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Properties

Label 16.8.52879738474...2289.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{4}\cdot 97^{14}$
Root discriminant $72.06$
Ramified primes $3, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-102043, -65579, 302724, 354256, -40253, -216222, -63521, 39690, 25804, 211, -4351, -688, 293, 109, -13, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 13*x^14 + 109*x^13 + 293*x^12 - 688*x^11 - 4351*x^10 + 211*x^9 + 25804*x^8 + 39690*x^7 - 63521*x^6 - 216222*x^5 - 40253*x^4 + 354256*x^3 + 302724*x^2 - 65579*x - 102043)
 
gp: K = bnfinit(x^16 - 7*x^15 - 13*x^14 + 109*x^13 + 293*x^12 - 688*x^11 - 4351*x^10 + 211*x^9 + 25804*x^8 + 39690*x^7 - 63521*x^6 - 216222*x^5 - 40253*x^4 + 354256*x^3 + 302724*x^2 - 65579*x - 102043, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 13 x^{14} + 109 x^{13} + 293 x^{12} - 688 x^{11} - 4351 x^{10} + 211 x^{9} + 25804 x^{8} + 39690 x^{7} - 63521 x^{6} - 216222 x^{5} - 40253 x^{4} + 354256 x^{3} + 302724 x^{2} - 65579 x - 102043 \)

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:  \(528797384743092167726859302289=3^{4}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.06$
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:  $3, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{206} a^{14} + \frac{7}{103} a^{13} - \frac{23}{103} a^{12} + \frac{12}{103} a^{11} - \frac{23}{206} a^{10} - \frac{29}{103} a^{9} + \frac{49}{103} a^{8} + \frac{17}{206} a^{7} + \frac{89}{206} a^{6} - \frac{25}{103} a^{5} - \frac{47}{206} a^{4} + \frac{47}{103} a^{3} + \frac{59}{206} a^{2} + \frac{101}{206} a - \frac{67}{206}$, $\frac{1}{41403430577619480674731418249500906} a^{15} - \frac{47341358068949216911708562594031}{20701715288809740337365709124750453} a^{14} - \frac{8044125721955591069162713874189803}{41403430577619480674731418249500906} a^{13} + \frac{2577103377075383589554176576932979}{20701715288809740337365709124750453} a^{12} + \frac{2354379829931200887368833666611033}{20701715288809740337365709124750453} a^{11} - \frac{9991987425334921292028768346336321}{41403430577619480674731418249500906} a^{10} + \frac{5822902404875426578900653975612641}{41403430577619480674731418249500906} a^{9} - \frac{3139516268194981374133626760357216}{20701715288809740337365709124750453} a^{8} + \frac{15661295498682892246845901883187039}{41403430577619480674731418249500906} a^{7} - \frac{1951117970503428370998801130380916}{20701715288809740337365709124750453} a^{6} + \frac{15679957128968870391776310500721441}{41403430577619480674731418249500906} a^{5} - \frac{13275043095573642560831595385542541}{41403430577619480674731418249500906} a^{4} - \frac{6488392946077888085703360473264933}{41403430577619480674731418249500906} a^{3} - \frac{18022876756223970483858648846147999}{41403430577619480674731418249500906} a^{2} - \frac{8843900674583793634242843015583883}{20701715288809740337365709124750453} a - \frac{8331763752156695923809745511158541}{41403430577619480674731418249500906}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 764009341.927 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 4.2.2738019.1, 4.2.28227.1, 8.4.727184560303017.1, 8.8.80798284478113.1, 8.4.7496748044361.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed