Properties

Label 16.0.536...753.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.364\times 10^{26}$
Root discriminant $46.84$
Ramified primes $3, 13, 17$
Class number $968$ (GRH)
Class group $[2, 22, 22]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 60*x^14 - 197*x^13 + 1713*x^12 - 4867*x^11 + 24121*x^10 - 51334*x^9 + 156212*x^8 - 228630*x^7 + 409997*x^6 - 399031*x^5 + 583274*x^4 - 446279*x^3 + 613192*x^2 - 266447*x + 270301)
 
gp: K = bnfinit(x^16 - 3*x^15 + 60*x^14 - 197*x^13 + 1713*x^12 - 4867*x^11 + 24121*x^10 - 51334*x^9 + 156212*x^8 - 228630*x^7 + 409997*x^6 - 399031*x^5 + 583274*x^4 - 446279*x^3 + 613192*x^2 - 266447*x + 270301, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![270301, -266447, 613192, -446279, 583274, -399031, 409997, -228630, 156212, -51334, 24121, -4867, 1713, -197, 60, -3, 1]);
 

\( x^{16} - 3 x^{15} + 60 x^{14} - 197 x^{13} + 1713 x^{12} - 4867 x^{11} + 24121 x^{10} - 51334 x^{9} + 156212 x^{8} - 228630 x^{7} + 409997 x^{6} - 399031 x^{5} + 583274 x^{4} - 446279 x^{3} + 613192 x^{2} - 266447 x + 270301 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(536385794583341500395870753\)\(\medspace = 3^{8}\cdot 13^{4}\cdot 17^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $46.84$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 13, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{101} a^{14} - \frac{11}{101} a^{13} + \frac{48}{101} a^{12} + \frac{14}{101} a^{11} + \frac{33}{101} a^{10} + \frac{34}{101} a^{9} + \frac{46}{101} a^{8} + \frac{44}{101} a^{7} - \frac{38}{101} a^{6} - \frac{22}{101} a^{5} - \frac{26}{101} a^{4} + \frac{4}{101} a^{3} + \frac{42}{101} a^{2} + \frac{11}{101} a - \frac{25}{101}$, $\frac{1}{125423509315090144432218887554136600883266923} a^{15} + \frac{369303363002659236468895888726974915247567}{125423509315090144432218887554136600883266923} a^{14} + \frac{118849484120913970177541010132592701152286}{1241816923911783608239790965882540602804623} a^{13} + \frac{48301145581325964982816904759467839742601627}{125423509315090144432218887554136600883266923} a^{12} - \frac{24304259067579845640561771110895728998731213}{125423509315090144432218887554136600883266923} a^{11} - \frac{27936333557482929429892157955651082671572479}{125423509315090144432218887554136600883266923} a^{10} - \frac{20464447875436917029704319310205755893302027}{125423509315090144432218887554136600883266923} a^{9} - \frac{12331753022254935534302690372851211005803000}{125423509315090144432218887554136600883266923} a^{8} - \frac{50545595380026860722427286618693894678228538}{125423509315090144432218887554136600883266923} a^{7} - \frac{38642096880500439627365242578674474714783454}{125423509315090144432218887554136600883266923} a^{6} + \frac{7940917575673730540012669587150245448820399}{125423509315090144432218887554136600883266923} a^{5} - \frac{34523920849234651753203490337380680655636415}{125423509315090144432218887554136600883266923} a^{4} + \frac{32844902764217800385407504641431254774605911}{125423509315090144432218887554136600883266923} a^{3} - \frac{18716429092636652904371504942661933341076589}{125423509315090144432218887554136600883266923} a^{2} - \frac{39154170219326128950371977958594362362754388}{125423509315090144432218887554136600883266923} a - \frac{30660929900361146906543845704635874954021600}{125423509315090144432218887554136600883266923}$

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

Class group and class number

$C_{2}\times C_{22}\times C_{22}$, which has order $968$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3640.01221338 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 968}{2\sqrt{536385794583341500395870753}}\approx 0.184777256899$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
17Data not computed