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Properties

Label 16.0.87823289190...6249.1
Degree $16$
Signature $[0, 8]$
Discriminant $67^{2}\cdot 89^{14}$
Root discriminant $85.90$
Ramified primes $67, 89$
Class number $339$ (GRH)
Class group $[339]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![472832, -1194368, 1821584, -1898648, 1271664, -578058, 290653, -142003, 43024, -12743, 3439, -722, 263, 19, -4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 4*x^14 + 19*x^13 + 263*x^12 - 722*x^11 + 3439*x^10 - 12743*x^9 + 43024*x^8 - 142003*x^7 + 290653*x^6 - 578058*x^5 + 1271664*x^4 - 1898648*x^3 + 1821584*x^2 - 1194368*x + 472832)
 
gp: K = bnfinit(x^16 - 5*x^15 - 4*x^14 + 19*x^13 + 263*x^12 - 722*x^11 + 3439*x^10 - 12743*x^9 + 43024*x^8 - 142003*x^7 + 290653*x^6 - 578058*x^5 + 1271664*x^4 - 1898648*x^3 + 1821584*x^2 - 1194368*x + 472832, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 4 x^{14} + 19 x^{13} + 263 x^{12} - 722 x^{11} + 3439 x^{10} - 12743 x^{9} + 43024 x^{8} - 142003 x^{7} + 290653 x^{6} - 578058 x^{5} + 1271664 x^{4} - 1898648 x^{3} + 1821584 x^{2} - 1194368 x + 472832 \)

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:  \(8782328919028941504502068196249=67^{2}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.90$
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:  $67, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{126016} a^{14} - \frac{7515}{126016} a^{13} - \frac{601}{63008} a^{12} + \frac{1415}{126016} a^{11} + \frac{813}{126016} a^{10} + \frac{181}{3938} a^{9} + \frac{1767}{126016} a^{8} + \frac{25215}{126016} a^{7} - \frac{1405}{63008} a^{6} + \frac{30241}{126016} a^{5} + \frac{20167}{126016} a^{4} + \frac{7259}{31504} a^{3} - \frac{3361}{7876} a^{2} - \frac{5947}{15752} a + \frac{1527}{3938}$, $\frac{1}{74819758239915265488215478193372672} a^{15} - \frac{215078930014418502718350598183}{74819758239915265488215478193372672} a^{14} + \frac{908553771123652545120181017011605}{37409879119957632744107739096686336} a^{13} + \frac{2294680634552658769279904432257727}{74819758239915265488215478193372672} a^{12} - \frac{1299695411367156286085547747809687}{74819758239915265488215478193372672} a^{11} - \frac{406688533492336709028892521684769}{18704939559978816372053869548343168} a^{10} - \frac{7360674958349193200627102344061513}{74819758239915265488215478193372672} a^{9} - \frac{2308233738652879454453987826863125}{74819758239915265488215478193372672} a^{8} + \frac{4476484864643758420386049557528621}{37409879119957632744107739096686336} a^{7} - \frac{8013282581301081461703286286084199}{74819758239915265488215478193372672} a^{6} - \frac{12816716321022690675653570398145205}{74819758239915265488215478193372672} a^{5} + \frac{191136003346714832491058798354929}{1169058722498676023253366846771448} a^{4} - \frac{2114116867251630129898430652497925}{4676234889994704093013467387085792} a^{3} - \frac{4369091593204939205617756892160607}{9352469779989408186026934774171584} a^{2} - \frac{105468341920109620048005456557611}{584529361249338011626683423385724} a + \frac{134483787720283576484417810513335}{584529361249338011626683423385724}$

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

Class group and class number

$C_{339}$, which has order $339$ (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:  \( 96345227.2282 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.2.2963499438000443.1, 8.0.44231334895529.1, 8.6.33297746494387.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
89Data not computed