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Properties

Label 16.0.32798502387...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{6}\cdot 109^{4}$
Root discriminant $25.54$
Ramified primes $5, 29, 109$
Class number $6$
Class group $[6]$
Galois group $C_2^4.C_2^3$ (as 16T392)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![475, -2650, 8355, -13300, 14746, -11766, 8081, -4615, 2398, -1279, 631, -321, 147, -59, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 20*x^14 - 59*x^13 + 147*x^12 - 321*x^11 + 631*x^10 - 1279*x^9 + 2398*x^8 - 4615*x^7 + 8081*x^6 - 11766*x^5 + 14746*x^4 - 13300*x^3 + 8355*x^2 - 2650*x + 475)
 
gp: K = bnfinit(x^16 - 5*x^15 + 20*x^14 - 59*x^13 + 147*x^12 - 321*x^11 + 631*x^10 - 1279*x^9 + 2398*x^8 - 4615*x^7 + 8081*x^6 - 11766*x^5 + 14746*x^4 - 13300*x^3 + 8355*x^2 - 2650*x + 475, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 20 x^{14} - 59 x^{13} + 147 x^{12} - 321 x^{11} + 631 x^{10} - 1279 x^{9} + 2398 x^{8} - 4615 x^{7} + 8081 x^{6} - 11766 x^{5} + 14746 x^{4} - 13300 x^{3} + 8355 x^{2} - 2650 x + 475 \)

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:  \(32798502387606516015625=5^{8}\cdot 29^{6}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.54$
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:  $5, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{10} a^{5} - \frac{1}{10} a^{4} - \frac{1}{10} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{3}{10} a^{4} - \frac{1}{10} a^{3} - \frac{3}{10} a^{2} - \frac{1}{2}$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{10} + \frac{3}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{20} a^{7} - \frac{1}{2} a^{6} - \frac{9}{20} a^{5} - \frac{1}{2} a^{4} + \frac{7}{20} a^{3} + \frac{1}{20} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{11} - \frac{1}{20} a^{10} + \frac{1}{5} a^{9} + \frac{3}{20} a^{8} - \frac{1}{10} a^{7} - \frac{7}{20} a^{6} - \frac{1}{5} a^{5} + \frac{1}{20} a^{4} + \frac{1}{20} a^{3} - \frac{1}{20} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{11} - \frac{1}{20} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{10} a^{6} + \frac{2}{5} a^{5} - \frac{1}{4} a^{4} + \frac{3}{10} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{668772221897948300} a^{15} - \frac{692287761646091}{133754444379589660} a^{14} - \frac{47777886965641}{2431898988719812} a^{13} - \frac{5989966294362769}{668772221897948300} a^{12} - \frac{8198453763719093}{668772221897948300} a^{11} + \frac{132837459290661}{167193055474487075} a^{10} - \frac{271596922756601}{15199368679498825} a^{9} - \frac{155964816169836039}{668772221897948300} a^{8} - \frac{118170054685517561}{334386110948974150} a^{7} + \frac{1677102019264969}{12159494943599060} a^{6} - \frac{124437602844578379}{668772221897948300} a^{5} + \frac{27174408783624106}{167193055474487075} a^{4} + \frac{218890612373028851}{668772221897948300} a^{3} - \frac{1526771860851866}{33438611094897415} a^{2} - \frac{12953732948495437}{133754444379589660} a + \frac{604165667954903}{26750888875917932}$

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

Class group and class number

$C_{6}$, which has order $6$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11170.5670423 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T392):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.79025.2, 4.0.2725.1, 4.4.725.1, 8.0.6244950625.1, 8.0.6244950625.3, 8.0.6244950625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$