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Properties

Label 16.8.17959483694...2113.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 89^{4}$
Root discriminant $43.74$
Ramified primes $17, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-98429, 3302, 440486, -615472, 385986, -173243, 74102, -19350, -5745, 9235, -4710, 1526, -305, -13, 16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^14 - 13*x^13 - 305*x^12 + 1526*x^11 - 4710*x^10 + 9235*x^9 - 5745*x^8 - 19350*x^7 + 74102*x^6 - 173243*x^5 + 385986*x^4 - 615472*x^3 + 440486*x^2 + 3302*x - 98429)
 
gp: K = bnfinit(x^16 - 4*x^15 + 16*x^14 - 13*x^13 - 305*x^12 + 1526*x^11 - 4710*x^10 + 9235*x^9 - 5745*x^8 - 19350*x^7 + 74102*x^6 - 173243*x^5 + 385986*x^4 - 615472*x^3 + 440486*x^2 + 3302*x - 98429, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 16 x^{14} - 13 x^{13} - 305 x^{12} + 1526 x^{11} - 4710 x^{10} + 9235 x^{9} - 5745 x^{8} - 19350 x^{7} + 74102 x^{6} - 173243 x^{5} + 385986 x^{4} - 615472 x^{3} + 440486 x^{2} + 3302 x - 98429 \)

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:  \(179594836941784276350012113=17^{15}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.74$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{397392799232882979711986937601694090297} a^{15} + \frac{188117738564076708015938607214779908941}{397392799232882979711986937601694090297} a^{14} + \frac{66895200723929842971740563136433204421}{397392799232882979711986937601694090297} a^{13} + \frac{127050491207566774014561980516378064070}{397392799232882979711986937601694090297} a^{12} - \frac{47534024913790655085040439988203255944}{397392799232882979711986937601694090297} a^{11} + \frac{181866755550037612377811531827102846354}{397392799232882979711986937601694090297} a^{10} - \frac{19572173511613418796548431302107640346}{397392799232882979711986937601694090297} a^{9} - \frac{98889366659409416139138325384588255189}{397392799232882979711986937601694090297} a^{8} + \frac{64804548943990283794179100896881272213}{397392799232882979711986937601694090297} a^{7} - \frac{59157866833534170990501641233247139253}{397392799232882979711986937601694090297} a^{6} + \frac{88618202554150759391921272007260932437}{397392799232882979711986937601694090297} a^{5} + \frac{80169463998915940790372466779886015434}{397392799232882979711986937601694090297} a^{4} + \frac{106085842826352878211471048059078300778}{397392799232882979711986937601694090297} a^{3} - \frac{30775695886723305174732873721607459069}{397392799232882979711986937601694090297} a^{2} - \frac{77748352545574845810516430987711969406}{397392799232882979711986937601694090297} a - \frac{32821640952010333723068718613975031773}{397392799232882979711986937601694090297}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 10557509.3305 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$