↑

Properties

Label 16.8.15194523093...3125.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 11^{4}\cdot 349^{3}$
Root discriminant $18.25$
Ramified primes $5, 11, 349$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T972

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29, 100, -111, -36, 240, -238, -8, 154, -84, 57, -41, -47, 48, 6, -10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 10*x^14 + 6*x^13 + 48*x^12 - 47*x^11 - 41*x^10 + 57*x^9 - 84*x^8 + 154*x^7 - 8*x^6 - 238*x^5 + 240*x^4 - 36*x^3 - 111*x^2 + 100*x - 29)
 
gp: K = bnfinit(x^16 - x^15 - 10*x^14 + 6*x^13 + 48*x^12 - 47*x^11 - 41*x^10 + 57*x^9 - 84*x^8 + 154*x^7 - 8*x^6 - 238*x^5 + 240*x^4 - 36*x^3 - 111*x^2 + 100*x - 29, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 10 x^{14} + 6 x^{13} + 48 x^{12} - 47 x^{11} - 41 x^{10} + 57 x^{9} - 84 x^{8} + 154 x^{7} - 8 x^{6} - 238 x^{5} + 240 x^{4} - 36 x^{3} - 111 x^{2} + 100 x - 29 \)

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:  \(151945230934814453125=5^{12}\cdot 11^{4}\cdot 349^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.25$
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:  $5, 11, 349$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} + \frac{2}{11} a^{13} - \frac{5}{11} a^{12} - \frac{2}{11} a^{10} + \frac{2}{11} a^{9} + \frac{4}{11} a^{6} + \frac{1}{11} a^{5} + \frac{2}{11} a^{4} - \frac{2}{11} a^{3} + \frac{1}{11} a^{2} + \frac{2}{11} a + \frac{4}{11}$, $\frac{1}{3523771507619} a^{15} - \frac{43062589555}{3523771507619} a^{14} - \frac{820293771991}{3523771507619} a^{13} - \frac{1111151784687}{3523771507619} a^{12} + \frac{790656181434}{3523771507619} a^{11} + \frac{255499776178}{3523771507619} a^{10} - \frac{340218625589}{3523771507619} a^{9} - \frac{27911156021}{320342864329} a^{8} + \frac{250422410924}{3523771507619} a^{7} - \frac{1165510672086}{3523771507619} a^{6} + \frac{879550151330}{3523771507619} a^{5} - \frac{1037138157889}{3523771507619} a^{4} - \frac{703962717450}{3523771507619} a^{3} + \frac{996961440325}{3523771507619} a^{2} - \frac{80543476753}{3523771507619} a - \frac{1545729736532}{3523771507619}$

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:  \( 13080.2427727 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T972:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 35 conjugacy class representatives for t16n972
Character table for t16n972 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.4.26393125.1

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R $16$ R $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
349Data not computed