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Properties

Label 16.8.13967711378...2033.2
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 47^{4}$
Root discriminant $37.29$
Ramified primes $17, 47$
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![6767, -27904, 36514, -19451, 18449, -15464, -16226, 26355, -12071, 4002, -882, -269, 183, -91, 25, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 25*x^14 - 91*x^13 + 183*x^12 - 269*x^11 - 882*x^10 + 4002*x^9 - 12071*x^8 + 26355*x^7 - 16226*x^6 - 15464*x^5 + 18449*x^4 - 19451*x^3 + 36514*x^2 - 27904*x + 6767)
 
gp: K = bnfinit(x^16 - 5*x^15 + 25*x^14 - 91*x^13 + 183*x^12 - 269*x^11 - 882*x^10 + 4002*x^9 - 12071*x^8 + 26355*x^7 - 16226*x^6 - 15464*x^5 + 18449*x^4 - 19451*x^3 + 36514*x^2 - 27904*x + 6767, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 25 x^{14} - 91 x^{13} + 183 x^{12} - 269 x^{11} - 882 x^{10} + 4002 x^{9} - 12071 x^{8} + 26355 x^{7} - 16226 x^{6} - 15464 x^{5} + 18449 x^{4} - 19451 x^{3} + 36514 x^{2} - 27904 x + 6767 \)

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:  \(13967711378414469438602033=17^{15}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.29$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{400961132878873962518154844410866249} a^{15} - \frac{44163699577082433326316921077994076}{400961132878873962518154844410866249} a^{14} - \frac{196069374909633818985825000625440843}{400961132878873962518154844410866249} a^{13} - \frac{194582702576936559266799148796695306}{400961132878873962518154844410866249} a^{12} - \frac{179533184358312157548422408757645826}{400961132878873962518154844410866249} a^{11} + \frac{166478646752429863774778245272764138}{400961132878873962518154844410866249} a^{10} + \frac{432823511197563922927743165268760}{400961132878873962518154844410866249} a^{9} - \frac{1016938398820281894987567452719389}{5984494520580208395793355886729347} a^{8} + \frac{74995459007391964211084757518179083}{400961132878873962518154844410866249} a^{7} + \frac{122347175262741784716299935240140908}{400961132878873962518154844410866249} a^{6} + \frac{59677456604101872574942799562916572}{400961132878873962518154844410866249} a^{5} + \frac{116573960582176002634514507112626961}{400961132878873962518154844410866249} a^{4} + \frac{16841541898495718845741926830833429}{400961132878873962518154844410866249} a^{3} + \frac{12949269831270795542665724323452211}{400961132878873962518154844410866249} a^{2} - \frac{171467815899580463407320313834123581}{400961132878873962518154844410866249} a - \frac{1562619986994775228791387960729257}{5984494520580208395793355886729347}$

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:  \( 3086565.7191 \) (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 ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$47$47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$