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Properties

Label 21.1.207...000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2.071\times 10^{61}$
Root discriminant \(831.42\)
Ramified primes $2,3,5,7,41$
Class number not computed
Class group not computed
Galois group $S_3\times F_7$ (as 21T15)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 49*x^19 + 539*x^18 + 161*x^17 - 16051*x^16 + 38241*x^15 + 209059*x^14 - 1086554*x^13 - 407106*x^12 + 13782440*x^11 - 27422584*x^10 - 41136060*x^9 + 183654100*x^8 + 84516280*x^7 - 1385748588*x^6 + 3217907756*x^5 - 2441792668*x^4 - 4210424932*x^3 + 11145838172*x^2 - 7005312580*x + 1393873588)
 
Copy content gp:K = bnfinit(y^21 - 7*y^20 - 49*y^19 + 539*y^18 + 161*y^17 - 16051*y^16 + 38241*y^15 + 209059*y^14 - 1086554*y^13 - 407106*y^12 + 13782440*y^11 - 27422584*y^10 - 41136060*y^9 + 183654100*y^8 + 84516280*y^7 - 1385748588*y^6 + 3217907756*y^5 - 2441792668*y^4 - 4210424932*y^3 + 11145838172*y^2 - 7005312580*y + 1393873588, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 7*x^20 - 49*x^19 + 539*x^18 + 161*x^17 - 16051*x^16 + 38241*x^15 + 209059*x^14 - 1086554*x^13 - 407106*x^12 + 13782440*x^11 - 27422584*x^10 - 41136060*x^9 + 183654100*x^8 + 84516280*x^7 - 1385748588*x^6 + 3217907756*x^5 - 2441792668*x^4 - 4210424932*x^3 + 11145838172*x^2 - 7005312580*x + 1393873588);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 7*x^20 - 49*x^19 + 539*x^18 + 161*x^17 - 16051*x^16 + 38241*x^15 + 209059*x^14 - 1086554*x^13 - 407106*x^12 + 13782440*x^11 - 27422584*x^10 - 41136060*x^9 + 183654100*x^8 + 84516280*x^7 - 1385748588*x^6 + 3217907756*x^5 - 2441792668*x^4 - 4210424932*x^3 + 11145838172*x^2 - 7005312580*x + 1393873588)
 

\( x^{21} - 7 x^{20} - 49 x^{19} + 539 x^{18} + 161 x^{17} - 16051 x^{16} + 38241 x^{15} + \cdots + 1393873588 \) 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:  $21$
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:  $[1, 10]$
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:   \(20714269215444879542778783404872527378677760000000000000000000\) \(\medspace = 2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{21}\cdot 41^{7}\) 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:  \(831.42\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{27/14}3^{13/14}5^{13/14}7^{47/42}41^{1/2}\approx 2659.129372520378$
Ramified primes:   \(2\), \(3\), \(5\), \(7\), \(41\) 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:  \(\Q(\sqrt{8610}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{4}-\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{9}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{5}+\frac{2}{5}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{6}+\frac{2}{5}a$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{7}+\frac{2}{5}a^{2}$, $\frac{1}{5}a^{13}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{150}a^{14}-\frac{3}{50}a^{13}-\frac{7}{150}a^{12}-\frac{1}{50}a^{11}+\frac{3}{50}a^{10}+\frac{1}{50}a^{9}+\frac{13}{150}a^{8}-\frac{17}{50}a^{7}-\frac{32}{75}a^{6}+\frac{12}{25}a^{5}+\frac{2}{25}a^{4}-\frac{3}{25}a^{3}-\frac{37}{75}a^{2}-\frac{6}{25}a+\frac{34}{75}$, $\frac{1}{150}a^{15}+\frac{1}{75}a^{13}-\frac{1}{25}a^{12}+\frac{2}{25}a^{11}-\frac{1}{25}a^{10}+\frac{1}{15}a^{9}+\frac{1}{25}a^{8}-\frac{43}{150}a^{7}-\frac{9}{25}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{75}a^{3}-\frac{12}{25}a^{2}-\frac{23}{75}a+\frac{2}{25}$, $\frac{1}{150}a^{16}+\frac{2}{25}a^{13}-\frac{2}{75}a^{12}-\frac{4}{75}a^{10}-\frac{3}{50}a^{8}-\frac{12}{25}a^{7}+\frac{4}{75}a^{6}-\frac{4}{25}a^{5}+\frac{4}{15}a^{4}-\frac{6}{25}a^{3}+\frac{12}{25}a^{2}-\frac{1}{25}a+\frac{37}{75}$, $\frac{1}{750}a^{17}-\frac{1}{750}a^{16}-\frac{1}{750}a^{15}+\frac{2}{25}a^{13}+\frac{32}{375}a^{12}+\frac{8}{375}a^{11}-\frac{2}{375}a^{10}-\frac{11}{150}a^{9}+\frac{3}{50}a^{8}+\frac{11}{50}a^{7}+\frac{31}{75}a^{6}-\frac{29}{75}a^{5}+\frac{2}{75}a^{4}-\frac{34}{75}a^{3}-\frac{8}{125}a^{2}+\frac{3}{125}a-\frac{31}{375}$, $\frac{1}{750}a^{18}-\frac{1}{375}a^{16}-\frac{1}{750}a^{15}+\frac{32}{375}a^{13}+\frac{1}{15}a^{12}+\frac{7}{125}a^{11}+\frac{1}{750}a^{10}-\frac{4}{75}a^{9}+\frac{1}{25}a^{8}-\frac{13}{150}a^{7}+\frac{11}{75}a^{6}+\frac{12}{25}a^{5}+\frac{31}{75}a^{4}-\frac{29}{375}a^{3}+\frac{7}{25}a^{2}+\frac{158}{375}a-\frac{46}{375}$, $\frac{1}{949500}a^{19}-\frac{49}{189900}a^{18}-\frac{121}{316500}a^{17}+\frac{191}{189900}a^{16}-\frac{2729}{949500}a^{15}+\frac{283}{316500}a^{14}+\frac{6577}{189900}a^{13}-\frac{64867}{949500}a^{12}-\frac{299}{15825}a^{11}+\frac{34997}{474750}a^{10}-\frac{103}{47475}a^{9}-\frac{3127}{31650}a^{8}+\frac{12919}{47475}a^{7}+\frac{21299}{94950}a^{6}-\frac{25}{1266}a^{5}+\frac{66523}{237375}a^{4}-\frac{8602}{47475}a^{3}+\frac{7144}{26375}a^{2}-\frac{21442}{47475}a-\frac{31712}{237375}$, $\frac{1}{55\cdots 00}a^{20}+\frac{79\cdots 03}{55\cdots 00}a^{19}-\frac{21\cdots 83}{61\cdots 00}a^{18}+\frac{42\cdots 49}{55\cdots 00}a^{17}-\frac{17\cdots 69}{55\cdots 00}a^{16}+\frac{51\cdots 31}{18\cdots 00}a^{15}-\frac{11\cdots 33}{55\cdots 00}a^{14}+\frac{13\cdots 27}{55\cdots 00}a^{13}+\frac{71\cdots 41}{92\cdots 50}a^{12}-\frac{76\cdots 69}{13\cdots 75}a^{11}-\frac{13\cdots 87}{27\cdots 50}a^{10}-\frac{45\cdots 97}{61\cdots 50}a^{9}+\frac{77\cdots 69}{11\cdots 30}a^{8}-\frac{56\cdots 52}{27\cdots 75}a^{7}-\frac{88\cdots 07}{18\cdots 50}a^{6}-\frac{60\cdots 67}{13\cdots 75}a^{5}-\frac{60\cdots 86}{13\cdots 75}a^{4}-\frac{57\cdots 77}{46\cdots 25}a^{3}-\frac{69\cdots 03}{13\cdots 75}a^{2}+\frac{59\cdots 93}{13\cdots 75}a+\frac{20\cdots 29}{46\cdots 25}$ 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:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  not computed
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:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

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:  $10$
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:   \( -1 \)  (order $2$) 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:  not computed
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:  not computed
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot R \cdot h}{2\cdot\sqrt{20714269215444879542778783404872527378677760000000000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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^21 - 7*x^20 - 49*x^19 + 539*x^18 + 161*x^17 - 16051*x^16 + 38241*x^15 + 209059*x^14 - 1086554*x^13 - 407106*x^12 + 13782440*x^11 - 27422584*x^10 - 41136060*x^9 + 183654100*x^8 + 84516280*x^7 - 1385748588*x^6 + 3217907756*x^5 - 2441792668*x^4 - 4210424932*x^3 + 11145838172*x^2 - 7005312580*x + 1393873588) 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^21 - 7*x^20 - 49*x^19 + 539*x^18 + 161*x^17 - 16051*x^16 + 38241*x^15 + 209059*x^14 - 1086554*x^13 - 407106*x^12 + 13782440*x^11 - 27422584*x^10 - 41136060*x^9 + 183654100*x^8 + 84516280*x^7 - 1385748588*x^6 + 3217907756*x^5 - 2441792668*x^4 - 4210424932*x^3 + 11145838172*x^2 - 7005312580*x + 1393873588, 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^21 - 7*x^20 - 49*x^19 + 539*x^18 + 161*x^17 - 16051*x^16 + 38241*x^15 + 209059*x^14 - 1086554*x^13 - 407106*x^12 + 13782440*x^11 - 27422584*x^10 - 41136060*x^9 + 183654100*x^8 + 84516280*x^7 - 1385748588*x^6 + 3217907756*x^5 - 2441792668*x^4 - 4210424932*x^3 + 11145838172*x^2 - 7005312580*x + 1393873588); 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 = polynomial_ring(QQ); K, a = number_field(x^21 - 7*x^20 - 49*x^19 + 539*x^18 + 161*x^17 - 16051*x^16 + 38241*x^15 + 209059*x^14 - 1086554*x^13 - 407106*x^12 + 13782440*x^11 - 27422584*x^10 - 41136060*x^9 + 183654100*x^8 + 84516280*x^7 - 1385748588*x^6 + 3217907756*x^5 - 2441792668*x^4 - 4210424932*x^3 + 11145838172*x^2 - 7005312580*x + 1393873588); 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

$S_3\times F_7$ (as 21T15):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$

Intermediate fields

3.1.4920.1, 7.1.600362847000000.35

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 42 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.2.0.1}{2} }^{10}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.3.0.1}{3} }^{7}$ ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.7.0.1}{7} }$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.3.0.1}{3} }^{7}$ R $21$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.3.0.1}{3} }^{7}$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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
\(2\) Copy content Toggle raw display 2.1.7.6a1.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$$[\ ]_{7}^{3}$$
2.1.14.27a1.17$x^{14} + 4 x^{7} + 2$$14$$1$$27$$(C_7:C_3) \times C_2$$$[3]_{7}^{3}$$
\(3\) Copy content Toggle raw display 3.1.7.6a1.1$x^{7} + 3$$7$$1$$6$$F_7$$$[\ ]_{7}^{6}$$
3.1.14.13a1.2$x^{14} + 6$$14$$1$$13$$F_7 \times C_2$$$[\ ]_{14}^{6}$$
\(5\) Copy content Toggle raw display 5.1.7.6a1.1$x^{7} + 5$$7$$1$$6$$F_7$$$[\ ]_{7}^{6}$$
5.1.14.13a1.1$x^{14} + 5$$14$$1$$13$$F_7 \times C_2$$$[\ ]_{14}^{6}$$
\(7\) Copy content Toggle raw display 7.3.7.21a6.1$x^{21} + 42 x^{20} + 756 x^{19} + 7588 x^{18} + 46368 x^{17} + 178416 x^{16} + 447888 x^{15} + 834336 x^{14} + 1427328 x^{13} + 2034368 x^{12} + 2231040 x^{11} + 3096576 x^{10} + 1944320 x^{9} + 3064320 x^{8} + 967680 x^{7} + 1956864 x^{6} + 258048 x^{5} + 774144 x^{4} + 28679 x^{3} + 172074 x^{2} + 16419$$7$$3$$21$21T9not computed
\(41\) Copy content Toggle raw display $\Q_{41}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.1.2.1a1.2$x^{2} + 246$$2$$1$$1$$C_2$$$[\ ]_{2}$$
41.2.2.2a1.2$x^{4} + 76 x^{3} + 1456 x^{2} + 456 x + 77$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
41.2.2.2a1.2$x^{4} + 76 x^{3} + 1456 x^{2} + 456 x + 77$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
41.2.2.2a1.2$x^{4} + 76 x^{3} + 1456 x^{2} + 456 x + 77$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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