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Properties

Label 16.0.56119412199...1744.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 17^{2}\cdot 41^{4}$
Root discriminant $17.15$
Ramified primes $2, 17, 41$
Class number $2$
Class group $[2]$
Galois group $C_2\times D_4^2.C_2$ (as 16T509)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -16, 106, -372, 748, -848, 520, -316, 507, -540, 232, -8, 4, -36, 26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 36*x^13 + 4*x^12 - 8*x^11 + 232*x^10 - 540*x^9 + 507*x^8 - 316*x^7 + 520*x^6 - 848*x^5 + 748*x^4 - 372*x^3 + 106*x^2 - 16*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 26*x^14 - 36*x^13 + 4*x^12 - 8*x^11 + 232*x^10 - 540*x^9 + 507*x^8 - 316*x^7 + 520*x^6 - 848*x^5 + 748*x^4 - 372*x^3 + 106*x^2 - 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 26 x^{14} - 36 x^{13} + 4 x^{12} - 8 x^{11} + 232 x^{10} - 540 x^{9} + 507 x^{8} - 316 x^{7} + 520 x^{6} - 848 x^{5} + 748 x^{4} - 372 x^{3} + 106 x^{2} - 16 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(56119412199987871744=2^{36}\cdot 17^{2}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.15$
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:  $2, 17, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{13} + \frac{3}{7} a^{12} - \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{798533435} a^{15} + \frac{1161718}{18570545} a^{14} + \frac{13963178}{34718845} a^{13} - \frac{946116}{18570545} a^{12} - \frac{114226572}{798533435} a^{11} - \frac{181732682}{798533435} a^{10} + \frac{9986101}{34718845} a^{9} - \frac{1274862}{114076205} a^{8} + \frac{11947203}{34718845} a^{7} + \frac{8356182}{798533435} a^{6} + \frac{72930219}{798533435} a^{5} - \frac{8042941}{22815241} a^{4} + \frac{21790639}{114076205} a^{3} - \frac{19073426}{798533435} a^{2} + \frac{56633369}{798533435} a + \frac{24659501}{114076205}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{296386149}{114076205} a^{15} + \frac{351525136}{18570545} a^{14} - \frac{1866659624}{34718845} a^{13} + \frac{986200703}{18570545} a^{12} + \frac{26858808741}{798533435} a^{11} + \frac{30230911761}{798533435} a^{10} - \frac{20071025208}{34718845} a^{9} + \frac{788741206952}{798533435} a^{8} - \frac{2729759307}{4959835} a^{7} + \frac{246484065459}{798533435} a^{6} - \frac{838605724977}{798533435} a^{5} + \frac{225677283948}{159706687} a^{4} - \frac{93324103267}{114076205} a^{3} + \frac{23624016789}{114076205} a^{2} - \frac{8957169477}{798533435} a - \frac{1824083806}{798533435} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4328.65884279 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4^2.C_2$ (as 16T509):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2\times D_4^2.C_2$
Character table for $C_2\times D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.4.2624.1, 4.0.10496.2, \(\Q(\zeta_{8})\), 8.4.1872822272.1, 8.4.1872822272.2, 8.0.110166016.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$