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Properties

Label 16.16.1141793086...6016.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{32}\cdot 113^{8}$
Root discriminant $42.52$
Ramified primes $2, 113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 48, -352, -3792, -740, 11384, 1452, -13240, 357, 6856, -906, -1596, 311, 148, -34, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 34*x^14 + 148*x^13 + 311*x^12 - 1596*x^11 - 906*x^10 + 6856*x^9 + 357*x^8 - 13240*x^7 + 1452*x^6 + 11384*x^5 - 740*x^4 - 3792*x^3 - 352*x^2 + 48*x + 4)
 
gp: K = bnfinit(x^16 - 4*x^15 - 34*x^14 + 148*x^13 + 311*x^12 - 1596*x^11 - 906*x^10 + 6856*x^9 + 357*x^8 - 13240*x^7 + 1452*x^6 + 11384*x^5 - 740*x^4 - 3792*x^3 - 352*x^2 + 48*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 34 x^{14} + 148 x^{13} + 311 x^{12} - 1596 x^{11} - 906 x^{10} + 6856 x^{9} + 357 x^{8} - 13240 x^{7} + 1452 x^{6} + 11384 x^{5} - 740 x^{4} - 3792 x^{3} - 352 x^{2} + 48 x + 4 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(114179308667742410575446016=2^{32}\cdot 113^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.52$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} - \frac{1}{8} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{368} a^{13} - \frac{3}{92} a^{12} - \frac{1}{46} a^{11} - \frac{11}{92} a^{10} + \frac{3}{16} a^{9} - \frac{13}{92} a^{8} - \frac{15}{92} a^{7} + \frac{7}{92} a^{6} + \frac{99}{368} a^{5} + \frac{45}{92} a^{4} + \frac{57}{184} a^{3} - \frac{35}{92} a^{2} + \frac{71}{184} a - \frac{13}{46}$, $\frac{1}{368} a^{14} - \frac{7}{184} a^{12} + \frac{11}{92} a^{11} + \frac{1}{368} a^{10} - \frac{13}{92} a^{9} + \frac{3}{184} a^{8} + \frac{17}{46} a^{7} - \frac{117}{368} a^{6} - \frac{3}{92} a^{5} + \frac{5}{92} a^{4} + \frac{31}{92} a^{3} + \frac{13}{184} a^{2} + \frac{8}{23} a + \frac{33}{92}$, $\frac{1}{9850040805008} a^{15} + \frac{2239292049}{9850040805008} a^{14} - \frac{12022288281}{9850040805008} a^{13} + \frac{5637237795}{107065660924} a^{12} - \frac{1072017958523}{9850040805008} a^{11} + \frac{198380734833}{9850040805008} a^{10} + \frac{2044540933979}{9850040805008} a^{9} + \frac{161450541907}{2462510201252} a^{8} - \frac{541736813489}{9850040805008} a^{7} - \frac{3565201639525}{9850040805008} a^{6} - \frac{1579685977097}{9850040805008} a^{5} + \frac{229367141233}{4925020402504} a^{4} - \frac{206904713617}{615627550313} a^{3} + \frac{1306875371121}{4925020402504} a^{2} - \frac{1619502877191}{4925020402504} a - \frac{601623657495}{1231255100626}$

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:  $15$
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:  \( 102949596.437 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7232.1, 8.8.5910106112.1, 8.8.94561697792.1, 8.8.10685471850496.2

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

Sibling fields

Degree 16 sibling: 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$113$113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.8.6.1$x^{8} - 18193 x^{4} + 127690000$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$