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Properties

Label 16.8.18738121436...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{10}\cdot 11^{2}\cdot 71^{8}\cdot 101^{6}\cdot 1579^{2}\cdot 4159^{2}\cdot 17881^{2}$
Root discriminant $12{,}010.07$
Ramified primes $2, 5, 11, 71, 101, 1579, 4159, 17881$
Class number Not computed
Class group Not computed
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43298734145286098398063387496619025, 0, 82597434109237025018114418481500, 0, -4195653086671836716753682900, 0, -403100026764513865076870, 0, 12945257956415288274, 0, 549770950990792, 0, -8522437947, 0, -86874, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 86874*x^14 - 8522437947*x^12 + 549770950990792*x^10 + 12945257956415288274*x^8 - 403100026764513865076870*x^6 - 4195653086671836716753682900*x^4 + 82597434109237025018114418481500*x^2 + 43298734145286098398063387496619025)
 
gp: K = bnfinit(x^16 - 86874*x^14 - 8522437947*x^12 + 549770950990792*x^10 + 12945257956415288274*x^8 - 403100026764513865076870*x^6 - 4195653086671836716753682900*x^4 + 82597434109237025018114418481500*x^2 + 43298734145286098398063387496619025, 1)
 

Normalized defining polynomial

\( x^{16} - 86874 x^{14} - 8522437947 x^{12} + 549770950990792 x^{10} + 12945257956415288274 x^{8} - 403100026764513865076870 x^{6} - 4195653086671836716753682900 x^{4} + 82597434109237025018114418481500 x^{2} + 43298734145286098398063387496619025 \)

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:  \(187381214367639282787301532684742893490174895091410698240000000000=2^{24}\cdot 5^{10}\cdot 11^{2}\cdot 71^{8}\cdot 101^{6}\cdot 1579^{2}\cdot 4159^{2}\cdot 17881^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12{,}010.07$
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:  $2, 5, 11, 71, 101, 1579, 4159, 17881$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} + \frac{1}{6} a$, $\frac{1}{1980} a^{12} - \frac{1}{12} a^{11} - \frac{29}{1980} a^{10} - \frac{1}{4} a^{9} + \frac{17}{220} a^{8} + \frac{103}{220} a^{6} - \frac{7}{660} a^{4} - \frac{1}{4} a^{3} + \frac{26}{99} a^{2} - \frac{1}{3} a - \frac{11}{36}$, $\frac{1}{631620} a^{13} + \frac{2344}{157905} a^{11} - \frac{4217}{17545} a^{9} - \frac{1}{4} a^{8} - \frac{15517}{70180} a^{7} + \frac{1}{4} a^{6} + \frac{44213}{210540} a^{5} + \frac{1195}{4356} a^{3} - \frac{839}{11484} a + \frac{1}{4}$, $\frac{1}{248351898730958082116597432324116233072506659528548347069311886747972201393818003890174475695040594175478366562052293260} a^{14} - \frac{29726083714568291799447476823531333597864976617394932367631304552330709970823642906400519553317334987116517682555193}{248351898730958082116597432324116233072506659528548347069311886747972201393818003890174475695040594175478366562052293260} a^{12} - \frac{1}{12} a^{11} - \frac{14142944658011403987669622038343553045028376257309050303093946980141388137447627995639234399568338625390002308667166051}{248351898730958082116597432324116233072506659528548347069311886747972201393818003890174475695040594175478366562052293260} a^{10} - \frac{3605426664734962265425256740553587155068149994736456513696501440745999011725607932639136020170962460908224318070665}{46770602397543894937212322471584977979756433056223794175011654754797024744598494141275795799442673102726622704717946} a^{8} - \frac{1}{4} a^{7} + \frac{8486439761838218426792100072300005930763649198482165404607561610324244212684644808421825831181417162904432340260029833}{20695991560913173509716452693676352756042221627379028922442657228997683449484833657514539641253382847956530546837691105} a^{6} - \frac{893321540058587802990020907123036467513311959561297576248616884409778119670667362403500762328190360674186141820989161}{2140964644232397259625839933828588216142298789039209888528550747827346563739810378363573066336556846340330746224588735} a^{4} - \frac{1}{4} a^{3} + \frac{18643919592547845450020388373425456311827303096415479125263720535484236287314454526959028334419245055160959179832361}{44707812552827737554743012119552877240775276242762978770353174932128209071794420142245630188126119563542460227192132} a^{2} + \frac{5}{12} a + \frac{41904254320883838159893265483500478031497505737009854360918975147600374709707557563442579185182194907}{108501864109930787800333049268617823308738279975497777510595253637810895010457013329360982034038858628}$, $\frac{1}{248351898730958082116597432324116233072506659528548347069311886747972201393818003890174475695040594175478366562052293260} a^{15} + \frac{31397139975678342830939344971334912265928752443142140915239799722226742078635560340889773141062674637858509882671}{49670379746191616423319486464823246614501331905709669413862377349594440278763600778034895139008118835095673312410458652} a^{13} - \frac{751794030007638747612381966612543593242124183501759212457734415870136656415388646034016459776491155649405670655011092}{62087974682739520529149358081029058268126664882137086767327971686993050348454500972543618923760148543869591640513073315} a^{11} - \frac{1}{12} a^{10} - \frac{43938225415701635722181062344818434809452069953849710265069714831359232105409402006518072431795216270599141821299601}{467706023975438949372123224715849779797564330562237941750116547547970247445984941412757957994426731027266227047179460} a^{9} + \frac{4397218877857380553379748610414568901586225454844762422657062877625159656206689923326267386762816398076486671979927157}{41391983121826347019432905387352705512084443254758057844885314457995366898969667315029079282506765695913061093675382210} a^{7} - \frac{1}{4} a^{6} - \frac{28398808569832961112930661281873736291058871808676996374824670766428570037491898594815698354993633195814273521498799166}{62087974682739520529149358081029058268126664882137086767327971686993050348454500972543618923760148543869591640513073315} a^{5} + \frac{183527366876378142060505946167442430507740773896030137476037442378650621973169236180836934484053766178807684280274}{11176953138206934388685753029888219310193819060690744692588293733032052267948605035561407547031529890885615056798033} a^{3} - \frac{1}{4} a^{2} + \frac{15699961501016874242745904237292568222410433121278141912681582765012174165472833597373026440005071943}{34612094651067921308306242716689085635487511312183791025879885910461675508335787252066153268858395902332} a + \frac{5}{12}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{355}) \), \(\Q(\sqrt{71}) \), 4.4.203656400.1, 4.4.2525.1, \(\Q(\sqrt{5}, \sqrt{71})\), 8.8.41475929260960000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
71Data not computed
101Data not computed
1579Data not computed
4159Data not computed
17881Data not computed