↑

Properties

Label 16.8.33930019567...1409.3
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 67^{4}$
Root discriminant $34.13$
Ramified primes $17, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![239, 1703, 2432, 329, -1156, -460, 275, -246, -338, 123, 90, -36, -17, 20, 4, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 4*x^14 + 20*x^13 - 17*x^12 - 36*x^11 + 90*x^10 + 123*x^9 - 338*x^8 - 246*x^7 + 275*x^6 - 460*x^5 - 1156*x^4 + 329*x^3 + 2432*x^2 + 1703*x + 239)
 
gp: K = bnfinit(x^16 - 6*x^15 + 4*x^14 + 20*x^13 - 17*x^12 - 36*x^11 + 90*x^10 + 123*x^9 - 338*x^8 - 246*x^7 + 275*x^6 - 460*x^5 - 1156*x^4 + 329*x^3 + 2432*x^2 + 1703*x + 239, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 4 x^{14} + 20 x^{13} - 17 x^{12} - 36 x^{11} + 90 x^{10} + 123 x^{9} - 338 x^{8} - 246 x^{7} + 275 x^{6} - 460 x^{5} - 1156 x^{4} + 329 x^{3} + 2432 x^{2} + 1703 x + 239 \)

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:  \(3393001956715501807791409=17^{14}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.13$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{34} a^{12} - \frac{2}{17} a^{11} - \frac{7}{17} a^{10} + \frac{7}{34} a^{9} + \frac{1}{17} a^{7} + \frac{7}{34} a^{6} - \frac{6}{17} a^{5} - \frac{3}{17} a^{4} - \frac{5}{34} a^{3} - \frac{7}{17} a^{2} + \frac{3}{17} a - \frac{1}{34}$, $\frac{1}{34} a^{13} + \frac{2}{17} a^{11} - \frac{15}{34} a^{10} - \frac{3}{17} a^{9} + \frac{1}{17} a^{8} + \frac{15}{34} a^{7} + \frac{8}{17} a^{6} + \frac{7}{17} a^{5} + \frac{5}{34} a^{4} - \frac{8}{17} a^{2} - \frac{11}{34} a - \frac{2}{17}$, $\frac{1}{578} a^{14} - \frac{5}{578} a^{13} + \frac{3}{578} a^{12} - \frac{201}{578} a^{11} + \frac{151}{578} a^{10} - \frac{43}{578} a^{9} - \frac{131}{578} a^{8} + \frac{75}{578} a^{7} - \frac{39}{578} a^{6} - \frac{19}{578} a^{5} - \frac{87}{578} a^{4} - \frac{249}{578} a^{3} + \frac{287}{578} a^{2} + \frac{147}{578} a + \frac{259}{578}$, $\frac{1}{288942302351889154} a^{15} + \frac{171478822231435}{288942302351889154} a^{14} + \frac{3151519680451579}{288942302351889154} a^{13} - \frac{4128485678849249}{288942302351889154} a^{12} + \frac{92133806861562363}{288942302351889154} a^{11} - \frac{114089918445495315}{288942302351889154} a^{10} - \frac{58974552593787043}{288942302351889154} a^{9} + \frac{104607012862586295}{288942302351889154} a^{8} + \frac{65545236091885341}{288942302351889154} a^{7} - \frac{67703460818297843}{288942302351889154} a^{6} - \frac{16573014598348807}{288942302351889154} a^{5} + \frac{7089117915650607}{288942302351889154} a^{4} + \frac{6964014217758423}{288942302351889154} a^{3} + \frac{4790302407337129}{288942302351889154} a^{2} + \frac{141718427107929371}{288942302351889154} a + \frac{64699839276119960}{144471151175944577}$

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:  \( 1876695.80684 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ 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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
67Data not computed