↑

Properties

Label 16.0.27391509508...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 29^{2}\cdot 89^{6}$
Root discriminant $51.86$
Ramified primes $2, 5, 29, 89$
Class number $160$ (GRH)
Class group $[2, 2, 40]$ (GRH)
Galois group 16T1429

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![126481, 65336, 94988, 43090, -22786, 30822, 12458, -9556, 6239, 1414, -1246, 764, 2, -50, 38, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 38*x^14 - 50*x^13 + 2*x^12 + 764*x^11 - 1246*x^10 + 1414*x^9 + 6239*x^8 - 9556*x^7 + 12458*x^6 + 30822*x^5 - 22786*x^4 + 43090*x^3 + 94988*x^2 + 65336*x + 126481)
 
gp: K = bnfinit(x^16 - 6*x^15 + 38*x^14 - 50*x^13 + 2*x^12 + 764*x^11 - 1246*x^10 + 1414*x^9 + 6239*x^8 - 9556*x^7 + 12458*x^6 + 30822*x^5 - 22786*x^4 + 43090*x^3 + 94988*x^2 + 65336*x + 126481, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 38 x^{14} - 50 x^{13} + 2 x^{12} + 764 x^{11} - 1246 x^{10} + 1414 x^{9} + 6239 x^{8} - 9556 x^{7} + 12458 x^{6} + 30822 x^{5} - 22786 x^{4} + 43090 x^{3} + 94988 x^{2} + 65336 x + 126481 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2739150950879730073600000000=2^{24}\cdot 5^{8}\cdot 29^{2}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.86$
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:  $2, 5, 29, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{38} a^{12} + \frac{3}{19} a^{11} + \frac{5}{38} a^{10} + \frac{4}{19} a^{9} - \frac{3}{19} a^{8} + \frac{5}{19} a^{7} - \frac{9}{38} a^{6} - \frac{1}{19} a^{5} + \frac{3}{19} a^{4} + \frac{8}{19} a^{3} - \frac{1}{38} a^{2} - \frac{2}{19} a - \frac{7}{38}$, $\frac{1}{38} a^{13} + \frac{7}{38} a^{11} - \frac{3}{38} a^{10} + \frac{3}{38} a^{9} + \frac{4}{19} a^{8} + \frac{7}{38} a^{7} - \frac{5}{38} a^{6} - \frac{1}{38} a^{5} + \frac{9}{19} a^{4} + \frac{17}{38} a^{3} - \frac{17}{38} a^{2} - \frac{1}{19} a + \frac{2}{19}$, $\frac{1}{38} a^{14} - \frac{7}{38} a^{11} + \frac{3}{19} a^{10} + \frac{9}{38} a^{9} - \frac{4}{19} a^{8} + \frac{1}{38} a^{7} - \frac{7}{19} a^{6} + \frac{13}{38} a^{5} - \frac{3}{19} a^{4} - \frac{15}{38} a^{3} + \frac{5}{38} a^{2} + \frac{13}{38} a - \frac{4}{19}$, $\frac{1}{186619534690177931017791112269122} a^{15} + \frac{56865620654677424160255088180}{93309767345088965508895556134561} a^{14} - \frac{1152428289727681671244807099961}{186619534690177931017791112269122} a^{13} - \frac{625693390807399297712183162924}{93309767345088965508895556134561} a^{12} + \frac{12658915559384952099186115633466}{93309767345088965508895556134561} a^{11} + \frac{29838979768619603000252904087255}{186619534690177931017791112269122} a^{10} + \frac{9147204281065129002918447001631}{93309767345088965508895556134561} a^{9} - \frac{13310391386657286946649539186300}{93309767345088965508895556134561} a^{8} + \frac{7819300817550794825666309757157}{93309767345088965508895556134561} a^{7} + \frac{84389272868684874864621657674249}{186619534690177931017791112269122} a^{6} - \frac{1187259225392086986375372123155}{93309767345088965508895556134561} a^{5} + \frac{1212945140167141012693549162024}{4911040386583629763626081901819} a^{4} + \frac{8653828551132320219357362572269}{186619534690177931017791112269122} a^{3} - \frac{20580993564829896079231924921673}{186619534690177931017791112269122} a^{2} - \frac{3927270716073390851939098995913}{9822080773167259527252163803638} a + \frac{46053663230128669919060907570759}{93309767345088965508895556134561}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{40}$, which has order $160$ (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:  $7$
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:  \( 120998.246464 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1429:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 77 conjugacy class representatives for t16n1429 are not computed
Character table for t16n1429 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.8.112795040000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$