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Properties

Label 16.0.61691432181...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 197^{4}$
Root discriminant $35.43$
Ramified primes $2, 5, 197$
Class number $24$ (GRH)
Class group $[2, 12]$ (GRH)
Galois group $C_2^4.(C_4\times S_3)$ (as 16T725)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 64, 384, 496, 616, 672, 896, -64, 136, 56, 200, -16, 72, -32, 20, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 20*x^14 - 32*x^13 + 72*x^12 - 16*x^11 + 200*x^10 + 56*x^9 + 136*x^8 - 64*x^7 + 896*x^6 + 672*x^5 + 616*x^4 + 496*x^3 + 384*x^2 + 64*x + 16)
 
gp: K = bnfinit(x^16 - 6*x^15 + 20*x^14 - 32*x^13 + 72*x^12 - 16*x^11 + 200*x^10 + 56*x^9 + 136*x^8 - 64*x^7 + 896*x^6 + 672*x^5 + 616*x^4 + 496*x^3 + 384*x^2 + 64*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 20 x^{14} - 32 x^{13} + 72 x^{12} - 16 x^{11} + 200 x^{10} + 56 x^{9} + 136 x^{8} - 64 x^{7} + 896 x^{6} + 672 x^{5} + 616 x^{4} + 496 x^{3} + 384 x^{2} + 64 x + 16 \)

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:  \(6169143218176000000000000=2^{24}\cdot 5^{12}\cdot 197^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.43$
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:  $2, 5, 197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{1361065629092259848} a^{15} + \frac{3600970583807767}{170133203636532481} a^{14} - \frac{3446778816390221}{1361065629092259848} a^{13} - \frac{72314732680846179}{1361065629092259848} a^{12} - \frac{9630758402210011}{680532814546129924} a^{11} + \frac{23495355068820913}{340266407273064962} a^{10} + \frac{14058057966813921}{170133203636532481} a^{9} - \frac{52398737974999305}{680532814546129924} a^{8} - \frac{73259186630441905}{340266407273064962} a^{7} + \frac{28241237770748179}{340266407273064962} a^{6} - \frac{10743832104584651}{340266407273064962} a^{5} + \frac{36204107992823200}{170133203636532481} a^{4} + \frac{40598665820526328}{170133203636532481} a^{3} + \frac{61781082551176896}{170133203636532481} a^{2} - \frac{1643277013149331}{170133203636532481} a - \frac{29977433772638107}{170133203636532481}$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (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:  \( -\frac{4748381644609069}{340266407273064962} a^{15} + \frac{54191820394978653}{680532814546129924} a^{14} - \frac{42672223688096969}{170133203636532481} a^{13} + \frac{465806374733939805}{1361065629092259848} a^{12} - \frac{271813456489229311}{340266407273064962} a^{11} - \frac{65611016218177133}{340266407273064962} a^{10} - \frac{417904567094369740}{170133203636532481} a^{9} - \frac{1137197858810859167}{680532814546129924} a^{8} - \frac{477657978093004407}{340266407273064962} a^{7} + \frac{82505174937136723}{170133203636532481} a^{6} - \frac{2010047423855191993}{170133203636532481} a^{5} - \frac{2302704095659254417}{170133203636532481} a^{4} - \frac{1382167425580197528}{170133203636532481} a^{3} - \frac{1230109080389459942}{170133203636532481} a^{2} - \frac{930719921881746162}{170133203636532481} a - \frac{157283344253338554}{170133203636532481} \) (order $10$)
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:  \( 106122.033859 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.(C_4\times S_3)$ (as 16T725):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 28 conjugacy class representatives for $C_2^4.(C_4\times S_3)$
Character table for $C_2^4.(C_4\times S_3)$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.8.99351040000.3

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
5Data not computed
$197$197.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
197.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
197.8.4.1$x^{8} + 1397124 x^{4} - 7645373 x^{2} + 487988867844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$