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Properties

Label 14.14.120...000.1
Degree $14$
Signature $[14, 0]$
Discriminant $1.208\times 10^{208}$
Root discriminant \(7.295\times 10^{14}\)
Ramified primes see page
Class number not computed
Class group not computed
Galois group $A_{14}$ (as 14T62)

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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^14 - 2900730*x^12 - 287910000*x^11 + 3313325898225*x^10 + 597401557200000*x^9 - 1855964346085631250*x^8 - 447630993322011000000*x^7 + 517062546566176619531250*x^6 + 144716607884926457572500000*x^5 - 63938673030863212150078125000*x^4 - 18637303238230533691855781250000*x^3 + 2587399700316109318860370458984375*x^2 + 773043792817583070706838466796875000*x + 40095511130962225240189804491943359375)
 
Copy content gp:K = bnfinit(y^14 - 2900730*y^12 - 287910000*y^11 + 3313325898225*y^10 + 597401557200000*y^9 - 1855964346085631250*y^8 - 447630993322011000000*y^7 + 517062546566176619531250*y^6 + 144716607884926457572500000*y^5 - 63938673030863212150078125000*y^4 - 18637303238230533691855781250000*y^3 + 2587399700316109318860370458984375*y^2 + 773043792817583070706838466796875000*y + 40095511130962225240189804491943359375, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 2900730*x^12 - 287910000*x^11 + 3313325898225*x^10 + 597401557200000*x^9 - 1855964346085631250*x^8 - 447630993322011000000*x^7 + 517062546566176619531250*x^6 + 144716607884926457572500000*x^5 - 63938673030863212150078125000*x^4 - 18637303238230533691855781250000*x^3 + 2587399700316109318860370458984375*x^2 + 773043792817583070706838466796875000*x + 40095511130962225240189804491943359375);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 2900730*x^12 - 287910000*x^11 + 3313325898225*x^10 + 597401557200000*x^9 - 1855964346085631250*x^8 - 447630993322011000000*x^7 + 517062546566176619531250*x^6 + 144716607884926457572500000*x^5 - 63938673030863212150078125000*x^4 - 18637303238230533691855781250000*x^3 + 2587399700316109318860370458984375*x^2 + 773043792817583070706838466796875000*x + 40095511130962225240189804491943359375)
 

\( x^{14} - 2900730 x^{12} - 287910000 x^{11} + 3313325898225 x^{10} + 597401557200000 x^{9} + \cdots + 40\!\cdots\!75 \) 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:  $14$
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:  $[14, 0]$
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:   \(120\!\cdots\!000\) \(\medspace = 2^{14}\cdot 3^{22}\cdot 5^{12}\cdot 7^{14}\cdot 19^{6}\cdot 197^{6}\cdot 199^{6}\cdot 2593^{6}\cdot 34963^{6}\cdot 560233^{6}\cdot 1072255949^{2}\cdot 6488895781911832501^{2}\) 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:  \(7.295\times 10^{14}\)
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:  not computed
Ramified primes:   \(2\), \(3\), \(5\), \(7\), \(19\), \(197\), \(199\), \(2593\), \(34963\), \(560233\), \(1072255949\), \(6488895781911832501\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{25}a^{3}+\frac{2}{5}a$, $\frac{1}{125}a^{4}+\frac{2}{25}a^{2}$, $\frac{1}{625}a^{5}+\frac{2}{125}a^{3}$, $\frac{1}{3125}a^{6}+\frac{2}{625}a^{4}$, $\frac{1}{244140625}a^{7}+\frac{22427}{48828125}a^{5}+\frac{922}{390625}a^{4}-\frac{4938}{390625}a^{3}-\frac{3542}{78125}a^{2}-\frac{877}{3125}a+\frac{2982}{15625}$, $\frac{1}{57373046875}a^{8}+\frac{23}{11474609375}a^{7}+\frac{569302}{11474609375}a^{6}+\frac{851371}{2294921875}a^{5}+\frac{144842}{91796875}a^{4}-\frac{201491}{18359375}a^{3}+\frac{242274}{3671875}a^{2}+\frac{1498707}{3671875}a+\frac{84211}{734375}$, $\frac{1}{286865234375}a^{9}+\frac{72}{57373046875}a^{7}-\frac{49078}{458984375}a^{6}-\frac{920807}{2294921875}a^{5}+\frac{35148}{91796875}a^{4}+\frac{289038}{18359375}a^{3}+\frac{1340642}{18359375}a^{2}+\frac{67317}{146875}a+\frac{167478}{734375}$, $\frac{1}{1434326171875}a^{10}+\frac{2}{286865234375}a^{8}+\frac{4}{11474609375}a^{7}-\frac{309457}{2294921875}a^{6}-\frac{1539472}{2294921875}a^{5}-\frac{14473}{18359375}a^{4}-\frac{767403}{91796875}a^{3}+\frac{231177}{3671875}a^{2}+\frac{5398}{15625}a+\frac{108658}{734375}$, $\frac{1}{7171630859375}a^{11}+\frac{2}{1434326171875}a^{9}-\frac{7}{11474609375}a^{7}-\frac{28961}{2294921875}a^{6}-\frac{12101}{18359375}a^{5}-\frac{1705518}{458984375}a^{4}+\frac{3749}{3671875}a^{3}-\frac{103779}{3671875}a^{2}+\frac{221721}{734375}a+\frac{13696}{29375}$, $\frac{1}{107574462890625}a^{12}-\frac{1}{7171630859375}a^{10}+\frac{1}{286865234375}a^{8}+\frac{2}{2294921875}a^{7}+\frac{1529692}{11474609375}a^{6}-\frac{268177}{2294921875}a^{5}-\frac{352148}{91796875}a^{4}-\frac{67199}{91796875}a^{3}+\frac{16308}{3671875}a^{2}+\frac{1200367}{3671875}a-\frac{303731}{734375}$, $\frac{1}{107574462890625}a^{13}+\frac{2}{1434326171875}a^{9}-\frac{67}{57373046875}a^{7}+\frac{130232}{2294921875}a^{6}-\frac{364836}{2294921875}a^{5}+\frac{1629172}{458984375}a^{4}+\frac{333564}{18359375}a^{3}+\frac{660938}{18359375}a^{2}-\frac{11863}{29375}a-\frac{186516}{734375}$ 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:  $13$
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{14}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{12082462805952683578833887544607885354840529275474033703346853825890523525763932307253207072895617326779140942034887593472026784080351332996854475050975704479877705399512513606726026903633577245444000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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^14 - 2900730*x^12 - 287910000*x^11 + 3313325898225*x^10 + 597401557200000*x^9 - 1855964346085631250*x^8 - 447630993322011000000*x^7 + 517062546566176619531250*x^6 + 144716607884926457572500000*x^5 - 63938673030863212150078125000*x^4 - 18637303238230533691855781250000*x^3 + 2587399700316109318860370458984375*x^2 + 773043792817583070706838466796875000*x + 40095511130962225240189804491943359375) 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^14 - 2900730*x^12 - 287910000*x^11 + 3313325898225*x^10 + 597401557200000*x^9 - 1855964346085631250*x^8 - 447630993322011000000*x^7 + 517062546566176619531250*x^6 + 144716607884926457572500000*x^5 - 63938673030863212150078125000*x^4 - 18637303238230533691855781250000*x^3 + 2587399700316109318860370458984375*x^2 + 773043792817583070706838466796875000*x + 40095511130962225240189804491943359375, 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^14 - 2900730*x^12 - 287910000*x^11 + 3313325898225*x^10 + 597401557200000*x^9 - 1855964346085631250*x^8 - 447630993322011000000*x^7 + 517062546566176619531250*x^6 + 144716607884926457572500000*x^5 - 63938673030863212150078125000*x^4 - 18637303238230533691855781250000*x^3 + 2587399700316109318860370458984375*x^2 + 773043792817583070706838466796875000*x + 40095511130962225240189804491943359375); 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^14 - 2900730*x^12 - 287910000*x^11 + 3313325898225*x^10 + 597401557200000*x^9 - 1855964346085631250*x^8 - 447630993322011000000*x^7 + 517062546566176619531250*x^6 + 144716607884926457572500000*x^5 - 63938673030863212150078125000*x^4 - 18637303238230533691855781250000*x^3 + 2587399700316109318860370458984375*x^2 + 773043792817583070706838466796875000*x + 40095511130962225240189804491943359375); 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

$A_{14}$ (as 14T62):

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)
 
A non-solvable group of order 43589145600
The 72 conjugacy class representatives for $A_{14}$
Character table for $A_{14}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 R R R R ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ R ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
2.6.2.12a7.1$x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{9} + 3 x^{8} + 4 x^{7} + 5 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + x^{2} + 2 x + 3$$2$$6$$12$12T134$$[2, 2, 2, 2, 2, 2]^{6}$$
\(3\) Copy content Toggle raw display 3.1.2.1a1.1$x^{2} + 3$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.12.21a1.23$x^{12} + 3 x^{11} + 3 x^{10} + 9 x^{2} + 6$$12$$1$$21$12T169$$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$$
\(5\) Copy content Toggle raw display 5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.5.5a1.2$x^{5} + 10 x + 5$$5$$1$$5$$F_5$$$[\frac{5}{4}]_{4}$$
5.1.5.5a1.2$x^{5} + 10 x + 5$$5$$1$$5$$F_5$$$[\frac{5}{4}]_{4}$$
\(7\) Copy content Toggle raw display 7.1.7.7a1.4$x^{7} + 28 x + 7$$7$$1$$7$$F_7$$$[\frac{7}{6}]_{6}$$
7.1.7.7a1.1$x^{7} + 7 x + 7$$7$$1$$7$$F_7$$$[\frac{7}{6}]_{6}$$
\(19\) Copy content Toggle raw display $\Q_{19}$$x + 17$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{19}$$x + 17$$1$$1$$0$Trivial$$[\ ]$$
19.6.2.6a1.2$x^{12} + 34 x^{9} + 34 x^{8} + 12 x^{7} + 293 x^{6} + 578 x^{5} + 493 x^{4} + 272 x^{3} + 104 x^{2} + 24 x + 23$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(197\) Copy content Toggle raw display $\Q_{197}$$x + 195$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{197}$$x + 195$$1$$1$$0$Trivial$$[\ ]$$
197.1.2.1a1.1$x^{2} + 197$$2$$1$$1$$C_2$$$[\ ]_{2}$$
197.5.2.5a1.2$x^{10} + 8 x^{6} + 390 x^{5} + 16 x^{2} + 1560 x + 38222$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(199\) Copy content Toggle raw display $\Q_{199}$$x + 196$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{199}$$x + 196$$1$$1$$0$Trivial$$[\ ]$$
199.6.2.6a1.2$x^{12} + 180 x^{9} + 116 x^{8} + 158 x^{7} + 8106 x^{6} + 10440 x^{5} + 17584 x^{4} + 9704 x^{3} + 6589 x^{2} + 474 x + 208$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(2593\) Copy content Toggle raw display $\Q_{2593}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2593}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$2$$2$
Deg $6$$2$$3$$3$
\(34963\) Copy content Toggle raw display $\Q_{34963}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{34963}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $4$$2$$2$$2$
Deg $8$$2$$4$$4$
\(560233\) Copy content Toggle raw display $\Q_{560233}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{560233}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $12$$2$$6$$6$
\(1072255949\) Copy content Toggle raw display $\Q_{1072255949}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1072255949}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$3$$1$$2$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(6488895781911832501\) Copy content Toggle raw display $\Q_{6488895781911832501}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $3$$3$$1$$2$
Deg $8$$1$$8$$0$$C_8$$$[\ ]^{8}$$

Spectrum of ring of integers

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