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Properties

Label 16.0.61168212980...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{12}\cdot 11^{8}\cdot 571^{4}$
Root discriminant $306.66$
Ramified primes $2, 5, 11, 571$
Class number $212713984$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 1661828]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T467)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11940059568116, -9033027884240, 6946284261904, -766557135840, 1004385747460, -23051035280, 63031672656, -364995040, 2023819954, -5455480, 33633576, -31440, 272760, -340, 924, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 924*x^14 - 340*x^13 + 272760*x^12 - 31440*x^11 + 33633576*x^10 - 5455480*x^9 + 2023819954*x^8 - 364995040*x^7 + 63031672656*x^6 - 23051035280*x^5 + 1004385747460*x^4 - 766557135840*x^3 + 6946284261904*x^2 - 9033027884240*x + 11940059568116)
 
gp: K = bnfinit(x^16 + 924*x^14 - 340*x^13 + 272760*x^12 - 31440*x^11 + 33633576*x^10 - 5455480*x^9 + 2023819954*x^8 - 364995040*x^7 + 63031672656*x^6 - 23051035280*x^5 + 1004385747460*x^4 - 766557135840*x^3 + 6946284261904*x^2 - 9033027884240*x + 11940059568116, 1)
 

Normalized defining polynomial

\( x^{16} + 924 x^{14} - 340 x^{13} + 272760 x^{12} - 31440 x^{11} + 33633576 x^{10} - 5455480 x^{9} + 2023819954 x^{8} - 364995040 x^{7} + 63031672656 x^{6} - 23051035280 x^{5} + 1004385747460 x^{4} - 766557135840 x^{3} + 6946284261904 x^{2} - 9033027884240 x + 11940059568116 \)

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:  \(6116821298063232221593993216000000000000=2^{40}\cdot 5^{12}\cdot 11^{8}\cdot 571^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $306.66$
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:  $2, 5, 11, 571$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{1142} a^{12} - \frac{267}{1142} a^{11} + \frac{27}{1142} a^{10} - \frac{110}{571} a^{9} - \frac{133}{1142} a^{8} - \frac{161}{571} a^{7} + \frac{37}{571} a^{6} - \frac{93}{571} a^{5} - \frac{63}{571} a^{4} + \frac{227}{571} a^{3} + \frac{181}{571} a^{2} + \frac{248}{571} a + \frac{275}{571}$, $\frac{1}{12562} a^{13} - \frac{3}{12562} a^{12} + \frac{1028}{6281} a^{11} + \frac{57}{1142} a^{10} + \frac{300}{6281} a^{9} - \frac{587}{6281} a^{8} + \frac{2071}{6281} a^{7} - \frac{603}{6281} a^{6} + \frac{1080}{6281} a^{5} + \frac{1867}{6281} a^{4} + \frac{2438}{6281} a^{3} + \frac{2923}{6281} a^{2} - \frac{489}{6281} a - \frac{2201}{6281}$, $\frac{1}{12562} a^{14} + \frac{1}{12562} a^{12} + \frac{349}{12562} a^{11} - \frac{2513}{12562} a^{10} - \frac{743}{6281} a^{9} + \frac{2655}{12562} a^{8} + \frac{193}{571} a^{7} - \frac{1059}{6281} a^{6} + \frac{674}{6281} a^{5} - \frac{1245}{6281} a^{4} - \frac{1973}{6281} a^{3} + \frac{2252}{6281} a^{2} - \frac{2315}{6281} a + \frac{2318}{6281}$, $\frac{1}{840169337895881014883645547475213033941136548567248164466837846222252183390372641471142} a^{15} + \frac{88333505186147543161111433628599769845815838563301031700024578153526250400323513}{38189515358903682494711161248873319724597115843965825657583538464647826517744210975961} a^{14} - \frac{2611147888712387522865279079006245989076027809613918867561847410858653151347200583}{420084668947940507441822773737606516970568274283624082233418923111126091695186320735571} a^{13} - \frac{1985845024433559039897763144146988519786433450125424345261406823445108302192278490}{420084668947940507441822773737606516970568274283624082233418923111126091695186320735571} a^{12} + \frac{7320451260842470556886719446421365181214083102720275176467963921631918010719795968403}{76379030717807364989422322497746639449194231687931651315167076929295653035488421951922} a^{11} + \frac{35117942807230672252417454216186679383634874256815280127525243224080506992399284187527}{840169337895881014883645547475213033941136548567248164466837846222252183390372641471142} a^{10} - \frac{162635455199626655097603580437353688853517346207686891150194751747328260751952787365385}{840169337895881014883645547475213033941136548567248164466837846222252183390372641471142} a^{9} - \frac{100777752481934204746007077459559907545885097058815878335217736729089337398952785379119}{840169337895881014883645547475213033941136548567248164466837846222252183390372641471142} a^{8} - \frac{63403342191469862852667657305445406912785260042527543950841576434369033527988887448681}{420084668947940507441822773737606516970568274283624082233418923111126091695186320735571} a^{7} + \frac{143866848667050402166756815383884603455021884994840215980833517417932406117969234063746}{420084668947940507441822773737606516970568274283624082233418923111126091695186320735571} a^{6} + \frac{139586740286393865622452335729115053566013803330309697302620384062522783547804390459934}{420084668947940507441822773737606516970568274283624082233418923111126091695186320735571} a^{5} - \frac{12467074806842562254172909114705373116533157881741132867286671729950030695308757651487}{38189515358903682494711161248873319724597115843965825657583538464647826517744210975961} a^{4} - \frac{203059362853330698177240688268170604317106423837414277515085350747467636956745653876384}{420084668947940507441822773737606516970568274283624082233418923111126091695186320735571} a^{3} - \frac{87353361193788463777185134371945814806613959320639808534426233703388231401946992644941}{420084668947940507441822773737606516970568274283624082233418923111126091695186320735571} a^{2} + \frac{4174253519010685947659552399547033424307691612623764034489852479575917683370648254101}{38189515358903682494711161248873319724597115843965825657583538464647826517744210975961} a - \frac{5338453349315334627214099308497646806724395906241368837385458647389548148123078298461}{420084668947940507441822773737606516970568274283624082233418923111126091695186320735571}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1661828}$, which has order $212713984$ (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:  \( -1 \) (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:  \( 130320.792052 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T467):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.17600.1, 4.4.8000.1, 4.4.22000.1, 8.8.123904000000.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
571Data not computed