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Properties

Label 16.0.11623374807...1313.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 67^{8}$
Root discriminant $116.57$
Ramified primes $17, 67$
Class number $128$ (GRH)
Class group $[8, 16]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![375, -2905, 8606, -11892, 10522, -16764, 30188, -33124, 24158, -11626, 2928, 212, -460, 168, -4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 460*x^12 + 212*x^11 + 2928*x^10 - 11626*x^9 + 24158*x^8 - 33124*x^7 + 30188*x^6 - 16764*x^5 + 10522*x^4 - 11892*x^3 + 8606*x^2 - 2905*x + 375)
 
gp: K = bnfinit(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 460*x^12 + 212*x^11 + 2928*x^10 - 11626*x^9 + 24158*x^8 - 33124*x^7 + 30188*x^6 - 16764*x^5 + 10522*x^4 - 11892*x^3 + 8606*x^2 - 2905*x + 375, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 4 x^{14} + 168 x^{13} - 460 x^{12} + 212 x^{11} + 2928 x^{10} - 11626 x^{9} + 24158 x^{8} - 33124 x^{7} + 30188 x^{6} - 16764 x^{5} + 10522 x^{4} - 11892 x^{3} + 8606 x^{2} - 2905 x + 375 \)

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:  \(1162337480711184271576898233831313=17^{15}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.57$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{12} + \frac{4}{15} a^{11} - \frac{1}{5} a^{10} + \frac{1}{3} a^{9} - \frac{2}{5} a^{8} + \frac{4}{15} a^{7} + \frac{1}{15} a^{6} + \frac{4}{15} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{4}{15} a$, $\frac{1}{445338622305} a^{14} - \frac{7}{445338622305} a^{13} + \frac{44130728636}{445338622305} a^{12} + \frac{36110850116}{89067724461} a^{11} + \frac{182505234164}{445338622305} a^{10} + \frac{178648036244}{445338622305} a^{9} + \frac{11741742772}{445338622305} a^{8} - \frac{23435676496}{445338622305} a^{7} + \frac{81741664511}{445338622305} a^{6} - \frac{6141800369}{148446207435} a^{5} + \frac{9974867885}{89067724461} a^{4} - \frac{40572006181}{89067724461} a^{3} + \frac{73055981093}{148446207435} a^{2} + \frac{187034413513}{445338622305} a + \frac{2094425615}{29689241487}$, $\frac{1}{27165655960605} a^{15} + \frac{23}{27165655960605} a^{14} - \frac{727789550236}{27165655960605} a^{13} + \frac{643488106537}{27165655960605} a^{12} - \frac{4079559973198}{27165655960605} a^{11} - \frac{3252967384936}{27165655960605} a^{10} - \frac{8731206876233}{27165655960605} a^{9} - \frac{2165079678244}{27165655960605} a^{8} - \frac{12081375844351}{27165655960605} a^{7} + \frac{668951356993}{1811043730707} a^{6} - \frac{6173532817802}{27165655960605} a^{5} - \frac{12452748656636}{27165655960605} a^{4} - \frac{1846683775838}{9055218653535} a^{3} - \frac{9477942381506}{27165655960605} a^{2} - \frac{108761024980}{1811043730707} a - \frac{266051744891}{603681243569}$

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

Class group and class number

$C_{8}\times C_{16}$, which has order $128$ (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:  \( 703580816.265 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1139}) \), 4.0.22054457.1, 8.0.8268784250602433.2

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
17Data not computed
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$