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Properties

Label 16.8.29630251663...0000.3
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $33.84$
Ramified primes $2, 5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 124, 1150, 1544, -1370, -2980, -3228, 916, 227, 308, 28, 92, -82, 40, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 6*x^14 + 40*x^13 - 82*x^12 + 92*x^11 + 28*x^10 + 308*x^9 + 227*x^8 + 916*x^7 - 3228*x^6 - 2980*x^5 - 1370*x^4 + 1544*x^3 + 1150*x^2 + 124*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 6*x^14 + 40*x^13 - 82*x^12 + 92*x^11 + 28*x^10 + 308*x^9 + 227*x^8 + 916*x^7 - 3228*x^6 - 2980*x^5 - 1370*x^4 + 1544*x^3 + 1150*x^2 + 124*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 6 x^{14} + 40 x^{13} - 82 x^{12} + 92 x^{11} + 28 x^{10} + 308 x^{9} + 227 x^{8} + 916 x^{7} - 3228 x^{6} - 2980 x^{5} - 1370 x^{4} + 1544 x^{3} + 1150 x^{2} + 124 x + 1 \)

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:  \(2963025166336000000000000=2^{32}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.84$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{40} a^{12} - \frac{1}{20} a^{11} + \frac{1}{40} a^{10} - \frac{1}{20} a^{9} + \frac{1}{5} a^{8} - \frac{7}{20} a^{7} + \frac{11}{40} a^{6} + \frac{1}{20} a^{5} + \frac{3}{10} a^{4} + \frac{7}{20} a^{3} - \frac{3}{40} a^{2} - \frac{9}{20} a + \frac{1}{40}$, $\frac{1}{40} a^{13} - \frac{3}{40} a^{11} + \frac{1}{10} a^{9} + \frac{1}{20} a^{8} - \frac{17}{40} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{20} a^{4} - \frac{3}{8} a^{3} + \frac{2}{5} a^{2} + \frac{1}{8} a + \frac{1}{20}$, $\frac{1}{80} a^{14} - \frac{1}{80} a^{13} + \frac{17}{80} a^{11} - \frac{13}{80} a^{10} + \frac{3}{20} a^{9} + \frac{1}{16} a^{8} - \frac{21}{80} a^{7} - \frac{7}{16} a^{6} + \frac{1}{10} a^{5} - \frac{17}{80} a^{4} + \frac{13}{80} a^{3} + \frac{3}{80} a - \frac{39}{80}$, $\frac{1}{26791849934881008424400} a^{15} - \frac{160650147976235970991}{26791849934881008424400} a^{14} - \frac{26113750860478785447}{13395924967440504212200} a^{13} - \frac{42036549314895335617}{26791849934881008424400} a^{12} - \frac{247595044098764138043}{26791849934881008424400} a^{11} + \frac{131250264009645331903}{1674490620930063026525} a^{10} - \frac{3353165295645620866623}{26791849934881008424400} a^{9} - \frac{6406609131354244646531}{26791849934881008424400} a^{8} - \frac{9033402978904066002701}{26791849934881008424400} a^{7} - \frac{1795507177566297139113}{6697962483720252106100} a^{6} + \frac{12575683974870856330971}{26791849934881008424400} a^{5} + \frac{6778718098142799564683}{26791849934881008424400} a^{4} + \frac{6615520002795091717}{326729877254646444200} a^{3} - \frac{4917757650054957740359}{26791849934881008424400} a^{2} - \frac{8463915386373269555337}{26791849934881008424400} a - \frac{417108239396614502619}{3348981241860126053050}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1400948.18183 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{20})^+\), 4.4.8000.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed