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Properties

Label 16.0.703687441776640000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{50}\cdot 5^{4}$
Root discriminant $13.05$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_2\wr C_2^2$ (as 16T150)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -16, 48, -56, -12, 72, 8, -128, 102, 8, -40, 0, 20, -8, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^13 + 20*x^12 - 40*x^10 + 8*x^9 + 102*x^8 - 128*x^7 + 8*x^6 + 72*x^5 - 12*x^4 - 56*x^3 + 48*x^2 - 16*x + 2)
 
gp: K = bnfinit(x^16 - 8*x^13 + 20*x^12 - 40*x^10 + 8*x^9 + 102*x^8 - 128*x^7 + 8*x^6 + 72*x^5 - 12*x^4 - 56*x^3 + 48*x^2 - 16*x + 2, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{13} + 20 x^{12} - 40 x^{10} + 8 x^{9} + 102 x^{8} - 128 x^{7} + 8 x^{6} + 72 x^{5} - 12 x^{4} - 56 x^{3} + 48 x^{2} - 16 x + 2 \)

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:  \(703687441776640000=2^{50}\cdot 5^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.05$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13} a^{13} + \frac{6}{13} a^{12} - \frac{2}{13} a^{11} - \frac{6}{13} a^{10} + \frac{4}{13} a^{9} + \frac{2}{13} a^{8} + \frac{6}{13} a^{7} - \frac{6}{13} a^{5} + \frac{1}{13} a^{4} - \frac{5}{13} a^{3} - \frac{5}{13} a^{2} + \frac{3}{13}$, $\frac{1}{169} a^{14} + \frac{4}{169} a^{13} + \frac{12}{169} a^{12} + \frac{24}{169} a^{11} + \frac{68}{169} a^{10} + \frac{7}{169} a^{9} + \frac{54}{169} a^{8} + \frac{27}{169} a^{7} - \frac{6}{169} a^{6} + \frac{6}{13} a^{5} + \frac{6}{169} a^{4} - \frac{47}{169} a^{3} - \frac{55}{169} a^{2} + \frac{81}{169} a - \frac{84}{169}$, $\frac{1}{2197} a^{15} + \frac{2}{2197} a^{14} + \frac{4}{2197} a^{13} + \frac{20}{2197} a^{11} + \frac{40}{2197} a^{10} + \frac{40}{2197} a^{9} + \frac{88}{2197} a^{8} + \frac{278}{2197} a^{7} + \frac{428}{2197} a^{6} + \frac{864}{2197} a^{5} - \frac{397}{2197} a^{4} - \frac{62}{169} a^{3} + \frac{529}{2197} a^{2} - \frac{1091}{2197} a - \frac{1}{2197}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{125166}{2197} a^{15} + \frac{67552}{2197} a^{14} + \frac{36226}{2197} a^{13} - \frac{75549}{169} a^{12} + \frac{1972842}{2197} a^{11} + \frac{1066132}{2197} a^{10} - \frac{4433492}{2197} a^{9} - \frac{1395751}{2197} a^{8} + \frac{12017730}{2197} a^{7} - \frac{9528612}{2197} a^{6} - \frac{4157414}{2197} a^{5} + \frac{6770269}{2197} a^{4} + \frac{166358}{169} a^{3} - \frac{5846992}{2197} a^{2} + \frac{2846144}{2197} a - \frac{460085}{2197} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1017.57695965 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T150):

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 $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.2.1024.1 x2, 4.0.512.1 x2, \(\Q(\zeta_{8})\), 8.0.4194304.1, 8.0.419430400.1, 8.4.419430400.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$