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Properties

Label 16.8.87995526059...5937.1
Degree $16$
Signature $[8, 4]$
Discriminant $61^{14}\cdot 73^{15}$
Root discriminant $2037.18$
Ramified primes $61, 73$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2223621705617209, -148420382825981, 945131876919130, -258874459402297, -69676368352736, 42632626532812, -8329715762855, 210269586483, 41697009299, -2295493716, 106534135, 600723, -564592, 9025, -29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 9025*x^13 - 564592*x^12 + 600723*x^11 + 106534135*x^10 - 2295493716*x^9 + 41697009299*x^8 + 210269586483*x^7 - 8329715762855*x^6 + 42632626532812*x^5 - 69676368352736*x^4 - 258874459402297*x^3 + 945131876919130*x^2 - 148420382825981*x - 2223621705617209)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 + 9025*x^13 - 564592*x^12 + 600723*x^11 + 106534135*x^10 - 2295493716*x^9 + 41697009299*x^8 + 210269586483*x^7 - 8329715762855*x^6 + 42632626532812*x^5 - 69676368352736*x^4 - 258874459402297*x^3 + 945131876919130*x^2 - 148420382825981*x - 2223621705617209, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 29 x^{14} + 9025 x^{13} - 564592 x^{12} + 600723 x^{11} + 106534135 x^{10} - 2295493716 x^{9} + 41697009299 x^{8} + 210269586483 x^{7} - 8329715762855 x^{6} + 42632626532812 x^{5} - 69676368352736 x^{4} - 258874459402297 x^{3} + 945131876919130 x^{2} - 148420382825981 x - 2223621705617209 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(87995526059768479187762363921103601182604493355285937=61^{14}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2037.18$
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:  $61, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{249185192636533015048} a^{14} - \frac{11795844862529787331}{249185192636533015048} a^{13} - \frac{18426852839432169245}{249185192636533015048} a^{12} - \frac{20702376403672048661}{249185192636533015048} a^{11} + \frac{823455851606631010}{31148149079566626881} a^{10} + \frac{2340350703263183897}{62296298159133253762} a^{9} + \frac{13570988715968295467}{249185192636533015048} a^{8} - \frac{25508556514340060553}{249185192636533015048} a^{7} - \frac{774040338437123281}{249185192636533015048} a^{6} - \frac{29218379394007017617}{249185192636533015048} a^{5} + \frac{50723399532939410109}{249185192636533015048} a^{4} - \frac{18615488410234880159}{124592596318266507524} a^{3} - \frac{39336030145098948831}{249185192636533015048} a^{2} + \frac{24892599044727333055}{124592596318266507524} a - \frac{122182444145571695785}{249185192636533015048}$, $\frac{1}{723735345101171730400515351953229064845809013490461808151933746983772635337297770917659853495780344} a^{15} + \frac{172751943086764075542312371636306753140106575004468956726919460617737496300813}{361867672550585865200257675976614532422904506745230904075966873491886317668648885458829926747890172} a^{14} + \frac{21924115310222229565073822331231013261454867889678386044140648775898633316473883394250879839450045}{180933836275292932600128837988307266211452253372615452037983436745943158834324442729414963373945086} a^{13} + \frac{38305528717751156164714960824437659159566120515837665324141184392198535215706800640871933041239573}{361867672550585865200257675976614532422904506745230904075966873491886317668648885458829926747890172} a^{12} + \frac{169357712845339352025980961551727987691931414070611367693408855782667624742167061808498551454772647}{723735345101171730400515351953229064845809013490461808151933746983772635337297770917659853495780344} a^{11} - \frac{40712440374485507695339690714748703863839309994038260456125007910069119138936979613491748755943909}{180933836275292932600128837988307266211452253372615452037983436745943158834324442729414963373945086} a^{10} + \frac{347956178655353465431857031156495729602654308481721877642099310104591229966029660462848746723315575}{723735345101171730400515351953229064845809013490461808151933746983772635337297770917659853495780344} a^{9} - \frac{159529599927771119344038367177884769068799193483672085732457923141320145465359347476359952984412001}{361867672550585865200257675976614532422904506745230904075966873491886317668648885458829926747890172} a^{8} + \frac{6232942559717775188221489665748773587878260195923648872757225338048256299743690570989026888368653}{361867672550585865200257675976614532422904506745230904075966873491886317668648885458829926747890172} a^{7} - \frac{136056702491482905019367961236221965143751805660009441035350046285207957437416748012378929100944915}{361867672550585865200257675976614532422904506745230904075966873491886317668648885458829926747890172} a^{6} - \frac{24274481904447329914706699994159273878115469913093882157617274554814405397797425134480786946030043}{90466918137646466300064418994153633105726126686307726018991718372971579417162221364707481686972543} a^{5} + \frac{55936658338687518920517140558509418321830646251503059906525939485871106509704618237825191305659131}{723735345101171730400515351953229064845809013490461808151933746983772635337297770917659853495780344} a^{4} + \frac{357995728173246980881128524423854861580172094546127696287729356304003553140266272532329015338442075}{723735345101171730400515351953229064845809013490461808151933746983772635337297770917659853495780344} a^{3} + \frac{261183644041264426782949618684471257794555073194709611530045419881942124308106923794780686516984931}{723735345101171730400515351953229064845809013490461808151933746983772635337297770917659853495780344} a^{2} - \frac{154032688816853592090736160910085617388994111545322898356968971326611541290422516607921275702587243}{723735345101171730400515351953229064845809013490461808151933746983772635337297770917659853495780344} a + \frac{356893918295583046721231941316893292588885471172049742363626561448608848909945596655195634237078547}{723735345101171730400515351953229064845809013490461808151933746983772635337297770917659853495780344}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  $11$
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:  \( 33165263323100000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.1447532257.1, 8.8.569166107419034447672017.2

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ $16$ $16$ $16$ $16$

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
$61$61.8.7.4$x^{8} + 488$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.4$x^{8} + 488$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
73Data not computed