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Properties

Label 16.0.18297527052795904.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{22}\cdot 257^{4}$
Root discriminant $10.38$
Ramified primes $2, 257$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^3:S_4$ (as 16T747)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -20, 50, -78, 77, -36, -22, 62, -67, 46, -16, -6, 14, -12, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 12*x^13 + 14*x^12 - 6*x^11 - 16*x^10 + 46*x^9 - 67*x^8 + 62*x^7 - 22*x^6 - 36*x^5 + 77*x^4 - 78*x^3 + 50*x^2 - 20*x + 4)
 
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 12*x^13 + 14*x^12 - 6*x^11 - 16*x^10 + 46*x^9 - 67*x^8 + 62*x^7 - 22*x^6 - 36*x^5 + 77*x^4 - 78*x^3 + 50*x^2 - 20*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 8 x^{14} - 12 x^{13} + 14 x^{12} - 6 x^{11} - 16 x^{10} + 46 x^{9} - 67 x^{8} + 62 x^{7} - 22 x^{6} - 36 x^{5} + 77 x^{4} - 78 x^{3} + 50 x^{2} - 20 x + 4 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18297527052795904=2^{22}\cdot 257^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.38$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{134518} a^{15} + \frac{8987}{134518} a^{14} + \frac{12033}{67259} a^{13} - \frac{31034}{67259} a^{12} + \frac{30904}{67259} a^{11} + \frac{10932}{67259} a^{10} + \frac{235}{653} a^{9} - \frac{22946}{67259} a^{8} - \frac{48333}{134518} a^{7} + \frac{65717}{134518} a^{6} - \frac{4395}{67259} a^{5} + \frac{32829}{67259} a^{4} + \frac{66171}{134518} a^{3} - \frac{29731}{134518} a^{2} + \frac{21577}{67259} a + \frac{23841}{67259}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{92401}{134518} a^{15} - \frac{154061}{67259} a^{14} + \frac{271740}{67259} a^{13} - \frac{388723}{67259} a^{12} + \frac{415954}{67259} a^{11} - \frac{35189}{67259} a^{10} - \frac{7157}{653} a^{9} + \frac{1653586}{67259} a^{8} - \frac{4189991}{134518} a^{7} + \frac{1667585}{67259} a^{6} - \frac{127071}{67259} a^{5} - \frac{1629946}{67259} a^{4} + \frac{5131601}{134518} a^{3} - \frac{2142426}{67259} a^{2} + \frac{1188502}{67259} a - \frac{338081}{67259} \) (order $4$)
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:  \( 50.6977746509 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^3:S_4$ (as 16T747):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 26 conjugacy class representatives for $C_2\times C_2^3:S_4$
Character table for $C_2\times C_2^3:S_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.257.1, 8.0.16908544.2, 8.0.4227136.1, 8.0.16908544.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.12.18.61$x^{12} - 6 x^{10} + 2 x^{8} - 4 x^{7} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 8$$4$$3$$18$$C_2^2 \times A_4$$[2, 2, 2]^{6}$
257Data not computed