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Properties

Label 16.0.31013248447...9929.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 31^{12}$
Root discriminant $123.94$
Ramified primes $13, 31$
Class number $75$ (GRH)
Class group $[5, 15]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![343991209, 0, 176923205, 0, -29140165, 0, -383778, 0, 520721, 0, 11667, 0, -721, 0, 22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 22*x^14 - 721*x^12 + 11667*x^10 + 520721*x^8 - 383778*x^6 - 29140165*x^4 + 176923205*x^2 + 343991209)
 
gp: K = bnfinit(x^16 + 22*x^14 - 721*x^12 + 11667*x^10 + 520721*x^8 - 383778*x^6 - 29140165*x^4 + 176923205*x^2 + 343991209, 1)
 

Normalized defining polynomial

\( x^{16} + 22 x^{14} - 721 x^{12} + 11667 x^{10} + 520721 x^{8} - 383778 x^{6} - 29140165 x^{4} + 176923205 x^{2} + 343991209 \)

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:  \(3101324844783200582771505158819929=13^{14}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.94$
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:  $13, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{34} a^{11} + \frac{2}{17} a^{9} + \frac{6}{17} a^{7} - \frac{1}{2} a^{5} - \frac{1}{17} a^{3} + \frac{8}{17} a - \frac{1}{2}$, $\frac{1}{34} a^{12} + \frac{2}{17} a^{10} - \frac{5}{34} a^{8} - \frac{1}{2} a^{5} + \frac{15}{34} a^{4} - \frac{1}{34} a^{2} - \frac{1}{2}$, $\frac{1}{34} a^{13} - \frac{2}{17} a^{9} + \frac{3}{34} a^{7} - \frac{1}{17} a^{5} - \frac{5}{17} a^{3} - \frac{1}{2} a^{2} + \frac{2}{17} a$, $\frac{1}{47724968218326686442378832022} a^{14} - \frac{211426139066687676591967489}{47724968218326686442378832022} a^{12} + \frac{88053812044715247360542487}{23862484109163343221189416011} a^{10} - \frac{1488827729561469063330912183}{23862484109163343221189416011} a^{8} - \frac{8343788460525602585996271944}{23862484109163343221189416011} a^{6} - \frac{1}{2} a^{5} - \frac{2751149040734374073863819021}{47724968218326686442378832022} a^{4} - \frac{1}{2} a^{3} - \frac{2863172264737748846790036241}{47724968218326686442378832022} a^{2} - \frac{36491989500810233512182027}{165138298333310333710653398}$, $\frac{1}{52067940326194414908635305736002} a^{15} - \frac{21862683874946980304674568931}{26033970163097207454317652868001} a^{13} - \frac{185197116917929479175992570069}{26033970163097207454317652868001} a^{11} - \frac{5976935583771229239216468792114}{26033970163097207454317652868001} a^{9} - \frac{3019149548068133037532237299625}{52067940326194414908635305736002} a^{7} - \frac{1}{2} a^{6} + \frac{9008845791008010536516492109839}{52067940326194414908635305736002} a^{5} - \frac{1}{2} a^{4} + \frac{16970381407029564970601587516995}{52067940326194414908635305736002} a^{3} - \frac{1}{2} a^{2} + \frac{617327148541733494775557920857}{3062820019187906759331488572706} a$

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

Class group and class number

$C_{5}\times C_{15}$, which has order $75$ (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:  \( 109545456.342 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-31}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-403}) \), \(\Q(\sqrt{13}, \sqrt{-31})\), 4.4.2111317.1, 4.0.2197.1, 8.0.4457659474489.1, 8.4.55689539814791077.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed
31Data not computed