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Properties

Label 16.0.24195399091...5136.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 229^{6}$
Root discriminant $21.70$
Ramified primes $2, 229$
Class number $1$
Class group Trivial
Galois group $\GL(2,Z/4)$ (as 16T186)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 22, 40, -19, -62, 114, 22, -48, 48, 24, -42, 21, 24, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 24*x^13 + 21*x^12 - 42*x^11 + 24*x^10 + 48*x^9 - 48*x^8 + 22*x^7 + 114*x^6 - 62*x^5 - 19*x^4 + 40*x^3 + 22*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 4*x^14 + 24*x^13 + 21*x^12 - 42*x^11 + 24*x^10 + 48*x^9 - 48*x^8 + 22*x^7 + 114*x^6 - 62*x^5 - 19*x^4 + 40*x^3 + 22*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 4 x^{14} + 24 x^{13} + 21 x^{12} - 42 x^{11} + 24 x^{10} + 48 x^{9} - 48 x^{8} + 22 x^{7} + 114 x^{6} - 62 x^{5} - 19 x^{4} + 40 x^{3} + 22 x^{2} + 6 x + 1 \)

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:  \(2419539909105613275136=2^{24}\cdot 229^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.70$
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:  $2, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{15325139647080} a^{15} - \frac{132140080579}{1702793294120} a^{14} + \frac{429408352013}{15325139647080} a^{13} - \frac{3595552123}{43911574920} a^{12} + \frac{362383832447}{1532513964708} a^{11} + \frac{1738343605817}{3831284911770} a^{10} + \frac{693647298887}{3831284911770} a^{9} - \frac{299772388327}{3831284911770} a^{8} + \frac{6951552047}{3831284911770} a^{7} + \frac{538986804331}{2554189941180} a^{6} + \frac{308086846381}{1277094970590} a^{5} + \frac{231559309667}{7662569823540} a^{4} + \frac{1184783018081}{5108379882360} a^{3} - \frac{1937590361261}{15325139647080} a^{2} - \frac{7435081801531}{15325139647080} a - \frac{6097760427697}{15325139647080}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{55594093}{69922890} a^{15} + \frac{53061189}{15538420} a^{14} + \frac{151819711}{69922890} a^{13} - \frac{2751186223}{139845780} a^{12} - \frac{77170964}{6992289} a^{11} + \frac{2543842381}{69922890} a^{10} - \frac{2027333339}{69922890} a^{9} - \frac{1975874561}{69922890} a^{8} + \frac{3185603251}{69922890} a^{7} - \frac{707803751}{23307630} a^{6} - \frac{932242286}{11653815} a^{5} + \frac{2496990079}{34961445} a^{4} - \frac{74728369}{11653815} a^{3} - \frac{3598808759}{139845780} a^{2} - \frac{366738856}{34961445} a - \frac{339395893}{139845780} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 188620.782456 \)
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

\(\Q(\sqrt{-1}) \), 4.4.14656.1, 4.0.14656.4, 8.0.214798336.3

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

Sibling fields

Degree 12 siblings: 12.6.36919249101342976.1, 12.0.2579510854242304.1, 12.0.2579510854242304.2, 12.0.10318043416969216.1
Degree 16 sibling: Deg 16
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
229Data not computed