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Properties

Label 16.2.23842965126...1875.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,3^{12}\cdot 5^{12}\cdot 179^{5}$
Root discriminant $38.56$
Ramified primes $3, 5, 179$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1354

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-72329, 75162, -84733, 42761, -31738, 11631, -12907, 10263, -9140, 4629, -1631, 106, 183, -119, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 - 119*x^13 + 183*x^12 + 106*x^11 - 1631*x^10 + 4629*x^9 - 9140*x^8 + 10263*x^7 - 12907*x^6 + 11631*x^5 - 31738*x^4 + 42761*x^3 - 84733*x^2 + 75162*x - 72329)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 - 119*x^13 + 183*x^12 + 106*x^11 - 1631*x^10 + 4629*x^9 - 9140*x^8 + 10263*x^7 - 12907*x^6 + 11631*x^5 - 31738*x^4 + 42761*x^3 - 84733*x^2 + 75162*x - 72329, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} - 119 x^{13} + 183 x^{12} + 106 x^{11} - 1631 x^{10} + 4629 x^{9} - 9140 x^{8} + 10263 x^{7} - 12907 x^{6} + 11631 x^{5} - 31738 x^{4} + 42761 x^{3} - 84733 x^{2} + 75162 x - 72329 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-23842965126465199951171875=-\,3^{12}\cdot 5^{12}\cdot 179^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.56$
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:  $3, 5, 179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} + \frac{1}{15} a^{10} - \frac{4}{15} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{7}{15} a^{6} + \frac{1}{5} a^{5} + \frac{4}{15} a^{4} - \frac{1}{3} a^{3} + \frac{2}{5} a^{2} - \frac{1}{15} a + \frac{1}{15}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{11} - \frac{4}{15} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{7}{15} a^{7} + \frac{1}{5} a^{6} + \frac{4}{15} a^{5} - \frac{1}{3} a^{4} + \frac{2}{5} a^{3} - \frac{1}{15} a^{2} + \frac{1}{15} a$, $\frac{1}{15} a^{14} - \frac{4}{15} a^{11} + \frac{1}{3} a^{10} + \frac{1}{15} a^{9} + \frac{1}{15} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{7}{15} a^{5} + \frac{2}{15} a^{4} + \frac{4}{15} a^{3} - \frac{1}{3} a^{2} + \frac{1}{15} a - \frac{1}{15}$, $\frac{1}{1361367302417622875993701697385} a^{15} - \frac{19748070823766595849138606584}{1361367302417622875993701697385} a^{14} - \frac{2580902594044220176458370498}{1361367302417622875993701697385} a^{13} + \frac{5228096518327193151812797639}{272273460483524575198740339477} a^{12} - \frac{100514650071393782013477991784}{453789100805874291997900565795} a^{11} - \frac{68642521237708910076220112408}{1361367302417622875993701697385} a^{10} - \frac{459005139472790041695146899522}{1361367302417622875993701697385} a^{9} + \frac{133899236952990442007123495714}{272273460483524575198740339477} a^{8} - \frac{106514921196987888517526936633}{272273460483524575198740339477} a^{7} - \frac{386149371902003983148394680912}{1361367302417622875993701697385} a^{6} + \frac{68225281074563275979761134449}{1361367302417622875993701697385} a^{5} + \frac{76482330314672508961595074472}{1361367302417622875993701697385} a^{4} - \frac{329567906651368188993885310279}{1361367302417622875993701697385} a^{3} - \frac{156557681619432497564344881064}{453789100805874291997900565795} a^{2} - \frac{6630496811482417368301374967}{1361367302417622875993701697385} a - \frac{23104019416973804748270945427}{1361367302417622875993701697385}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $8$
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:  \( 366900.531256 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1354:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 59 conjugacy class representatives for t16n1354 are not computed
Character table for t16n1354 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.226546875.1

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 $16$ R R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$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}$
179Data not computed