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Properties

Label 16.0.10297981845...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 59^{6}$
Root discriminant $15.43$
Ramified primes $5, 59$
Class number $1$
Class group Trivial
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 37, -124, 213, -194, 51, 103, -137, 71, -6, -10, 8, -8, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 8*x^13 + 8*x^12 - 10*x^11 - 6*x^10 + 71*x^9 - 137*x^8 + 103*x^7 + 51*x^6 - 194*x^5 + 213*x^4 - 124*x^3 + 37*x^2 + x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 8*x^13 + 8*x^12 - 10*x^11 - 6*x^10 + 71*x^9 - 137*x^8 + 103*x^7 + 51*x^6 - 194*x^5 + 213*x^4 - 124*x^3 + 37*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} - 8 x^{13} + 8 x^{12} - 10 x^{11} - 6 x^{10} + 71 x^{9} - 137 x^{8} + 103 x^{7} + 51 x^{6} - 194 x^{5} + 213 x^{4} - 124 x^{3} + 37 x^{2} + x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10297981845947265625=5^{12}\cdot 59^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.43$
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:  $5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{22592480419} a^{15} + \frac{397076423}{22592480419} a^{14} - \frac{2828720454}{22592480419} a^{13} + \frac{5973300131}{22592480419} a^{12} + \frac{5071993066}{22592480419} a^{11} - \frac{7908439260}{22592480419} a^{10} + \frac{2007821899}{22592480419} a^{9} + \frac{1572359281}{22592480419} a^{8} - \frac{5916163190}{22592480419} a^{7} - \frac{3006449652}{22592480419} a^{6} - \frac{5634726313}{22592480419} a^{5} + \frac{4359637712}{22592480419} a^{4} + \frac{60351811}{22592480419} a^{3} + \frac{8237297912}{22592480419} a^{2} + \frac{10052361464}{22592480419} a - \frac{7436319314}{22592480419}$

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:  \( \frac{833636690}{22592480419} a^{15} - \frac{2263968184}{22592480419} a^{14} + \frac{5869917190}{22592480419} a^{13} - \frac{9428642146}{22592480419} a^{12} + \frac{14614390223}{22592480419} a^{11} - \frac{8080348884}{22592480419} a^{10} - \frac{3033658798}{22592480419} a^{9} + \frac{28694129788}{22592480419} a^{8} - \frac{85926926912}{22592480419} a^{7} + \frac{160637929179}{22592480419} a^{6} - \frac{110747899082}{22592480419} a^{5} - \frac{95883854070}{22592480419} a^{4} + \frac{289312474490}{22592480419} a^{3} - \frac{259282473116}{22592480419} a^{2} + \frac{131699487676}{22592480419} a - \frac{6354477578}{22592480419} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3370.63382124 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.1475.1, 4.2.7375.1, 8.2.128361875.1, 8.2.3209046875.2, 8.0.54390625.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$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}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$