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Properties

Label 16.8.15231185783...5001.2
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 67^{6}$
Root discriminant $57.73$
Ramified primes $17, 67$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![129097, 2092986, 123563, -317749, -195407, -198713, -49851, -10817, 6173, 5065, 1119, 357, -121, -39, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 - 39*x^13 - 121*x^12 + 357*x^11 + 1119*x^10 + 5065*x^9 + 6173*x^8 - 10817*x^7 - 49851*x^6 - 198713*x^5 - 195407*x^4 - 317749*x^3 + 123563*x^2 + 2092986*x + 129097)
 
gp: K = bnfinit(x^16 - 2*x^15 - 5*x^14 - 39*x^13 - 121*x^12 + 357*x^11 + 1119*x^10 + 5065*x^9 + 6173*x^8 - 10817*x^7 - 49851*x^6 - 198713*x^5 - 195407*x^4 - 317749*x^3 + 123563*x^2 + 2092986*x + 129097, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 5 x^{14} - 39 x^{13} - 121 x^{12} + 357 x^{11} + 1119 x^{10} + 5065 x^{9} + 6173 x^{8} - 10817 x^{7} - 49851 x^{6} - 198713 x^{5} - 195407 x^{4} - 317749 x^{3} + 123563 x^{2} + 2092986 x + 129097 \)

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:  \(15231185783695887615175635001=17^{14}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{8} a^{14} + \frac{3}{8} a^{13} + \frac{3}{8} a^{12} + \frac{3}{8} a^{11} + \frac{1}{8} a^{10} - \frac{3}{8} a^{9} + \frac{1}{8} a^{8} + \frac{3}{8} a^{7} - \frac{3}{8} a^{6} + \frac{3}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{3}{8} a^{2} + \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{36869566498355527306336282189738338865336712} a^{15} + \frac{102631348626142179425301961108715443986971}{2836120499873502100487406322287564528102824} a^{14} + \frac{6728065198791399522645831334219671335401199}{36869566498355527306336282189738338865336712} a^{13} - \frac{9160615406477888632690802539853009026149945}{36869566498355527306336282189738338865336712} a^{12} - \frac{16799582784974172154736345285679207095765875}{36869566498355527306336282189738338865336712} a^{11} + \frac{12640927500826479857064680796515017580541561}{36869566498355527306336282189738338865336712} a^{10} - \frac{12413575118986603606850943121074920145027379}{36869566498355527306336282189738338865336712} a^{9} - \frac{11578793530380602064718186637082629584795337}{36869566498355527306336282189738338865336712} a^{8} + \frac{7248152793825818492157469043584407911251753}{36869566498355527306336282189738338865336712} a^{7} - \frac{613681037876541062596402158233207174163805}{2836120499873502100487406322287564528102824} a^{6} - \frac{11740188888185024864107858644644686066014491}{36869566498355527306336282189738338865336712} a^{5} - \frac{1158563062805578480545551708494144244098305}{2836120499873502100487406322287564528102824} a^{4} - \frac{9320256127421992656833324469765916307726791}{36869566498355527306336282189738338865336712} a^{3} + \frac{14559586858490780850526653711929856172013247}{36869566498355527306336282189738338865336712} a^{2} - \frac{2359055246680035658163983510599151572592949}{36869566498355527306336282189738338865336712} a + \frac{2799355293577423139363742844890993268854079}{9217391624588881826584070547434584716334178}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 69740717.6846 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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