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Properties

Label 16.0.76127245922...625.17
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 89^{10}$
Root discriminant $55.28$
Ramified primes $5, 89$
Class number $148$ (GRH)
Class group $[2, 74]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21492496, 22364064, 10797536, 7824574, 850907, -5502447, -489347, 843231, 98681, -62838, -7659, 1665, 436, 6, -10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 10*x^14 + 6*x^13 + 436*x^12 + 1665*x^11 - 7659*x^10 - 62838*x^9 + 98681*x^8 + 843231*x^7 - 489347*x^6 - 5502447*x^5 + 850907*x^4 + 7824574*x^3 + 10797536*x^2 + 22364064*x + 21492496)
 
gp: K = bnfinit(x^16 - 5*x^15 - 10*x^14 + 6*x^13 + 436*x^12 + 1665*x^11 - 7659*x^10 - 62838*x^9 + 98681*x^8 + 843231*x^7 - 489347*x^6 - 5502447*x^5 + 850907*x^4 + 7824574*x^3 + 10797536*x^2 + 22364064*x + 21492496, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 10 x^{14} + 6 x^{13} + 436 x^{12} + 1665 x^{11} - 7659 x^{10} - 62838 x^{9} + 98681 x^{8} + 843231 x^{7} - 489347 x^{6} - 5502447 x^{5} + 850907 x^{4} + 7824574 x^{3} + 10797536 x^{2} + 22364064 x + 21492496 \)

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:  \(7612724592276900293212890625=5^{12}\cdot 89^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.28$
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:  $5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6868} a^{14} - \frac{1145}{6868} a^{13} - \frac{125}{3434} a^{12} - \frac{477}{3434} a^{11} - \frac{28}{1717} a^{10} - \frac{315}{6868} a^{9} + \frac{1933}{6868} a^{8} + \frac{1489}{3434} a^{7} - \frac{3027}{6868} a^{6} + \frac{2643}{6868} a^{5} + \frac{2089}{6868} a^{4} + \frac{197}{404} a^{3} + \frac{2895}{6868} a^{2} + \frac{1443}{3434} a - \frac{395}{1717}$, $\frac{1}{119246978691402969154046126386677565246568588142219464} a^{15} - \frac{2049210182834512166267316985821952675957525710175}{119246978691402969154046126386677565246568588142219464} a^{14} + \frac{2267981176569797554678501601218307751733816798088403}{29811744672850742288511531596669391311642147035554866} a^{13} + \frac{12883244569241073107307645169020931199583978692622473}{59623489345701484577023063193338782623284294071109732} a^{12} + \frac{1970498154272881415995244738208106885492104049512278}{14905872336425371144255765798334695655821073517777433} a^{11} - \frac{14660183095609088797670209916069346481272841859891487}{119246978691402969154046126386677565246568588142219464} a^{10} - \frac{20179990190257133793071951047671753102174887895014181}{119246978691402969154046126386677565246568588142219464} a^{9} - \frac{4633240735275853000079154176037011049902294554965328}{14905872336425371144255765798334695655821073517777433} a^{8} + \frac{17661310571143043595201783621527900147727527289696593}{119246978691402969154046126386677565246568588142219464} a^{7} + \frac{8280412153459224815371366681606502875543573949995937}{119246978691402969154046126386677565246568588142219464} a^{6} + \frac{51574608966883351327222442596087336239747405821267231}{119246978691402969154046126386677565246568588142219464} a^{5} + \frac{1831530832543536350477344493096695591615365130473767}{7014528158317821714943889787451621485092269890718792} a^{4} - \frac{17649357902048480540797326436796256008470590148499691}{119246978691402969154046126386677565246568588142219464} a^{3} - \frac{7783378692633552679349914456157658532959401036795005}{29811744672850742288511531596669391311642147035554866} a^{2} - \frac{11685653376303458510881057963629894007687123688865}{147582894420053179646096691072620749067535381364133} a - \frac{204298616480008133555530553441159027244896757434}{756528058489842721628978622460269789160080876911}$

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

Class group and class number

$C_{2}\times C_{74}$, which has order $148$ (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:  \( -\frac{5982196028234438103099283120943453919873213}{295165788840106359292193382145241498135070762728266} a^{15} + \frac{2303038459360479791551991278155191625947165}{8681346730591363508593923004271808768678551844949} a^{14} - \frac{53397173065855368309403897798563952793372659}{147582894420053179646096691072620749067535381364133} a^{13} - \frac{570027512391785001266011190414155149731512343}{295165788840106359292193382145241498135070762728266} a^{12} - \frac{1744220727618072643627606293088840545395956858}{147582894420053179646096691072620749067535381364133} a^{11} + \frac{6552403694847433761659062166452814528631025565}{295165788840106359292193382145241498135070762728266} a^{10} + \frac{71360372536106119497466180086544258804572854222}{147582894420053179646096691072620749067535381364133} a^{9} + \frac{108562081594727880154674942849575658545811978868}{147582894420053179646096691072620749067535381364133} a^{8} - \frac{98265551003440584831676956667089185458423935563}{8681346730591363508593923004271808768678551844949} a^{7} - \frac{2161998195249059060327984560621264697887191759894}{147582894420053179646096691072620749067535381364133} a^{6} + \frac{32505967888688948609844637604473416430943738300043}{295165788840106359292193382145241498135070762728266} a^{5} + \frac{35744168158983109572752470824982736686535016967839}{295165788840106359292193382145241498135070762728266} a^{4} - \frac{155061671848828763118921711888951190021318304365625}{295165788840106359292193382145241498135070762728266} a^{3} + \frac{9635786861809997863181515086988366030490173233851}{147582894420053179646096691072620749067535381364133} a^{2} - \frac{24972366873777950664956511190789283556410903259775}{295165788840106359292193382145241498135070762728266} a - \frac{111965119672874336973007233813434372945618232809}{127336405884428972947451847344797885304172028787} \) (order $10$)
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:  \( 505489.246116 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, \(\Q(\zeta_{5})\), 4.0.11125.1, 8.0.3490037155625.1, 8.8.87250928890625.1, 8.0.123765625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$89$89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$