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Properties

Label 16.0.45748960181...0649.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 47^{12}$
Root discriminant $169.35$
Ramified primes $13, 47$
Class number $25$ (GRH)
Class group $[5, 5]$ (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![749062161, 0, 2304643077, 0, 835715203, 0, 101508906, 0, 2774661, 0, -77889, 0, -2853, 0, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 30*x^14 - 2853*x^12 - 77889*x^10 + 2774661*x^8 + 101508906*x^6 + 835715203*x^4 + 2304643077*x^2 + 749062161)
 
gp: K = bnfinit(x^16 + 30*x^14 - 2853*x^12 - 77889*x^10 + 2774661*x^8 + 101508906*x^6 + 835715203*x^4 + 2304643077*x^2 + 749062161, 1)
 

Normalized defining polynomial

\( x^{16} + 30 x^{14} - 2853 x^{12} - 77889 x^{10} + 2774661 x^{8} + 101508906 x^{6} + 835715203 x^{4} + 2304643077 x^{2} + 749062161 \)

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:  \(457489601812551940177211781114570649=13^{14}\cdot 47^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $169.35$
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:  $13, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{18} a^{10} - \frac{1}{6} a^{7} + \frac{1}{9} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{3} - \frac{4}{9} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{18} a^{11} + \frac{1}{9} a^{7} - \frac{1}{6} a^{5} - \frac{4}{9} a^{3} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{8} - \frac{1}{6} a^{5} + \frac{7}{18} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{9} - \frac{1}{6} a^{6} + \frac{1}{18} a^{5} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{333858749301475666321953967686} a^{14} + \frac{390038821129798377665462798}{166929374650737833160976983843} a^{12} + \frac{9006464005177388391682852387}{333858749301475666321953967686} a^{10} - \frac{1112411596408396275611516017}{333858749301475666321953967686} a^{8} - \frac{1}{6} a^{7} + \frac{15429726254211724671874149883}{166929374650737833160976983843} a^{6} - \frac{1}{6} a^{5} + \frac{34598437290100688171842074791}{166929374650737833160976983843} a^{4} + \frac{1}{3} a^{3} + \frac{42257276398562822479893085270}{166929374650737833160976983843} a^{2} - \frac{1}{6} a - \frac{14821135633834657413443453873}{37095416589052851813550440854}$, $\frac{1}{3045793369877362503855186047199378} a^{15} - \frac{13916703764445575039798563418527}{1015264456625787501285062015733126} a^{13} + \frac{5469983739503737469579507964625}{507632228312893750642531007866563} a^{11} - \frac{52947895415306906620632866417615}{1015264456625787501285062015733126} a^{9} - \frac{54326930479565749192330154201915}{338421485541929167095020671911042} a^{7} - \frac{66939511561378855152099006643064}{507632228312893750642531007866563} a^{5} - \frac{1}{2} a^{4} + \frac{470968570874424516595502739726800}{1522896684938681251927593023599689} a^{3} - \frac{1}{2} a^{2} + \frac{80869871450465562837786626944997}{169210742770964583547510335955521} a$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( 11281256571.3 \) (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

\(\Q(\sqrt{-611}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{13}, \sqrt{-47})\), 4.4.4853173.1, 4.0.2197.1, 8.0.23553288167929.1, 8.4.676379776318417093.1 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
47Data not computed