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Properties

Label 16.16.1709204261...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{24}\cdot 5^{12}\cdot 29^{6}\cdot 89^{6}\cdot 109^{4}$
Root discriminant $581.50$
Ramified primes $2, 5, 29, 89, 109$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T790

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-171107553307089379, -127514159553381278, -18027969452365224, 5554771415609822, 1300696765087662, -64296287880780, -24824036076810, 244194385350, 206515921983, 185218176, -875311194, -3143498, 1973254, 6482, -2242, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 2242*x^14 + 6482*x^13 + 1973254*x^12 - 3143498*x^11 - 875311194*x^10 + 185218176*x^9 + 206515921983*x^8 + 244194385350*x^7 - 24824036076810*x^6 - 64296287880780*x^5 + 1300696765087662*x^4 + 5554771415609822*x^3 - 18027969452365224*x^2 - 127514159553381278*x - 171107553307089379)
 
gp: K = bnfinit(x^16 - 4*x^15 - 2242*x^14 + 6482*x^13 + 1973254*x^12 - 3143498*x^11 - 875311194*x^10 + 185218176*x^9 + 206515921983*x^8 + 244194385350*x^7 - 24824036076810*x^6 - 64296287880780*x^5 + 1300696765087662*x^4 + 5554771415609822*x^3 - 18027969452365224*x^2 - 127514159553381278*x - 171107553307089379, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 2242 x^{14} + 6482 x^{13} + 1973254 x^{12} - 3143498 x^{11} - 875311194 x^{10} + 185218176 x^{9} + 206515921983 x^{8} + 244194385350 x^{7} - 24824036076810 x^{6} - 64296287880780 x^{5} + 1300696765087662 x^{4} + 5554771415609822 x^{3} - 18027969452365224 x^{2} - 127514159553381278 x - 171107553307089379 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(170920426163982849921157269262336000000000000=2^{24}\cdot 5^{12}\cdot 29^{6}\cdot 89^{6}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $581.50$
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:  $2, 5, 29, 89, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{58} a^{12} - \frac{7}{29} a^{11} - \frac{4}{29} a^{10} - \frac{4}{29} a^{9} + \frac{7}{29} a^{8} + \frac{11}{29} a^{7} + \frac{3}{29} a^{6} - \frac{11}{29} a^{5} - \frac{10}{29} a^{3} - \frac{12}{29} a^{2} - \frac{6}{29} a - \frac{1}{2}$, $\frac{1}{58} a^{13} - \frac{1}{58} a^{11} - \frac{2}{29} a^{10} - \frac{11}{58} a^{9} - \frac{7}{29} a^{8} - \frac{5}{58} a^{7} + \frac{2}{29} a^{6} + \frac{11}{58} a^{5} - \frac{10}{29} a^{4} + \frac{15}{58} a^{3} + \frac{3}{29} a$, $\frac{1}{134456122} a^{14} + \frac{477712}{67228061} a^{13} - \frac{427513}{67228061} a^{12} + \frac{25538615}{134456122} a^{11} + \frac{9117309}{134456122} a^{10} + \frac{20087583}{134456122} a^{9} - \frac{16621800}{67228061} a^{8} - \frac{22638293}{134456122} a^{7} + \frac{20832245}{134456122} a^{6} - \frac{38619411}{134456122} a^{5} - \frac{12703067}{67228061} a^{4} + \frac{6443517}{134456122} a^{3} + \frac{5790446}{67228061} a^{2} - \frac{18628545}{134456122} a - \frac{986089}{2318209}$, $\frac{1}{82369904215157210430981857513434904522192710954401607912453284056036543039190728073996239519182892556718} a^{15} - \frac{74149998677420550053279079833030899822421514383471263292934813231419287158011336153920866794207}{82369904215157210430981857513434904522192710954401607912453284056036543039190728073996239519182892556718} a^{14} + \frac{76087528352763086453519113923314552058799671371916077632171214282658332449628039242114638730864979459}{41184952107578605215490928756717452261096355477200803956226642028018271519595364036998119759591446278359} a^{13} - \frac{8420195794697917858009090382530098520018141153357216808306106827843083920715460203163165465526241055}{1420170762330296731568652715748877664175736395765544964007815242345457638606736690930969646882463664771} a^{12} - \frac{9797641027953339866008648681727697915407639984625821257936546017746426200905641549992873248703472895215}{82369904215157210430981857513434904522192710954401607912453284056036543039190728073996239519182892556718} a^{11} + \frac{941272490594538176498425655763731345208896259541008232722185332396299917006381073172057545680460780399}{41184952107578605215490928756717452261096355477200803956226642028018271519595364036998119759591446278359} a^{10} + \frac{4183930060255738288712760035470598929766681355454429637135631386794882630618089079085214646752744110601}{82369904215157210430981857513434904522192710954401607912453284056036543039190728073996239519182892556718} a^{9} - \frac{20211138703718267299125807286663331784834179477113710877160280602130960863020308209816657510176840618603}{82369904215157210430981857513434904522192710954401607912453284056036543039190728073996239519182892556718} a^{8} + \frac{6147326803033543797602543710606224954723627310375655385117246592668924084638213519497143693010263288339}{82369904215157210430981857513434904522192710954401607912453284056036543039190728073996239519182892556718} a^{7} - \frac{6925160227754518277960038391377170719403072445294038545271034813291552506132187453585397022224770917164}{41184952107578605215490928756717452261096355477200803956226642028018271519595364036998119759591446278359} a^{6} - \frac{7404241080750713735315435133137767368794675285758526633173250917686668297126730172421671083279463088957}{82369904215157210430981857513434904522192710954401607912453284056036543039190728073996239519182892556718} a^{5} - \frac{37844285785196536308784240682455864916489836160179107043724811908600795488107204275293964115806625197993}{82369904215157210430981857513434904522192710954401607912453284056036543039190728073996239519182892556718} a^{4} + \frac{13270501898310828523656771557246484632128920085660671457295348934954271292042634816952070553872234873552}{41184952107578605215490928756717452261096355477200803956226642028018271519595364036998119759591446278359} a^{3} + \frac{23757634472194138916694987985698265720764020507649521520018550455369453333560070386304906111284481072221}{82369904215157210430981857513434904522192710954401607912453284056036543039190728073996239519182892556718} a^{2} + \frac{37277184758521445974643734958491325366564142819944250372476864820161195832920182539351767159645209888227}{82369904215157210430981857513434904522192710954401607912453284056036543039190728073996239519182892556718} a + \frac{506626703674466131066804720547417722464622231616150315733817817389409306919064634367820511950918568389}{2840341524660593463137305431497755328351472791531089928015630484690915277213473381861939293764927329542}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  $15$
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:  \( 7300222779600000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T790:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n790 are not computed
Character table for t16n790 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5162000.1, 4.4.2225.1, 4.4.58000.1, 8.8.26646244000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
5Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$