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Properties

Label 16.8.10729765818...1088.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{67}\cdot 17^{6}\cdot 548837153^{2}$
Root discriminant $652.25$
Ramified primes $2, 17, 548837153$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1113

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![174106443456598046402, 0, -6455576794872506880, 0, -54996144938520724, 0, 179252047961656, 0, 1263201268434, 0, 332530944, 0, -3656200, 0, -2536, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2536*x^14 - 3656200*x^12 + 332530944*x^10 + 1263201268434*x^8 + 179252047961656*x^6 - 54996144938520724*x^4 - 6455576794872506880*x^2 + 174106443456598046402)
 
gp: K = bnfinit(x^16 - 2536*x^14 - 3656200*x^12 + 332530944*x^10 + 1263201268434*x^8 + 179252047961656*x^6 - 54996144938520724*x^4 - 6455576794872506880*x^2 + 174106443456598046402, 1)
 

Normalized defining polynomial

\( x^{16} - 2536 x^{14} - 3656200 x^{12} + 332530944 x^{10} + 1263201268434 x^{8} + 179252047961656 x^{6} - 54996144938520724 x^{4} - 6455576794872506880 x^{2} + 174106443456598046402 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1072976581893572471636613847437598787096281088=2^{67}\cdot 17^{6}\cdot 548837153^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $652.25$
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:  $2, 17, 548837153$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4557837203873523641090858438705894536261425672868980222002554500151347258023} a^{14} + \frac{1155326572690647383035159266288095714828468095293308675769669035876563881992}{4557837203873523641090858438705894536261425672868980222002554500151347258023} a^{12} - \frac{1353499338185306921145144231826545855311609095136810966477840914347679291218}{4557837203873523641090858438705894536261425672868980222002554500151347258023} a^{10} + \frac{124622432781560031500500284855036987093471982534165081338123820832164844976}{651119600553360520155836919815127790894489381838425746000364928593049608289} a^{8} - \frac{155305555666972313260326007359347296274200557184858242685337658319383124202}{651119600553360520155836919815127790894489381838425746000364928593049608289} a^{6} - \frac{2150825056438331797430288444635052696916014121930480477429461786646011982761}{4557837203873523641090858438705894536261425672868980222002554500151347258023} a^{4} + \frac{256625226598190517462300161608362583015848479036650536958406144579888006056}{651119600553360520155836919815127790894489381838425746000364928593049608289} a^{2} - \frac{184220144456783526217319052546399060346698696605376298929437529162}{488502043548953446937148376013382793225416063588771382525358011223}$, $\frac{1}{4557837203873523641090858438705894536261425672868980222002554500151347258023} a^{15} + \frac{1155326572690647383035159266288095714828468095293308675769669035876563881992}{4557837203873523641090858438705894536261425672868980222002554500151347258023} a^{13} - \frac{1353499338185306921145144231826545855311609095136810966477840914347679291218}{4557837203873523641090858438705894536261425672868980222002554500151347258023} a^{11} + \frac{124622432781560031500500284855036987093471982534165081338123820832164844976}{651119600553360520155836919815127790894489381838425746000364928593049608289} a^{9} - \frac{155305555666972313260326007359347296274200557184858242685337658319383124202}{651119600553360520155836919815127790894489381838425746000364928593049608289} a^{7} - \frac{2150825056438331797430288444635052696916014121930480477429461786646011982761}{4557837203873523641090858438705894536261425672868980222002554500151347258023} a^{5} + \frac{256625226598190517462300161608362583015848479036650536958406144579888006056}{651119600553360520155836919815127790894489381838425746000364928593049608289} a^{3} - \frac{184220144456783526217319052546399060346698696605376298929437529162}{488502043548953446937148376013382793225416063588771382525358011223} a$

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:  $11$
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:  \( 11153553466900000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1113:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1113
Character table for t16n1113 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4352.1, 8.8.9697230848.1

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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
548837153Data not computed