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Properties

Label 16.8.24027704049...2081.1
Degree $16$
Signature $[8, 4]$
Discriminant $53^{14}\cdot 89^{15}$
Root discriminant $2169.17$
Ramified primes $53, 89$
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![-1115465623645375, 263445165203961, 852542410999822, 153070746965979, -22238144524836, 8721064590146, 1070375916313, -185731157449, 17745867159, -1318149458, -122064213, 3159219, -621434, 9581, -37, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 37*x^14 + 9581*x^13 - 621434*x^12 + 3159219*x^11 - 122064213*x^10 - 1318149458*x^9 + 17745867159*x^8 - 185731157449*x^7 + 1070375916313*x^6 + 8721064590146*x^5 - 22238144524836*x^4 + 153070746965979*x^3 + 852542410999822*x^2 + 263445165203961*x - 1115465623645375)
 
gp: K = bnfinit(x^16 - 4*x^15 - 37*x^14 + 9581*x^13 - 621434*x^12 + 3159219*x^11 - 122064213*x^10 - 1318149458*x^9 + 17745867159*x^8 - 185731157449*x^7 + 1070375916313*x^6 + 8721064590146*x^5 - 22238144524836*x^4 + 153070746965979*x^3 + 852542410999822*x^2 + 263445165203961*x - 1115465623645375, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 37 x^{14} + 9581 x^{13} - 621434 x^{12} + 3159219 x^{11} - 122064213 x^{10} - 1318149458 x^{9} + 17745867159 x^{8} - 185731157449 x^{7} + 1070375916313 x^{6} + 8721064590146 x^{5} - 22238144524836 x^{4} + 153070746965979 x^{3} + 852542410999822 x^{2} + 263445165203961 x - 1115465623645375 \)

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:  \(240277040497518512877081049134097240815202879947452081=53^{14}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2169.17$
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:  $53, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{2} a^{6} + \frac{1}{10} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{9} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} + \frac{3}{10} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{3}{10} a - \frac{1}{2}$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{9} + \frac{3}{20} a^{8} - \frac{1}{10} a^{7} + \frac{2}{5} a^{6} + \frac{1}{20} a^{5} + \frac{3}{10} a^{4} + \frac{1}{10} a^{3} - \frac{3}{20} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{100} a^{13} - \frac{1}{50} a^{11} - \frac{1}{20} a^{10} + \frac{1}{100} a^{9} - \frac{3}{50} a^{8} + \frac{19}{50} a^{7} - \frac{19}{100} a^{6} + \frac{4}{25} a^{5} + \frac{3}{50} a^{4} + \frac{49}{100} a^{3} + \frac{49}{100} a^{2} - \frac{3}{100} a - \frac{1}{2}$, $\frac{1}{65810464927594123000} a^{14} + \frac{278846687189562719}{65810464927594123000} a^{13} + \frac{770792275568298273}{65810464927594123000} a^{12} + \frac{682305670974248047}{65810464927594123000} a^{11} + \frac{220468168023435219}{16452616231898530750} a^{10} + \frac{2996559589743268799}{32905232463797061500} a^{9} + \frac{31790296825509838689}{65810464927594123000} a^{8} - \frac{29175187410877815057}{65810464927594123000} a^{7} + \frac{1343331107971172289}{13162092985518824600} a^{6} + \frac{197259225856536421}{2632418597103764920} a^{5} - \frac{22188313288053287427}{65810464927594123000} a^{4} - \frac{55950316496779469}{1316209298551882460} a^{3} + \frac{18095593191630906043}{65810464927594123000} a^{2} - \frac{5416270960643683891}{32905232463797061500} a + \frac{122102015912310971}{526483719420752984}$, $\frac{1}{977613382634955318358184070496504649526081896916897759136642229991583861418328956307658975000} a^{15} + \frac{967179647527980628842352266984952340578087832507626043474680172593486469}{488806691317477659179092035248252324763040948458448879568321114995791930709164478153829487500} a^{14} + \frac{940038882389086893071275696119724934142467720134113044520509485133232401779072874165044867}{488806691317477659179092035248252324763040948458448879568321114995791930709164478153829487500} a^{13} + \frac{5458617126350814277830206335064186425211329812314961034311783257982967070166328932909160167}{488806691317477659179092035248252324763040948458448879568321114995791930709164478153829487500} a^{12} + \frac{22611228052000338210297109696905823003709285192393442028605167588961096486478131376440971969}{977613382634955318358184070496504649526081896916897759136642229991583861418328956307658975000} a^{11} + \frac{14083015096894745851770887201252366163761546338491552237425542059496534281553783753918400121}{488806691317477659179092035248252324763040948458448879568321114995791930709164478153829487500} a^{10} - \frac{58607041486200284817701725576374174839103651769101661018850737699885525378052491468480725449}{977613382634955318358184070496504649526081896916897759136642229991583861418328956307658975000} a^{9} + \frac{69434868296453880860668661841809204042844260909490936557855401984859695375347221892041277867}{488806691317477659179092035248252324763040948458448879568321114995791930709164478153829487500} a^{8} + \frac{111943097948523383307745362135659985618329799275090173444231721647915286456852739478830425081}{488806691317477659179092035248252324763040948458448879568321114995791930709164478153829487500} a^{7} - \frac{6355800262128162017738971791482884499954557739738104576466740747786415846650654497149887491}{48880669131747765917909203524825232476304094845844887956832111499579193070916447815382948750} a^{6} + \frac{38781068075122919557602516943405298648098044972203379239009791013038201743367989761972614031}{122201672829369414794773008812063081190760237114612219892080278748947982677291119538457371875} a^{5} + \frac{139136759174969083854463490510420051068173131583018364493924535810064807395027617909554626637}{977613382634955318358184070496504649526081896916897759136642229991583861418328956307658975000} a^{4} - \frac{102020643920928106656298603634107023942435205688734618204959486903766196690613721933270639507}{977613382634955318358184070496504649526081896916897759136642229991583861418328956307658975000} a^{3} + \frac{66579487012750729091198952393844321451266039318584002770313206571420805661507671650892461827}{195522676526991063671636814099300929905216379383379551827328445998316772283665791261531795000} a^{2} + \frac{427426420798873734279783628675871753669125596990579509779160234306279209655353349623388717017}{977613382634955318358184070496504649526081896916897759136642229991583861418328956307658975000} a - \frac{341029949821380413682677841965144716944558157204859300078005225290151267665612912394873801}{710991551007240231533224778542912472382605015939562006644830712721151899213330150041933800}$

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:  \( 337089472222000000000 \) (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

\(\Q(\sqrt{89}) \), 4.4.1980257921.1, 8.8.980359279842244243492241.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ R $16$

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
$53$53.8.7.3$x^{8} + 106$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.3$x^{8} + 106$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
89Data not computed