↑

Properties

Label 16.0.79202442563...6656.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{38}\cdot 257^{6}$
Root discriminant $41.56$
Ramified primes $2, 257$
Class number $48$ (GRH)
Class group $[2, 24]$ (GRH)
Galois group $\GL(2,Z/4)$ (as 16T186)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![866, -3368, 8812, -11896, 15216, -9200, 7152, 228, 652, 1236, 284, 80, 72, -44, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 44*x^13 + 72*x^12 + 80*x^11 + 284*x^10 + 1236*x^9 + 652*x^8 + 228*x^7 + 7152*x^6 - 9200*x^5 + 15216*x^4 - 11896*x^3 + 8812*x^2 - 3368*x + 866)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 44*x^13 + 72*x^12 + 80*x^11 + 284*x^10 + 1236*x^9 + 652*x^8 + 228*x^7 + 7152*x^6 - 9200*x^5 + 15216*x^4 - 11896*x^3 + 8812*x^2 - 3368*x + 866, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 44 x^{13} + 72 x^{12} + 80 x^{11} + 284 x^{10} + 1236 x^{9} + 652 x^{8} + 228 x^{7} + 7152 x^{6} - 9200 x^{5} + 15216 x^{4} - 11896 x^{3} + 8812 x^{2} - 3368 x + 866 \)

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:  \(79202442563427805645766656=2^{38}\cdot 257^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.56$
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:  $2, 257$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{5128508548302751613307967989} a^{15} + \frac{396683415203540904325820468}{5128508548302751613307967989} a^{14} + \frac{36319285330467720824835370}{1709502849434250537769322663} a^{13} + \frac{1687782223805913955174216244}{5128508548302751613307967989} a^{12} - \frac{643939789341604021477523392}{1709502849434250537769322663} a^{11} + \frac{764065174621137713903509028}{1709502849434250537769322663} a^{10} - \frac{2329385029447713832998586903}{5128508548302751613307967989} a^{9} - \frac{722827262579866009320213703}{1709502849434250537769322663} a^{8} + \frac{1488545296651112361189472388}{5128508548302751613307967989} a^{7} + \frac{701631114414892646540184360}{1709502849434250537769322663} a^{6} + \frac{49166018889039924423489880}{5128508548302751613307967989} a^{5} - \frac{419316921626455512425682068}{5128508548302751613307967989} a^{4} + \frac{1815559019889921908842168283}{5128508548302751613307967989} a^{3} - \frac{1687778263184955782733160700}{5128508548302751613307967989} a^{2} - \frac{217067068458612314929744048}{5128508548302751613307967989} a + \frac{525878178127092631295449422}{1709502849434250537769322663}$

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

Class group and class number

$C_{2}\times C_{24}$, which has order $48$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 53054.6396271 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,Z/4)$ (as 16T186):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 14 conjugacy class representatives for $\GL(2,Z/4)$
Character table for $\GL(2,Z/4)$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.16448.1, 4.0.65792.3, 8.8.17314349056.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
257Data not computed