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Properties

Label 16.0.22168695080...1376.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{50}\cdot 13^{8}\cdot 17^{6}$
Root discriminant $91.01$
Ramified primes $2, 13, 17$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![611524, 0, 1008032, 0, 926840, 0, 244528, 0, -3928, 0, -3584, 0, 324, 0, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 + 324*x^12 - 3584*x^10 - 3928*x^8 + 244528*x^6 + 926840*x^4 + 1008032*x^2 + 611524)
 
gp: K = bnfinit(x^16 - 24*x^14 + 324*x^12 - 3584*x^10 - 3928*x^8 + 244528*x^6 + 926840*x^4 + 1008032*x^2 + 611524, 1)
 

Normalized defining polynomial

\( x^{16} - 24 x^{14} + 324 x^{12} - 3584 x^{10} - 3928 x^{8} + 244528 x^{6} + 926840 x^{4} + 1008032 x^{2} + 611524 \)

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:  \(22168695080663051109329457381376=2^{50}\cdot 13^{8}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.01$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{40} a^{10} + \frac{1}{20} a^{6} - \frac{1}{10} a^{4} + \frac{1}{20} a^{2} + \frac{1}{5}$, $\frac{1}{40} a^{11} + \frac{1}{20} a^{7} - \frac{1}{10} a^{5} + \frac{1}{20} a^{3} + \frac{1}{5} a$, $\frac{1}{44200} a^{12} - \frac{483}{44200} a^{10} - \frac{543}{44200} a^{8} - \frac{77}{340} a^{6} + \frac{3591}{11050} a^{4} - \frac{237}{1300} a^{2} + \frac{499}{1300}$, $\frac{1}{44200} a^{13} - \frac{483}{44200} a^{11} - \frac{543}{44200} a^{9} - \frac{77}{340} a^{7} + \frac{3591}{11050} a^{5} - \frac{237}{1300} a^{3} + \frac{499}{1300} a$, $\frac{1}{1610945950874000} a^{14} + \frac{1541856477}{805472975437000} a^{12} + \frac{1184817958959}{402736487718500} a^{10} + \frac{13526412655137}{805472975437000} a^{8} + \frac{386194253196147}{805472975437000} a^{6} + \frac{1816147834782}{20136824385925} a^{4} + \frac{1897973119503}{4738076326100} a^{2} + \frac{1740622781369}{23690381630500}$, $\frac{1}{37051756870102000} a^{15} + \frac{46329354451}{9262939217525500} a^{13} - \frac{22824696813104}{2315734804381375} a^{11} - \frac{130348907188486}{2315734804381375} a^{9} - \frac{6446111809040053}{18525878435051000} a^{7} + \frac{117400797607419}{926293921752550} a^{5} - \frac{27769674030077}{108975755500300} a^{3} + \frac{107704820858997}{272439388750750} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 96690766.2217 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n869
Character table for t16n869 is not computed

Intermediate fields

\(\Q(\sqrt{26}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), 4.0.735488.4, 4.0.735488.5, \(\Q(\sqrt{2}, \sqrt{13})\), 8.0.540942598144.29

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$