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Properties

Label 14.0.119...791.1
Degree $14$
Signature $[0, 7]$
Discriminant $-1.193\times 10^{35}$
Root discriminant \(320.23\)
Ramified primes $31,113$
Class number $1524264$ (GRH)
Class group [2, 2, 381066] (GRH)
Galois group $C_{14}$ (as 14T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 31*x^12 + 309*x^11 + 262*x^10 - 8842*x^9 + 67732*x^8 - 350971*x^7 + 1534032*x^6 - 4949688*x^5 + 19407809*x^4 - 39113311*x^3 + 126961596*x^2 - 125321277*x + 300251687)
 
Copy content gp:K = bnfinit(y^14 - 5*y^13 - 31*y^12 + 309*y^11 + 262*y^10 - 8842*y^9 + 67732*y^8 - 350971*y^7 + 1534032*y^6 - 4949688*y^5 + 19407809*y^4 - 39113311*y^3 + 126961596*y^2 - 125321277*y + 300251687, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 5*x^13 - 31*x^12 + 309*x^11 + 262*x^10 - 8842*x^9 + 67732*x^8 - 350971*x^7 + 1534032*x^6 - 4949688*x^5 + 19407809*x^4 - 39113311*x^3 + 126961596*x^2 - 125321277*x + 300251687);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 5*x^13 - 31*x^12 + 309*x^11 + 262*x^10 - 8842*x^9 + 67732*x^8 - 350971*x^7 + 1534032*x^6 - 4949688*x^5 + 19407809*x^4 - 39113311*x^3 + 126961596*x^2 - 125321277*x + 300251687)
 

\( x^{14} - 5 x^{13} - 31 x^{12} + 309 x^{11} + 262 x^{10} - 8842 x^{9} + 67732 x^{8} - 350971 x^{7} + \cdots + 300251687 \) 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:  $[0, 7]$
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:   \(-119254061410789267361233458448997791\) \(\medspace = -\,31^{7}\cdot 113^{12}\) 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:  \(320.23\)
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:  $31^{1/2}113^{6/7}\approx 320.2302614292546$
Ramified primes:   \(31\), \(113\) 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:  \(\Q(\sqrt{-31}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{14}$
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:  \(3503=31\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{3503}(1,·)$, $\chi_{3503}(30,·)$, $\chi_{3503}(340,·)$, $\chi_{3503}(807,·)$, $\chi_{3503}(900,·)$, $\chi_{3503}(1146,·)$, $\chi_{3503}(1179,·)$, $\chi_{3503}(1239,·)$, $\chi_{3503}(1518,·)$, $\chi_{3503}(2140,·)$, $\chi_{3503}(2479,·)$, $\chi_{3503}(2853,·)$, $\chi_{3503}(3192,·)$, $\chi_{3503}(3194,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{64}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11461}a^{12}-\frac{4666}{11461}a^{11}-\frac{3507}{11461}a^{10}-\frac{2507}{11461}a^{9}+\frac{1488}{11461}a^{8}-\frac{357}{11461}a^{7}-\frac{2740}{11461}a^{6}-\frac{647}{11461}a^{5}-\frac{953}{11461}a^{4}-\frac{3851}{11461}a^{3}+\frac{4759}{11461}a^{2}+\frac{1311}{11461}a+\frac{4395}{11461}$, $\frac{1}{41\cdots 39}a^{13}-\frac{16\cdots 75}{41\cdots 39}a^{12}+\frac{49\cdots 14}{41\cdots 39}a^{11}+\frac{92\cdots 41}{41\cdots 39}a^{10}-\frac{11\cdots 82}{41\cdots 39}a^{9}+\frac{54\cdots 77}{41\cdots 39}a^{8}+\frac{20\cdots 24}{41\cdots 39}a^{7}-\frac{66\cdots 19}{26\cdots 27}a^{6}+\frac{20\cdots 78}{41\cdots 39}a^{5}-\frac{99\cdots 28}{41\cdots 39}a^{4}-\frac{37\cdots 02}{41\cdots 39}a^{3}+\frac{51\cdots 18}{41\cdots 39}a^{2}+\frac{20\cdots 80}{41\cdots 39}a+\frac{12\cdots 57}{41\cdots 39}$ 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:  $C_{2}\times C_{2}\times C_{381066}$, which has order $1524264$ (assuming GRH)
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:  $C_{2}\times C_{2}\times C_{381066}$, which has order $1524264$ (assuming GRH)
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:   $1524264$ (assuming GRH)

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:  $6$
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:   $\frac{89\cdots 20}{41\cdots 39}a^{13}-\frac{67\cdots 17}{41\cdots 39}a^{12}-\frac{20\cdots 90}{41\cdots 39}a^{11}+\frac{38\cdots 84}{41\cdots 39}a^{10}-\frac{41\cdots 13}{41\cdots 39}a^{9}-\frac{10\cdots 32}{41\cdots 39}a^{8}+\frac{85\cdots 82}{41\cdots 39}a^{7}-\frac{40\cdots 20}{41\cdots 39}a^{6}+\frac{16\cdots 87}{41\cdots 39}a^{5}-\frac{53\cdots 42}{41\cdots 39}a^{4}+\frac{18\cdots 42}{41\cdots 39}a^{3}-\frac{44\cdots 02}{41\cdots 39}a^{2}+\frac{86\cdots 49}{41\cdots 39}a-\frac{11\cdots 68}{41\cdots 39}$, $\frac{26\cdots 84}{41\cdots 39}a^{13}+\frac{27\cdots 39}{41\cdots 39}a^{12}-\frac{25\cdots 62}{41\cdots 39}a^{11}-\frac{94\cdots 92}{41\cdots 39}a^{10}+\frac{13\cdots 15}{41\cdots 39}a^{9}+\frac{12\cdots 87}{41\cdots 39}a^{8}-\frac{25\cdots 06}{41\cdots 39}a^{7}+\frac{11\cdots 44}{41\cdots 39}a^{6}-\frac{54\cdots 11}{41\cdots 39}a^{5}+\frac{26\cdots 62}{41\cdots 39}a^{4}-\frac{48\cdots 38}{41\cdots 39}a^{3}+\frac{33\cdots 41}{41\cdots 39}a^{2}-\frac{23\cdots 43}{41\cdots 39}a+\frac{12\cdots 40}{41\cdots 39}$, $\frac{28\cdots 40}{41\cdots 39}a^{13}+\frac{40\cdots 97}{41\cdots 39}a^{12}-\frac{17\cdots 06}{41\cdots 39}a^{11}+\frac{12\cdots 33}{41\cdots 39}a^{10}+\frac{68\cdots 30}{41\cdots 39}a^{9}-\frac{10\cdots 47}{41\cdots 39}a^{8}-\frac{58\cdots 30}{41\cdots 39}a^{7}+\frac{16\cdots 27}{41\cdots 39}a^{6}-\frac{63\cdots 78}{41\cdots 39}a^{5}+\frac{78\cdots 60}{41\cdots 39}a^{4}-\frac{11\cdots 28}{41\cdots 39}a^{3}+\frac{12\cdots 62}{41\cdots 39}a^{2}-\frac{95\cdots 40}{41\cdots 39}a+\frac{73\cdots 48}{41\cdots 39}$, $\frac{41\cdots 62}{36\cdots 99}a^{13}-\frac{17\cdots 17}{36\cdots 99}a^{12}-\frac{13\cdots 78}{36\cdots 99}a^{11}+\frac{11\cdots 36}{36\cdots 99}a^{10}+\frac{16\cdots 71}{36\cdots 99}a^{9}-\frac{33\cdots 69}{36\cdots 99}a^{8}+\frac{25\cdots 22}{36\cdots 99}a^{7}-\frac{13\cdots 24}{36\cdots 99}a^{6}+\frac{58\cdots 59}{36\cdots 99}a^{5}-\frac{18\cdots 48}{36\cdots 99}a^{4}+\frac{80\cdots 02}{36\cdots 99}a^{3}-\frac{14\cdots 62}{36\cdots 99}a^{2}+\frac{37\cdots 30}{36\cdots 99}a-\frac{26\cdots 62}{36\cdots 99}$, $\frac{17\cdots 00}{41\cdots 39}a^{13}+\frac{35\cdots 27}{41\cdots 39}a^{12}-\frac{23\cdots 58}{41\cdots 39}a^{11}-\frac{14\cdots 76}{41\cdots 39}a^{10}+\frac{13\cdots 43}{41\cdots 39}a^{9}+\frac{25\cdots 41}{41\cdots 39}a^{8}-\frac{34\cdots 50}{41\cdots 39}a^{7}+\frac{16\cdots 32}{41\cdots 39}a^{6}-\frac{70\cdots 51}{41\cdots 39}a^{5}+\frac{31\cdots 53}{41\cdots 39}a^{4}-\frac{64\cdots 48}{41\cdots 39}a^{3}+\frac{36\cdots 86}{41\cdots 39}a^{2}-\frac{31\cdots 47}{41\cdots 39}a+\frac{14\cdots 16}{41\cdots 39}$, $\frac{41\cdots 78}{41\cdots 39}a^{13}-\frac{11\cdots 58}{41\cdots 39}a^{12}+\frac{24\cdots 98}{41\cdots 39}a^{11}+\frac{54\cdots 04}{41\cdots 39}a^{10}-\frac{27\cdots 96}{41\cdots 39}a^{9}-\frac{12\cdots 86}{41\cdots 39}a^{8}+\frac{12\cdots 38}{41\cdots 39}a^{7}-\frac{62\cdots 51}{41\cdots 39}a^{6}+\frac{26\cdots 55}{41\cdots 39}a^{5}-\frac{10\cdots 07}{41\cdots 39}a^{4}+\frac{28\cdots 63}{41\cdots 39}a^{3}-\frac{10\cdots 48}{41\cdots 39}a^{2}+\frac{13\cdots 76}{41\cdots 39}a-\frac{37\cdots 88}{41\cdots 39}$ Copy content Toggle raw display (assuming GRH)
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:  \( 222748.97284811488 \) (assuming GRH)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 222748.97284811488 \cdot 1524264}{2\cdot\sqrt{119254061410789267361233458448997791}}\cr\approx \mathstrut & 0.190050228543587 \end{aligned}\] (assuming GRH)

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 - 5*x^13 - 31*x^12 + 309*x^11 + 262*x^10 - 8842*x^9 + 67732*x^8 - 350971*x^7 + 1534032*x^6 - 4949688*x^5 + 19407809*x^4 - 39113311*x^3 + 126961596*x^2 - 125321277*x + 300251687) 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 - 5*x^13 - 31*x^12 + 309*x^11 + 262*x^10 - 8842*x^9 + 67732*x^8 - 350971*x^7 + 1534032*x^6 - 4949688*x^5 + 19407809*x^4 - 39113311*x^3 + 126961596*x^2 - 125321277*x + 300251687, 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 - 5*x^13 - 31*x^12 + 309*x^11 + 262*x^10 - 8842*x^9 + 67732*x^8 - 350971*x^7 + 1534032*x^6 - 4949688*x^5 + 19407809*x^4 - 39113311*x^3 + 126961596*x^2 - 125321277*x + 300251687); 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^14 - 5*x^13 - 31*x^12 + 309*x^11 + 262*x^10 - 8842*x^9 + 67732*x^8 - 350971*x^7 + 1534032*x^6 - 4949688*x^5 + 19407809*x^4 - 39113311*x^3 + 126961596*x^2 - 125321277*x + 300251687); 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

$C_{14}$ (as 14T1):

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 cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-31}) \), 7.7.2081951752609.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.7.0.1}{7} }^{2}$ ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.7.0.1}{7} }^{2}$ ${\href{/padicField/7.7.0.1}{7} }^{2}$ ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.14.0.1}{14} }$ ${\href{/padicField/17.14.0.1}{14} }$ ${\href{/padicField/19.7.0.1}{7} }^{2}$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.14.0.1}{14} }$ R ${\href{/padicField/37.14.0.1}{14} }$ ${\href{/padicField/41.7.0.1}{7} }^{2}$ ${\href{/padicField/43.14.0.1}{14} }$ ${\href{/padicField/47.7.0.1}{7} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }$ ${\href{/padicField/59.7.0.1}{7} }^{2}$

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
\(31\) Copy content Toggle raw display 31.7.2.7a1.2$x^{14} + 2 x^{8} + 56 x^{7} + x^{2} + 56 x + 815$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$
\(113\) Copy content Toggle raw display 113.1.7.6a1.1$x^{7} + 113$$7$$1$$6$$C_7$$$[\ ]_{7}$$
113.1.7.6a1.1$x^{7} + 113$$7$$1$$6$$C_7$$$[\ ]_{7}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*14 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*14 1.31.2t1.a.a$1$ $ 31 $ \(\Q(\sqrt{-31}) \) $C_2$ (as 2T1) $1$ $-1$
*14 1.113.7t1.a.d$1$ $ 113 $ 7.7.2081951752609.1 $C_7$ (as 7T1) $0$ $1$
*14 1.3503.14t1.a.a$1$ $ 31 \cdot 113 $ 14.0.119254061410789267361233458448997791.1 $C_{14}$ (as 14T1) $0$ $-1$
*14 1.113.7t1.a.a$1$ $ 113 $ 7.7.2081951752609.1 $C_7$ (as 7T1) $0$ $1$
*14 1.3503.14t1.a.c$1$ $ 31 \cdot 113 $ 14.0.119254061410789267361233458448997791.1 $C_{14}$ (as 14T1) $0$ $-1$
*14 1.113.7t1.a.e$1$ $ 113 $ 7.7.2081951752609.1 $C_7$ (as 7T1) $0$ $1$
*14 1.3503.14t1.a.e$1$ $ 31 \cdot 113 $ 14.0.119254061410789267361233458448997791.1 $C_{14}$ (as 14T1) $0$ $-1$
*14 1.113.7t1.a.b$1$ $ 113 $ 7.7.2081951752609.1 $C_7$ (as 7T1) $0$ $1$
*14 1.3503.14t1.a.b$1$ $ 31 \cdot 113 $ 14.0.119254061410789267361233458448997791.1 $C_{14}$ (as 14T1) $0$ $-1$
*14 1.113.7t1.a.f$1$ $ 113 $ 7.7.2081951752609.1 $C_7$ (as 7T1) $0$ $1$
*14 1.3503.14t1.a.f$1$ $ 31 \cdot 113 $ 14.0.119254061410789267361233458448997791.1 $C_{14}$ (as 14T1) $0$ $-1$
*14 1.113.7t1.a.c$1$ $ 113 $ 7.7.2081951752609.1 $C_7$ (as 7T1) $0$ $1$
*14 1.3503.14t1.a.d$1$ $ 31 \cdot 113 $ 14.0.119254061410789267361233458448997791.1 $C_{14}$ (as 14T1) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

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