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Properties

Label 16.8.31130283153...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 41^{6}$
Root discriminant $25.46$
Ramified primes $2, 5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_4:D_4$ (as 16T141)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-256, -1280, 256, 3616, 4, -3352, 510, 1044, -269, -106, -91, 70, 30, -6, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 - 6*x^13 + 30*x^12 + 70*x^11 - 91*x^10 - 106*x^9 - 269*x^8 + 1044*x^7 + 510*x^6 - 3352*x^5 + 4*x^4 + 3616*x^3 + 256*x^2 - 1280*x - 256)
 
gp: K = bnfinit(x^16 - 2*x^15 - 5*x^14 - 6*x^13 + 30*x^12 + 70*x^11 - 91*x^10 - 106*x^9 - 269*x^8 + 1044*x^7 + 510*x^6 - 3352*x^5 + 4*x^4 + 3616*x^3 + 256*x^2 - 1280*x - 256, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 5 x^{14} - 6 x^{13} + 30 x^{12} + 70 x^{11} - 91 x^{10} - 106 x^{9} - 269 x^{8} + 1044 x^{7} + 510 x^{6} - 3352 x^{5} + 4 x^{4} + 3616 x^{3} + 256 x^{2} - 1280 x - 256 \)

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:  \(31130283153817600000000=2^{24}\cdot 5^{8}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.46$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{64} a^{14} - \frac{1}{32} a^{13} + \frac{3}{64} a^{12} + \frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{32} a^{9} + \frac{5}{64} a^{8} + \frac{7}{32} a^{7} + \frac{3}{64} a^{6} - \frac{5}{16} a^{5} + \frac{11}{32} a^{4} - \frac{1}{8} a^{3} + \frac{5}{16} a^{2} - \frac{1}{2}$, $\frac{1}{8511364812628366976} a^{15} + \frac{13826009942269771}{2127841203157091744} a^{14} + \frac{468664218419698999}{8511364812628366976} a^{13} + \frac{38688109600616013}{2127841203157091744} a^{12} + \frac{225938962777281673}{4255682406314183488} a^{11} - \frac{273785504451358091}{4255682406314183488} a^{10} - \frac{97218967069113751}{8511364812628366976} a^{9} - \frac{28665375281034515}{303977314736727392} a^{8} - \frac{1378191282048199489}{8511364812628366976} a^{7} - \frac{971236079263914609}{4255682406314183488} a^{6} - \frac{1069682242818533}{12193932396315712} a^{5} - \frac{115703390557006401}{2127841203157091744} a^{4} + \frac{377168777424009329}{2127841203157091744} a^{3} + \frac{299257388001318487}{1063920601578545872} a^{2} - \frac{30089519767375387}{265980150394636468} a - \frac{25014174232356691}{132990075197318234}$

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:  \( 222611.967473 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:D_4$ (as 16T141):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $D_4:D_4$
Character table for $D_4:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.65600.2, 4.4.2624.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.176437760000.3 x2, 8.4.7057510400.1 x2, 8.8.4303360000.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$