↑

Properties

Label 16.0.24968628224...2241.6
Degree $16$
Signature $[0, 8]$
Discriminant $37^{12}\cdot 41^{14}$
Root discriminant $386.67$
Ramified primes $37, 41$
Class number $4672$ (GRH)
Class group $[2, 2, 2, 584]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16797845967299, -11366868502319, 4687448410199, -1573313537696, 431348541875, -93432130616, 17803648743, -2767976084, 379919624, -45946626, 4721837, -486437, 39881, -3065, 218, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 218*x^14 - 3065*x^13 + 39881*x^12 - 486437*x^11 + 4721837*x^10 - 45946626*x^9 + 379919624*x^8 - 2767976084*x^7 + 17803648743*x^6 - 93432130616*x^5 + 431348541875*x^4 - 1573313537696*x^3 + 4687448410199*x^2 - 11366868502319*x + 16797845967299)
 
gp: K = bnfinit(x^16 - 7*x^15 + 218*x^14 - 3065*x^13 + 39881*x^12 - 486437*x^11 + 4721837*x^10 - 45946626*x^9 + 379919624*x^8 - 2767976084*x^7 + 17803648743*x^6 - 93432130616*x^5 + 431348541875*x^4 - 1573313537696*x^3 + 4687448410199*x^2 - 11366868502319*x + 16797845967299, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 218 x^{14} - 3065 x^{13} + 39881 x^{12} - 486437 x^{11} + 4721837 x^{10} - 45946626 x^{9} + 379919624 x^{8} - 2767976084 x^{7} + 17803648743 x^{6} - 93432130616 x^{5} + 431348541875 x^{4} - 1573313537696 x^{3} + 4687448410199 x^{2} - 11366868502319 x + 16797845967299 \)

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:  \(249686282242731792647174886492450339042241=37^{12}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $386.67$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{124} a^{13} + \frac{3}{62} a^{12} - \frac{11}{124} a^{11} - \frac{3}{31} a^{10} + \frac{15}{124} a^{9} + \frac{29}{124} a^{8} - \frac{39}{124} a^{7} + \frac{15}{62} a^{6} + \frac{4}{31} a^{5} - \frac{25}{62} a^{4} - \frac{9}{31} a^{3} + \frac{1}{124} a^{2} - \frac{35}{124} a + \frac{23}{124}$, $\frac{1}{124} a^{14} + \frac{15}{124} a^{12} - \frac{2}{31} a^{11} + \frac{25}{124} a^{10} + \frac{1}{124} a^{9} - \frac{27}{124} a^{8} - \frac{23}{62} a^{7} - \frac{10}{31} a^{6} - \frac{11}{62} a^{5} - \frac{23}{62} a^{4} - \frac{1}{4} a^{3} - \frac{41}{124} a^{2} - \frac{15}{124} a + \frac{12}{31}$, $\frac{1}{3409883070326095821907123173238155556025915153680484909924145125345527552289681531116} a^{15} - \frac{4647329704954327096561029683488136434978879840328115113652793184353840947765494057}{1704941535163047910953561586619077778012957576840242454962072562672763776144840765558} a^{14} + \frac{56574813313244485687575181851562970330704483099551756994097317343033851840979653}{31868066077813979643991805357365939775943132277387709438543412386406799554109173188} a^{13} - \frac{181587478948304740656852936324617428634779594281319053813542961444386248979711835391}{1704941535163047910953561586619077778012957576840242454962072562672763776144840765558} a^{12} + \frac{190133783943227396285760919131687414710479100683526032194653471361600673904003086853}{3409883070326095821907123173238155556025915153680484909924145125345527552289681531116} a^{11} - \frac{606104010610883463926596711868770273288384481230553782957891596612365476095062714875}{3409883070326095821907123173238155556025915153680484909924145125345527552289681531116} a^{10} - \frac{635371363819115075642836112124907073483486539799630938871951755388600116986276483625}{3409883070326095821907123173238155556025915153680484909924145125345527552289681531116} a^{9} + \frac{142606122226660710027681822960043487206171540270079090100264519607153050602216982669}{852470767581523955476780793309538889006478788420121227481036281336381888072420382779} a^{8} + \frac{73647074051364520807486679452270940453602201079650406793046952005844691073384606193}{852470767581523955476780793309538889006478788420121227481036281336381888072420382779} a^{7} + \frac{842696755076341319439638733398660553941865263771783559180376324041749730218201741887}{1704941535163047910953561586619077778012957576840242454962072562672763776144840765558} a^{6} + \frac{12581622264335565999568045405373261261292480962259990355006411081901682757223718046}{27499057018758837273444541719662544806660606078068426692936654236657480260400657509} a^{5} - \frac{1349934614804243621221999684392341173209659823091505762108649258572253294410710929243}{3409883070326095821907123173238155556025915153680484909924145125345527552289681531116} a^{4} + \frac{412917129541394315801247667224222637725002582475703943723181056314348658398333606377}{3409883070326095821907123173238155556025915153680484909924145125345527552289681531116} a^{3} + \frac{633420144782671210505975636746414497452305133663779820999027505285057668840343667087}{3409883070326095821907123173238155556025915153680484909924145125345527552289681531116} a^{2} - \frac{78216710213574290011604170533834898464803859663498139933138293356482040657643173521}{1704941535163047910953561586619077778012957576840242454962072562672763776144840765558} a + \frac{85174144143345991261172727099495745078598994662717895033806934497675115989012259}{893107142568385495523080977799412141442094068538628839686784998780913450049680862}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{584}$, which has order $4672$ (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:  \( 31435858697.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.0.365000864691088841.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
37Data not computed
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$