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Properties

Label 16.16.8271710657...1489.2
Degree $16$
Signature $[16, 0]$
Discriminant $17^{14}\cdot 53^{12}$
Root discriminant $234.34$
Ramified primes $17, 53$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $Q_{16}$ (as 16T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1104680656, -174707192, -4676812808, 2475352300, 2530948457, -805577043, -504661128, 73975139, 43855038, -2050498, -1712220, 6818, 32164, 351, -288, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 288*x^14 + 351*x^13 + 32164*x^12 + 6818*x^11 - 1712220*x^10 - 2050498*x^9 + 43855038*x^8 + 73975139*x^7 - 504661128*x^6 - 805577043*x^5 + 2530948457*x^4 + 2475352300*x^3 - 4676812808*x^2 - 174707192*x + 1104680656)
 
gp: K = bnfinit(x^16 - 3*x^15 - 288*x^14 + 351*x^13 + 32164*x^12 + 6818*x^11 - 1712220*x^10 - 2050498*x^9 + 43855038*x^8 + 73975139*x^7 - 504661128*x^6 - 805577043*x^5 + 2530948457*x^4 + 2475352300*x^3 - 4676812808*x^2 - 174707192*x + 1104680656, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 288 x^{14} + 351 x^{13} + 32164 x^{12} + 6818 x^{11} - 1712220 x^{10} - 2050498 x^{9} + 43855038 x^{8} + 73975139 x^{7} - 504661128 x^{6} - 805577043 x^{5} + 2530948457 x^{4} + 2475352300 x^{3} - 4676812808 x^{2} - 174707192 x + 1104680656 \)

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:  \(82717106576700383216526055870773061489=17^{14}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $234.34$
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:  $17, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{12} - \frac{3}{20} a^{11} + \frac{1}{10} a^{10} + \frac{3}{20} a^{9} + \frac{3}{10} a^{8} + \frac{1}{20} a^{7} - \frac{1}{10} a^{6} + \frac{1}{4} a^{5} - \frac{1}{20} a^{4} - \frac{3}{20} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{11800} a^{14} + \frac{6}{295} a^{13} + \frac{119}{2950} a^{12} + \frac{2379}{11800} a^{11} - \frac{79}{2360} a^{10} - \frac{5091}{11800} a^{9} - \frac{3313}{11800} a^{8} + \frac{39}{11800} a^{7} - \frac{1057}{11800} a^{6} + \frac{251}{2950} a^{5} + \frac{642}{1475} a^{4} + \frac{4581}{11800} a^{3} + \frac{3}{1475} a^{2} - \frac{438}{1475} a - \frac{428}{1475}$, $\frac{1}{255761287648858184865675965990410247573015833823922963179041000} a^{15} + \frac{7252525927889299826219604501923435480658458293414029298611}{255761287648858184865675965990410247573015833823922963179041000} a^{14} + \frac{163994284870570811324891406315332084813273979473779867783202}{31970160956107273108209495748801280946626979227990370397380125} a^{13} + \frac{505195798540041570858613730732336568787445680176321859416583}{10230451505954327394627038639616409902920633352956918527161640} a^{12} + \frac{26247281076607544835635968469675061704112895589376332109091357}{127880643824429092432837982995205123786507916911961481589520500} a^{11} - \frac{15534533528049949913263432217339116172574562518461137153582759}{63940321912214546216418991497602561893253958455980740794760250} a^{10} + \frac{18209720572015080904743810825129122084440999008638560688631463}{127880643824429092432837982995205123786507916911961481589520500} a^{9} + \frac{14151036757238053792279874484474673196222167261634927180390852}{31970160956107273108209495748801280946626979227990370397380125} a^{8} + \frac{9960497964714486422874429285377262832610672518462060943375453}{63940321912214546216418991497602561893253958455980740794760250} a^{7} - \frac{18884261281585324861653603848075687869574828539498489665805043}{255761287648858184865675965990410247573015833823922963179041000} a^{6} + \frac{4689704233735410011366412073519348916719618279082339808679511}{12788064382442909243283798299520512378650791691196148158952050} a^{5} - \frac{27495017877372886477435630320895706635937369405102539224325963}{255761287648858184865675965990410247573015833823922963179041000} a^{4} - \frac{2194032408558700517232041427786174980728896189330035967097493}{10230451505954327394627038639616409902920633352956918527161640} a^{3} - \frac{18988234162451825829825181633435107850224715691847412049374}{255761287648858184865675965990410247573015833823922963179041} a^{2} - \frac{25416206034176036922859451085498550406773038384237908791583077}{63940321912214546216418991497602561893253958455980740794760250} a - \frac{5834250689094094967337233128641496628967148922485884922211163}{31970160956107273108209495748801280946626979227990370397380125}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 11079174313200 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_{16}$ (as 16T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $Q_{16}$
Character table for $Q_{16}$

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{901}) \), \(\Q(\sqrt{17}, \sqrt{53})\), 4.4.260389.1 x2, 4.4.13800617.1 x2, 8.8.190457029580689.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$
$53$53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} - 53$$4$$1$$3$$C_4$$[\ ]_{4}$