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Properties

Label 16.4.43690605516...1401.1
Degree $16$
Signature $[4, 6]$
Discriminant $29^{14}\cdot 59^{8}$
Root discriminant $146.23$
Ramified primes $29, 59$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![279841, 0, -187795, 0, -447367, 0, -138943, 0, 33057, 0, 4736, 0, 110, 0, 23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 23*x^14 + 110*x^12 + 4736*x^10 + 33057*x^8 - 138943*x^6 - 447367*x^4 - 187795*x^2 + 279841)
 
gp: K = bnfinit(x^16 + 23*x^14 + 110*x^12 + 4736*x^10 + 33057*x^8 - 138943*x^6 - 447367*x^4 - 187795*x^2 + 279841, 1)
 

Normalized defining polynomial

\( x^{16} + 23 x^{14} + 110 x^{12} + 4736 x^{10} + 33057 x^{8} - 138943 x^{6} - 447367 x^{4} - 187795 x^{2} + 279841 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(43690605516556003276152578767301401=29^{14}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $146.23$
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:  $29, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{14} a^{8} - \frac{1}{2} a^{7} + \frac{3}{7} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{14} a^{2} - \frac{1}{2} a - \frac{1}{14}$, $\frac{1}{14} a^{9} - \frac{1}{14} a^{7} - \frac{1}{2} a^{5} - \frac{3}{7} a^{3} + \frac{3}{7} a - \frac{1}{2}$, $\frac{1}{14} a^{10} - \frac{1}{2} a^{7} - \frac{1}{14} a^{6} + \frac{1}{14} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{14}$, $\frac{1}{14} a^{11} + \frac{3}{7} a^{7} + \frac{1}{14} a^{5} - \frac{1}{2} a^{2} + \frac{3}{7} a - \frac{1}{2}$, $\frac{1}{14} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a + \frac{3}{7}$, $\frac{1}{322} a^{13} - \frac{5}{322} a^{9} - \frac{1}{161} a^{7} - \frac{9}{46} a^{5} - \frac{1}{2} a^{4} - \frac{3}{7} a^{3} - \frac{1}{2} a^{2} + \frac{72}{161} a - \frac{1}{2}$, $\frac{1}{6370826552163129854} a^{14} + \frac{4110203316431998}{138496229394850649} a^{12} + \frac{211872211252646641}{6370826552163129854} a^{10} - \frac{1969037306670968}{455059039440223561} a^{8} - \frac{602987574699193373}{6370826552163129854} a^{6} - \frac{1}{2} a^{5} - \frac{18083926000523935}{138496229394850649} a^{4} - \frac{1}{2} a^{3} + \frac{949849553987762809}{3185413276081564927} a^{2} - \frac{1}{2} a + \frac{1604688979976385}{6021575191080463}$, $\frac{1}{146529010699751986642} a^{15} + \frac{4110203316431998}{3185413276081564927} a^{13} - \frac{3428600104269141847}{146529010699751986642} a^{11} - \frac{131985905718163414}{10466357907125141903} a^{9} - \frac{58850544623047809181}{146529010699751986642} a^{7} - \frac{1}{2} a^{6} + \frac{1564730124226340625}{3185413276081564927} a^{5} - \frac{1}{2} a^{4} + \frac{9595971303352010468}{73264505349875993321} a^{3} - \frac{1}{2} a^{2} - \frac{53449712767044991}{138496229394850649} a$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $9$
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:  \( 30374182208.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.84898109.1, 4.2.49619.1, 4.2.1438951.1, 8.4.209022978441500549.1 x2, 8.4.7207688911775881.1

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 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ 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
29Data not computed
$59$59.4.2.1$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$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$