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Properties

Label 16.0.923...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $9.234\times 10^{18}$
Root discriminant \(15.32\)
Ramified primes $3,5,7$
Class number $1$
Class group trivial
Galois group $C_4\times C_2^2$ (as 16T2)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 2*x^14 - 5*x^13 + 5*x^12 + x^11 + 6*x^10 + 5*x^9 - 21*x^8 + 10*x^7 + 24*x^6 + 8*x^5 + 80*x^4 - 160*x^3 + 128*x^2 - 128*x + 256)
 
Copy content gp:K = bnfinit(y^16 - y^15 + 2*y^14 - 5*y^13 + 5*y^12 + y^11 + 6*y^10 + 5*y^9 - 21*y^8 + 10*y^7 + 24*y^6 + 8*y^5 + 80*y^4 - 160*y^3 + 128*y^2 - 128*y + 256, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 2*x^14 - 5*x^13 + 5*x^12 + x^11 + 6*x^10 + 5*x^9 - 21*x^8 + 10*x^7 + 24*x^6 + 8*x^5 + 80*x^4 - 160*x^3 + 128*x^2 - 128*x + 256);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 + 2*x^14 - 5*x^13 + 5*x^12 + x^11 + 6*x^10 + 5*x^9 - 21*x^8 + 10*x^7 + 24*x^6 + 8*x^5 + 80*x^4 - 160*x^3 + 128*x^2 - 128*x + 256)
 

\( x^{16} - x^{15} + 2 x^{14} - 5 x^{13} + 5 x^{12} + x^{11} + 6 x^{10} + 5 x^{9} - 21 x^{8} + 10 x^{7} + \cdots + 256 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[0, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(9234096523681640625\) \(\medspace = 3^{8}\cdot 5^{12}\cdot 7^{8}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(15.32\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}5^{3/4}7^{1/2}\approx 15.322765339111537$
Ramified primes:   \(3\), \(5\), \(7\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_2^2\times C_4$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(105=3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(71,·)$, $\chi_{105}(8,·)$, $\chi_{105}(76,·)$, $\chi_{105}(13,·)$, $\chi_{105}(83,·)$, $\chi_{105}(22,·)$, $\chi_{105}(92,·)$, $\chi_{105}(29,·)$, $\chi_{105}(97,·)$, $\chi_{105}(34,·)$, $\chi_{105}(104,·)$, $\chi_{105}(41,·)$, $\chi_{105}(43,·)$, $\chi_{105}(62,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{8}a^{8}-\frac{3}{8}a^{7}-\frac{3}{8}a^{6}+\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{1}{8}a^{10}-\frac{1}{16}a^{9}-\frac{3}{16}a^{8}+\frac{5}{16}a^{7}+\frac{1}{8}a^{6}+\frac{1}{16}a^{5}+\frac{3}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{2848}a^{13}-\frac{73}{2848}a^{12}-\frac{51}{1424}a^{11}+\frac{27}{2848}a^{10}+\frac{13}{2848}a^{9}+\frac{777}{2848}a^{8}+\frac{151}{1424}a^{7}-\frac{1083}{2848}a^{6}-\frac{221}{2848}a^{5}+\frac{561}{1424}a^{4}-\frac{37}{356}a^{3}-\frac{27}{356}a^{2}-\frac{1}{178}a-\frac{39}{89}$, $\frac{1}{5696}a^{14}-\frac{1}{5696}a^{13}-\frac{9}{2848}a^{12}+\frac{159}{5696}a^{11}-\frac{179}{5696}a^{10}+\frac{645}{5696}a^{9}-\frac{535}{2848}a^{8}+\frac{1793}{5696}a^{7}+\frac{835}{5696}a^{6}+\frac{259}{2848}a^{5}+\frac{275}{1424}a^{4}+\frac{123}{356}a^{3}-\frac{83}{356}a^{2}+\frac{7}{89}a+\frac{20}{89}$, $\frac{1}{11392}a^{15}-\frac{1}{11392}a^{14}-\frac{1}{5696}a^{13}-\frac{297}{11392}a^{12}+\frac{325}{11392}a^{11}-\frac{347}{11392}a^{10}+\frac{637}{5696}a^{9}-\frac{2151}{11392}a^{8}+\frac{3531}{11392}a^{7}+\frac{851}{5696}a^{6}+\frac{281}{2848}a^{5}+\frac{265}{1424}a^{4}+\frac{61}{178}a^{3}-\frac{47}{178}a^{2}+\frac{8}{89}a+\frac{22}{89}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  Trivial group, which has order $1$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   $1$

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $7$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( \frac{1}{356} a^{15} - \frac{3}{356} a^{14} + \frac{3}{178} a^{12} + \frac{7}{356} a^{11} + \frac{11}{356} a^{10} - \frac{4}{89} a^{9} - \frac{11}{356} a^{8} + \frac{17}{178} a^{7} + \frac{8}{89} a^{6} + \frac{22}{89} a^{5} - \frac{20}{89} a^{4} - \frac{8}{89} a^{3} + \frac{271}{356} a^{2} + \frac{32}{89} a + \frac{64}{89} \)  (order $30$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{71}{5696}a^{15}-\frac{47}{2848}a^{14}+\frac{187}{5696}a^{13}-\frac{291}{5696}a^{12}+\frac{155}{2848}a^{11}-\frac{77}{2848}a^{10}-\frac{147}{5696}a^{9}+\frac{715}{5696}a^{8}-\frac{363}{2848}a^{7}+\frac{215}{5696}a^{6}+\frac{33}{712}a^{5}-\frac{219}{356}a^{4}+\frac{1025}{712}a^{3}-\frac{110}{89}a^{2}+\frac{142}{89}a-\frac{188}{89}$, $\frac{39}{5696}a^{15}-\frac{51}{5696}a^{14}-\frac{3}{712}a^{13}-\frac{193}{5696}a^{12}+\frac{75}{5696}a^{11}+\frac{429}{5696}a^{10}-\frac{57}{1424}a^{9}-\frac{679}{5696}a^{8}-\frac{1419}{5696}a^{7}+\frac{3}{89}a^{6}+\frac{1353}{2848}a^{5}-\frac{219}{1424}a^{4}-\frac{159}{712}a^{3}-\frac{495}{356}a^{2}-\frac{9}{178}a+\frac{34}{89}$, $\frac{1}{11392}a^{15}-\frac{135}{11392}a^{14}-\frac{353}{11392}a^{12}+\frac{315}{11392}a^{11}+\frac{139}{11392}a^{10}-\frac{45}{712}a^{9}-\frac{495}{11392}a^{8}-\frac{2475}{11392}a^{7}+\frac{45}{356}a^{6}+\frac{11}{2848}a^{5}-\frac{225}{712}a^{4}-\frac{45}{356}a^{3}-\frac{495}{356}a^{2}+\frac{45}{89}a-\frac{31}{89}$, $\frac{49}{5696}a^{15}-\frac{117}{5696}a^{14}+\frac{49}{2848}a^{13}-\frac{245}{5696}a^{12}+\frac{245}{5696}a^{11}+\frac{49}{5696}a^{10}-\frac{387}{2848}a^{9}+\frac{245}{5696}a^{8}-\frac{1029}{5696}a^{7}+\frac{245}{2848}a^{6}+\frac{147}{712}a^{5}-\frac{1415}{1424}a^{4}+\frac{245}{356}a^{3}-\frac{245}{178}a^{2}+\frac{98}{89}a-\frac{98}{89}$, $\frac{1}{11392}a^{15}-\frac{3}{11392}a^{14}+\frac{13}{2848}a^{13}+\frac{215}{11392}a^{12}+\frac{399}{11392}a^{11}-\frac{9}{11392}a^{10}-\frac{191}{2848}a^{9}+\frac{521}{11392}a^{8}+\frac{1409}{11392}a^{7}+\frac{525}{2848}a^{6}+\frac{353}{2848}a^{5}-\frac{549}{1424}a^{4}+\frac{15}{712}a^{3}+\frac{261}{356}a^{2}+\frac{167}{178}a+\frac{29}{89}$, $\frac{31}{11392}a^{15}-\frac{9}{11392}a^{14}+\frac{1}{89}a^{13}-\frac{159}{11392}a^{12}-\frac{219}{11392}a^{11}-\frac{427}{11392}a^{10}+\frac{3}{178}a^{9}+\frac{1167}{11392}a^{8}+\frac{43}{11392}a^{7}-\frac{57}{356}a^{6}-\frac{331}{2848}a^{5}+\frac{75}{356}a^{4}+\frac{237}{356}a^{3}-\frac{24}{89}a^{2}-\frac{13}{89}a-\frac{97}{89}$, $\frac{261}{11392}a^{15}+\frac{103}{11392}a^{14}+\frac{161}{5696}a^{13}-\frac{1373}{11392}a^{12}-\frac{451}{11392}a^{11}-\frac{643}{11392}a^{10}+\frac{1003}{5696}a^{9}+\frac{1965}{11392}a^{8}-\frac{6125}{11392}a^{7}-\frac{2371}{5696}a^{6}-\frac{375}{2848}a^{5}+\frac{633}{1424}a^{4}+\frac{601}{356}a^{3}-\frac{579}{178}a^{2}-\frac{253}{178}a-\frac{527}{89}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 7204.63964805 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 7204.63964805 \cdot 1}{30\cdot\sqrt{9234096523681640625}}\cr\approx \mathstrut & 0.191969728614 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 2*x^14 - 5*x^13 + 5*x^12 + x^11 + 6*x^10 + 5*x^9 - 21*x^8 + 10*x^7 + 24*x^6 + 8*x^5 + 80*x^4 - 160*x^3 + 128*x^2 - 128*x + 256) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - x^15 + 2*x^14 - 5*x^13 + 5*x^12 + x^11 + 6*x^10 + 5*x^9 - 21*x^8 + 10*x^7 + 24*x^6 + 8*x^5 + 80*x^4 - 160*x^3 + 128*x^2 - 128*x + 256, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 2*x^14 - 5*x^13 + 5*x^12 + x^11 + 6*x^10 + 5*x^9 - 21*x^8 + 10*x^7 + 24*x^6 + 8*x^5 + 80*x^4 - 160*x^3 + 128*x^2 - 128*x + 256); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 + 2*x^14 - 5*x^13 + 5*x^12 + x^11 + 6*x^10 + 5*x^9 - 21*x^8 + 10*x^7 + 24*x^6 + 8*x^5 + 80*x^4 - 160*x^3 + 128*x^2 - 128*x + 256); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^2\times C_4$ (as 16T2):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), 4.4.6125.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), 4.0.55125.1, 8.0.121550625.1, 8.0.37515625.1, 8.0.3038765625.2, 8.8.3038765625.1, 8.0.3038765625.3, 8.0.3038765625.1, \(\Q(\zeta_{15})\)

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ R R R ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(7\) Copy content Toggle raw display 7.4.2.4a1.2$x^{8} + 10 x^{6} + 8 x^{5} + 31 x^{4} + 40 x^{3} + 46 x^{2} + 24 x + 16$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
7.4.2.4a1.2$x^{8} + 10 x^{6} + 8 x^{5} + 31 x^{4} + 40 x^{3} + 46 x^{2} + 24 x + 16$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)