Properties

Label 16.8.385...689.1
Degree $16$
Signature $[8, 4]$
Discriminant $3.854\times 10^{44}$
Root discriminant $611.81$
Ramified primes $19, 89$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 432*x^14 + 3189*x^13 + 45677*x^12 - 683348*x^11 + 4486191*x^10 - 28234100*x^9 - 528567471*x^8 + 16652185199*x^7 - 118567987102*x^6 - 764164602577*x^5 + 14583743055018*x^4 - 43451566069395*x^3 - 200403824876743*x^2 + 689590561853787*x + 1412711161908947)
 
gp: K = bnfinit(x^16 - 2*x^15 - 432*x^14 + 3189*x^13 + 45677*x^12 - 683348*x^11 + 4486191*x^10 - 28234100*x^9 - 528567471*x^8 + 16652185199*x^7 - 118567987102*x^6 - 764164602577*x^5 + 14583743055018*x^4 - 43451566069395*x^3 - 200403824876743*x^2 + 689590561853787*x + 1412711161908947, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1412711161908947, 689590561853787, -200403824876743, -43451566069395, 14583743055018, -764164602577, -118567987102, 16652185199, -528567471, -28234100, 4486191, -683348, 45677, 3189, -432, -2, 1]);
 

\( x^{16} - 2 x^{15} - 432 x^{14} + 3189 x^{13} + 45677 x^{12} - 683348 x^{11} + 4486191 x^{10} - 28234100 x^{9} - 528567471 x^{8} + 16652185199 x^{7} - 118567987102 x^{6} - 764164602577 x^{5} + 14583743055018 x^{4} - 43451566069395 x^{3} - 200403824876743 x^{2} + 689590561853787 x + 1412711161908947 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(385383672585505073080666077824760451253265689\)\(\medspace = 19^{12}\cdot 89^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $611.81$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $19, 89$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{1844} a^{14} - \frac{59}{1844} a^{13} + \frac{77}{922} a^{12} + \frac{131}{1844} a^{11} + \frac{97}{1844} a^{10} - \frac{201}{1844} a^{9} + \frac{909}{1844} a^{8} - \frac{215}{922} a^{7} + \frac{597}{1844} a^{6} + \frac{107}{1844} a^{5} - \frac{673}{1844} a^{4} + \frac{875}{1844} a^{3} + \frac{183}{922} a^{2} - \frac{385}{1844} a - \frac{99}{1844}$, $\frac{1}{16136173788585619090930383430263493903190818181809124423332302644659906013160025644356309759121316} a^{15} + \frac{605813462741867889374633068143117952549205356967862841788503103843406658525595669583557617611}{8068086894292809545465191715131746951595409090904562211666151322329953006580012822178154879560658} a^{14} + \frac{869843018658793943869261661765583646819660471039136435830160583676166406246761805016778802603297}{16136173788585619090930383430263493903190818181809124423332302644659906013160025644356309759121316} a^{13} + \frac{403811840486980158500737947726827641859397900947810675922938258422939035104546086763234098929696}{4034043447146404772732595857565873475797704545452281105833075661164976503290006411089077439780329} a^{12} - \frac{1964102184939992589945179085550690040707493140199556898174725227525952966108372356812221510025789}{16136173788585619090930383430263493903190818181809124423332302644659906013160025644356309759121316} a^{11} + \frac{872489663352584824784279840783769315475546111185344406262219106364393275822488549292662580178955}{16136173788585619090930383430263493903190818181809124423332302644659906013160025644356309759121316} a^{10} - \frac{828536043374377606171779906432778858216618046293915591377552007446738131658715130549961515002912}{4034043447146404772732595857565873475797704545452281105833075661164976503290006411089077439780329} a^{9} - \frac{3782534644544730530183652401312553140239849804711304729034403836265286431569661841207277693005741}{16136173788585619090930383430263493903190818181809124423332302644659906013160025644356309759121316} a^{8} + \frac{1014869659436568702416132222126910429938311364669986099821136667913242694605718762517312755657061}{4034043447146404772732595857565873475797704545452281105833075661164976503290006411089077439780329} a^{7} + \frac{2343066594457200608661274497763274726223965557161667881007977052941633952028652774067118267331157}{16136173788585619090930383430263493903190818181809124423332302644659906013160025644356309759121316} a^{6} - \frac{819517421738914331642190530235881465168086536217060393086446107664198653808895205308083802363441}{16136173788585619090930383430263493903190818181809124423332302644659906013160025644356309759121316} a^{5} - \frac{3759311334276970787376019539838799951197341816096327229131225676376146488490310148512601860899037}{8068086894292809545465191715131746951595409090904562211666151322329953006580012822178154879560658} a^{4} + \frac{2987221246746188562293576826037572418843539441670817983854830270511737643510542182219387421976829}{16136173788585619090930383430263493903190818181809124423332302644659906013160025644356309759121316} a^{3} - \frac{3631239045985869161142631043817323314589880155442797509689633751321184519426456599570169578334625}{8068086894292809545465191715131746951595409090904562211666151322329953006580012822178154879560658} a^{2} - \frac{4831389669979117598963772765862073979140614874967134099445313274760271911930647365251362922903813}{16136173788585619090930383430263493903190818181809124423332302644659906013160025644356309759121316} a + \frac{787625709955014620213042924227152762139045725572995746302987302989844951352274194023957460812249}{8068086894292809545465191715131746951595409090904562211666151322329953006580012822178154879560658}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 13898886654100000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 13898886654100000 \cdot 1}{2\sqrt{385383672585505073080666077824760451253265689}}\approx 0.141241655978276$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.5764271794920234809.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
19Data not computed
89Data not computed