{2} - 5929 x + 4781 \)

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:  \(9934686090879658520416349720249=7^{7}\cdot 47^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.56$
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:  $7, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{45} a^{10} + \frac{2}{15} a^{9} + \frac{4}{45} a^{8} + \frac{7}{45} a^{7} - \frac{2}{15} a^{6} - \frac{19}{45} a^{4} - \frac{1}{5} a^{3} - \frac{2}{15} a^{2} - \frac{8}{45} a + \frac{4}{45}$, $\frac{1}{45} a^{11} - \frac{2}{45} a^{9} - \frac{2}{45} a^{8} - \frac{1}{15} a^{7} + \frac{2}{15} a^{6} - \frac{4}{45} a^{5} + \frac{2}{5} a^{3} - \frac{17}{45} a^{2} + \frac{22}{45} a - \frac{1}{5}$, $\frac{1}{45} a^{12} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{45} a^{6} - \frac{4}{9} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a - \frac{22}{45}$, $\frac{1}{7155} a^{13} - \frac{13}{2385} a^{12} - \frac{4}{7155} a^{11} - \frac{2}{795} a^{10} + \frac{41}{1431} a^{9} - \frac{46}{795} a^{8} - \frac{160}{1431} a^{7} - \frac{254}{2385} a^{6} + \frac{56}{7155} a^{5} - \frac{568}{7155} a^{4} + \frac{191}{1431} a^{3} - \frac{1769}{7155} a^{2} + \frac{248}{2385} a + \frac{2702}{7155}$, $\frac{1}{135945} a^{14} + \frac{8}{135945} a^{13} - \frac{1201}{135945} a^{12} - \frac{842}{135945} a^{11} + \frac{1108}{135945} a^{10} - \frac{3658}{135945} a^{9} + \frac{146}{27189} a^{8} - \frac{13876}{135945} a^{7} + \frac{18461}{135945} a^{6} + \frac{1307}{15105} a^{5} - \frac{62152}{135945} a^{4} - \frac{21191}{45315} a^{3} + \frac{7754}{135945} a^{2} - \frac{63454}{135945} a - \frac{26918}{135945}$, $\frac{1}{1544173020512113155345} a^{15} - \frac{199629383184651}{57191593352300487235} a^{14} + \frac{5290037690125666}{1544173020512113155345} a^{13} - \frac{889748243325771665}{102944868034140877023} a^{12} + \frac{9951748877481998816}{1544173020512113155345} a^{11} + \frac{174578081270694227}{171574780056901461705} a^{10} - \frac{4840356935710712303}{34314956011380292341} a^{9} - \frac{67034956908955676686}{514724340170704385115} a^{8} + \frac{105455668957919725942}{1544173020512113155345} a^{7} + \frac{195612642871907996453}{1544173020512113155345} a^{6} + \frac{107993222939554104641}{1544173020512113155345} a^{5} + \frac{114630337256430520268}{1544173020512113155345} a^{4} - \frac{71200973909581503286}{1544173020512113155345} a^{3} - \frac{173846541301157598701}{1544173020512113155345} a^{2} - \frac{4132645862719226582}{57191593352300487235} a + \frac{209642541513452826343}{1544173020512113155345}$

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

Class group and class number

$C_{15}$, which has order $15$ (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:  \( 321984542.695 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-47}) \), 4.0.726761.1, 8.0.173771730318809.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ $16$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ R ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47Data not computed