Properties

Label 16.8.279...897.1
Degree $16$
Signature $[8, 4]$
Discriminant $2.796\times 10^{40}$
Root discriminant $337.22$
Ramified primes $11, 73$
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 - 4*x^15 - 29*x^14 - 684*x^13 - 14245*x^12 - 8754*x^11 + 214599*x^10 + 581416*x^9 + 11811196*x^8 + 46895440*x^7 + 110537580*x^6 + 509315106*x^5 + 677109343*x^4 + 1464634672*x^3 + 1252749264*x^2 - 229349504*x - 18436864)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 - 684*x^13 - 14245*x^12 - 8754*x^11 + 214599*x^10 + 581416*x^9 + 11811196*x^8 + 46895440*x^7 + 110537580*x^6 + 509315106*x^5 + 677109343*x^4 + 1464634672*x^3 + 1252749264*x^2 - 229349504*x - 18436864, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18436864, -229349504, 1252749264, 1464634672, 677109343, 509315106, 110537580, 46895440, 11811196, 581416, 214599, -8754, -14245, -684, -29, -4, 1]);
 

\( x^{16} - 4 x^{15} - 29 x^{14} - 684 x^{13} - 14245 x^{12} - 8754 x^{11} + 214599 x^{10} + 581416 x^{9} + 11811196 x^{8} + 46895440 x^{7} + 110537580 x^{6} + 509315106 x^{5} + 677109343 x^{4} + 1464634672 x^{3} + 1252749264 x^{2} - 229349504 x - 18436864 \)

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:  \(27961156076874708483172971177878992693897\)\(\medspace = 11^{12}\cdot 73^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $337.22$
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:  $11, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{164} a^{13} + \frac{5}{82} a^{12} - \frac{11}{164} a^{11} + \frac{17}{82} a^{10} - \frac{35}{164} a^{9} + \frac{1}{41} a^{8} + \frac{19}{164} a^{7} + \frac{33}{82} a^{6} - \frac{20}{41} a^{5} - \frac{21}{82} a^{4} + \frac{19}{41} a^{3} - \frac{14}{41} a^{2} - \frac{11}{164} a - \frac{4}{41}$, $\frac{1}{1312} a^{14} - \frac{29}{1312} a^{12} + \frac{9}{82} a^{11} - \frac{293}{1312} a^{10} + \frac{13}{656} a^{9} + \frac{143}{1312} a^{8} - \frac{31}{328} a^{7} - \frac{21}{328} a^{6} - \frac{9}{82} a^{5} - \frac{81}{328} a^{4} + \frac{289}{656} a^{3} - \frac{25}{1312} a^{2} - \frac{161}{328} a - \frac{31}{82}$, $\frac{1}{318418398670884169291461845977193634786862318858019584572858154438784} a^{15} - \frac{2452261166541473615403845134114806628376066392212103488168870997}{79604599667721042322865461494298408696715579714504896143214538609696} a^{14} + \frac{401086591337124249408243055329446525181122357015658948971136810003}{318418398670884169291461845977193634786862318858019584572858154438784} a^{13} - \frac{7963515055525626317946320674330101478113765078553129306511261156663}{79604599667721042322865461494298408696715579714504896143214538609696} a^{12} + \frac{1360145404593790031663816463064587820416611366216886349257113196267}{318418398670884169291461845977193634786862318858019584572858154438784} a^{11} - \frac{19476052804935309343133656514218033738546059576169622913247496837009}{159209199335442084645730922988596817393431159429009792286429077219392} a^{10} + \frac{77789210395111863324009753731956360415134450405803652996776685751415}{318418398670884169291461845977193634786862318858019584572858154438784} a^{9} + \frac{7250533580168099874532986984291905432231137500300823298465965890707}{39802299833860521161432730747149204348357789857252448071607269304848} a^{8} - \frac{11914721432305196743656301381796143202005459299642033275933488910893}{79604599667721042322865461494298408696715579714504896143214538609696} a^{7} - \frac{6372488798375999733704069094518071943058755939556056817395077516371}{19901149916930260580716365373574602174178894928626224035803634652424} a^{6} + \frac{10517988328185033310554742990826677578896927138916532204072809630203}{79604599667721042322865461494298408696715579714504896143214538609696} a^{5} - \frac{26665203991057134082123188421347818151483154555517567297208502177583}{159209199335442084645730922988596817393431159429009792286429077219392} a^{4} - \frac{38746110608875978799611721293579205050736874247955853439259657578049}{318418398670884169291461845977193634786862318858019584572858154438784} a^{3} - \frac{4164882868657475740226818593438317942090285325207243446362593286575}{9950574958465130290358182686787301087089447464313112017901817326212} a^{2} + \frac{4279638417574353985076191282011892294394582161714973323119674750197}{9950574958465130290358182686787301087089447464313112017901817326212} a + \frac{1450460692698016876955969975067692775187093758438863849142084199343}{4975287479232565145179091343393650543544723732156556008950908663106}$

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:  \( 188399221895000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 188399221895000 \cdot 1}{2\sqrt{27961156076874708483172971177878992693897}}\approx 0.224766191073677$ (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

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.161744961718099177.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ R $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$

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
11Data not computed
73Data not computed