↑

Properties

Label 16.12.1734798430...9629.2
Degree $16$
Signature $[12, 2]$
Discriminant $17^{14}\cdot 101^{3}$
Root discriminant $28.34$
Ramified primes $17, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1351

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 30, -242, 522, 306, -1693, 414, 1779, -935, -665, 590, -18, -136, 50, 2, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 2*x^14 + 50*x^13 - 136*x^12 - 18*x^11 + 590*x^10 - 665*x^9 - 935*x^8 + 1779*x^7 + 414*x^6 - 1693*x^5 + 306*x^4 + 522*x^3 - 242*x^2 + 30*x - 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 2*x^14 + 50*x^13 - 136*x^12 - 18*x^11 + 590*x^10 - 665*x^9 - 935*x^8 + 1779*x^7 + 414*x^6 - 1693*x^5 + 306*x^4 + 522*x^3 - 242*x^2 + 30*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 2 x^{14} + 50 x^{13} - 136 x^{12} - 18 x^{11} + 590 x^{10} - 665 x^{9} - 935 x^{8} + 1779 x^{7} + 414 x^{6} - 1693 x^{5} + 306 x^{4} + 522 x^{3} - 242 x^{2} + 30 x - 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(173479843081977336549629=17^{14}\cdot 101^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.34$
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:  $17, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1291} a^{14} + \frac{373}{1291} a^{13} - \frac{131}{1291} a^{12} - \frac{256}{1291} a^{11} + \frac{438}{1291} a^{10} + \frac{175}{1291} a^{9} + \frac{353}{1291} a^{8} - \frac{599}{1291} a^{7} + \frac{427}{1291} a^{6} - \frac{379}{1291} a^{5} + \frac{522}{1291} a^{4} + \frac{394}{1291} a^{3} - \frac{479}{1291} a^{2} - \frac{468}{1291} a + \frac{212}{1291}$, $\frac{1}{52822567619} a^{15} - \frac{9785224}{52822567619} a^{14} - \frac{26043223232}{52822567619} a^{13} - \frac{9571924259}{52822567619} a^{12} - \frac{15298950249}{52822567619} a^{11} - \frac{14375784330}{52822567619} a^{10} + \frac{2039284359}{52822567619} a^{9} + \frac{2247045627}{52822567619} a^{8} - \frac{20852763347}{52822567619} a^{7} - \frac{4295217375}{52822567619} a^{6} - \frac{10901099841}{52822567619} a^{5} + \frac{1363049805}{52822567619} a^{4} + \frac{21671183632}{52822567619} a^{3} + \frac{14463213382}{52822567619} a^{2} + \frac{10892413355}{52822567619} a + \frac{2611059185}{52822567619}$

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:  $13$
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:  \( 859130.927919 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1351:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 56 conjugacy class representatives for t16n1351 are not computed
Character table for t16n1351 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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
101Data not computed