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Properties

Label 16.8.21634605854...1424.3
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 223^{6}$
Root discriminant $51.10$
Ramified primes $2, 223$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3:S_4.C_2$ (as 16T764)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1893457, 5566660, -5330000, 983556, 1352106, -787088, 124248, -7596, -7767, 12816, -4928, 524, 92, -36, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 36*x^13 + 92*x^12 + 524*x^11 - 4928*x^10 + 12816*x^9 - 7767*x^8 - 7596*x^7 + 124248*x^6 - 787088*x^5 + 1352106*x^4 + 983556*x^3 - 5330000*x^2 + 5566660*x - 1893457)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 36*x^13 + 92*x^12 + 524*x^11 - 4928*x^10 + 12816*x^9 - 7767*x^8 - 7596*x^7 + 124248*x^6 - 787088*x^5 + 1352106*x^4 + 983556*x^3 - 5330000*x^2 + 5566660*x - 1893457, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 36 x^{13} + 92 x^{12} + 524 x^{11} - 4928 x^{10} + 12816 x^{9} - 7767 x^{8} - 7596 x^{7} + 124248 x^{6} - 787088 x^{5} + 1352106 x^{4} + 983556 x^{3} - 5330000 x^{2} + 5566660 x - 1893457 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2163460585448341410282471424=2^{44}\cdot 223^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.10$
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:  $2, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{9109020421787145195419673815811659987392489} a^{15} - \frac{2610479474833484710350612133984126033622953}{9109020421787145195419673815811659987392489} a^{14} + \frac{3279790012809768173158185109502765117129228}{9109020421787145195419673815811659987392489} a^{13} - \frac{447361341261921115962088585562718412426949}{9109020421787145195419673815811659987392489} a^{12} - \frac{1695566639990949907380786181682587785467770}{9109020421787145195419673815811659987392489} a^{11} - \frac{1284955471542472329056601731105091901174302}{9109020421787145195419673815811659987392489} a^{10} - \frac{2396408693629940937271045486857798067302242}{9109020421787145195419673815811659987392489} a^{9} - \frac{4381130234330021668676341221750733750955441}{9109020421787145195419673815811659987392489} a^{8} - \frac{2227603071743948009260982619451604450616729}{9109020421787145195419673815811659987392489} a^{7} - \frac{3722293979006575349460853271853583563886265}{9109020421787145195419673815811659987392489} a^{6} + \frac{3200490833887512751119450963021576781011913}{9109020421787145195419673815811659987392489} a^{5} + \frac{1086672261219361401304457514156349679790874}{9109020421787145195419673815811659987392489} a^{4} + \frac{818096369495258393308469955778713808471090}{9109020421787145195419673815811659987392489} a^{3} - \frac{3004713778911594793950647919193609844755291}{9109020421787145195419673815811659987392489} a^{2} - \frac{2406404308280850430500824037600166853184167}{9109020421787145195419673815811659987392489} a - \frac{1930988265065864494527838190712063957096715}{9109020421787145195419673815811659987392489}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 67005150.0329 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:S_4.C_2$ (as 16T764):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $C_2^3:S_4.C_2$
Character table for $C_2^3:S_4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.14272.1, 8.8.3259039744.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
223Data not computed