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Properties

Label 18.8.440...336.2
Degree $18$
Signature $[8, 5]$
Discriminant $-4.405\times 10^{30}$
Root discriminant \(50.40\)
Ramified primes $2,32009$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^3:A_4^2.S_4$ (as 18T662)

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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^18 + 9*x^16 - 8*x^14 - 191*x^12 - 107*x^10 + 893*x^8 + 181*x^6 - 780*x^4 + 104*x^2 + 1)
 
Copy content gp:K = bnfinit(y^18 + 9*y^16 - 8*y^14 - 191*y^12 - 107*y^10 + 893*y^8 + 181*y^6 - 780*y^4 + 104*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 9*x^16 - 8*x^14 - 191*x^12 - 107*x^10 + 893*x^8 + 181*x^6 - 780*x^4 + 104*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + 9*x^16 - 8*x^14 - 191*x^12 - 107*x^10 + 893*x^8 + 181*x^6 - 780*x^4 + 104*x^2 + 1)
 

\( x^{18} + 9x^{16} - 8x^{14} - 191x^{12} - 107x^{10} + 893x^{8} + 181x^{6} - 780x^{4} + 104x^{2} + 1 \) 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:  $18$
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:  $[8, 5]$
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:   \(-4405473434934059943108359950336\) \(\medspace = -\,2^{12}\cdot 32009^{6}\) 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:  \(50.40\)
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:  not computed
Ramified primes:   \(2\), \(32009\) 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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a$, $\frac{1}{6886857178}a^{16}-\frac{334471770}{3443428589}a^{14}-\frac{281050833}{3443428589}a^{12}+\frac{1038665305}{6886857178}a^{10}-\frac{1279766775}{6886857178}a^{8}-\frac{1}{2}a^{7}-\frac{1485020054}{3443428589}a^{6}+\frac{746893824}{3443428589}a^{4}-\frac{1}{2}a^{3}+\frac{2159340103}{6886857178}a^{2}-\frac{1}{2}a+\frac{2487584835}{6886857178}$, $\frac{1}{6886857178}a^{17}-\frac{334471770}{3443428589}a^{15}-\frac{281050833}{3443428589}a^{13}+\frac{1038665305}{6886857178}a^{11}-\frac{1279766775}{6886857178}a^{9}-\frac{1}{2}a^{8}-\frac{1485020054}{3443428589}a^{7}+\frac{746893824}{3443428589}a^{5}-\frac{1}{2}a^{4}+\frac{2159340103}{6886857178}a^{3}-\frac{1}{2}a^{2}+\frac{2487584835}{6886857178}a$ 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:  Trivial group, which has order $1$ (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:  $C_{2}\times C_{2}$, which has order $4$ (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);
 

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:  $12$
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:   $a$, $\frac{523679109}{3443428589}a^{17}+\frac{9442191273}{6886857178}a^{15}-\frac{8244981571}{6886857178}a^{13}-\frac{200307402921}{6886857178}a^{11}-\frac{115420400019}{6886857178}a^{9}+\frac{935878699841}{6886857178}a^{7}+\frac{212368067903}{6886857178}a^{5}-\frac{817418119851}{6886857178}a^{3}+\frac{34198628605}{3443428589}a$, $\frac{3843657}{112899298}a^{16}+\frac{35345999}{112899298}a^{14}-\frac{11766584}{56449649}a^{12}-\frac{367773762}{56449649}a^{10}-\frac{275723572}{56449649}a^{8}+\frac{1637668217}{56449649}a^{6}+\frac{607578613}{56449649}a^{4}-\frac{2658858299}{112899298}a^{2}+\frac{170704361}{112899298}$, $\frac{673240381}{6886857178}a^{17}+\frac{6029618115}{6886857178}a^{15}-\frac{5661205707}{6886857178}a^{13}-\frac{128506622841}{6886857178}a^{11}-\frac{66765827097}{6886857178}a^{9}+\frac{607621428199}{6886857178}a^{7}+\frac{104820751923}{6886857178}a^{5}-\frac{273336168593}{3443428589}a^{3}+\frac{36762171273}{3443428589}a$, $\frac{374117837}{6886857178}a^{16}+\frac{1706286579}{3443428589}a^{14}-\frac{1291887932}{3443428589}a^{12}-\frac{35900390040}{3443428589}a^{10}-\frac{24327286461}{3443428589}a^{8}+\frac{164128635821}{3443428589}a^{6}+\frac{53773657990}{3443428589}a^{4}-\frac{270745782665}{6886857178}a^{2}+\frac{879885921}{3443428589}$, $\frac{123639587}{6886857178}a^{16}+\frac{549637002}{3443428589}a^{14}-\frac{541948549}{3443428589}a^{12}-\frac{11699799470}{3443428589}a^{10}-\frac{5835517899}{3443428589}a^{8}+\frac{54774792655}{3443428589}a^{6}+\frac{10185552740}{3443428589}a^{4}-\frac{91401768627}{6886857178}a^{2}+\frac{7849842328}{3443428589}$, $\frac{69803449}{3443428589}a^{16}+\frac{1233158177}{6886857178}a^{14}-\frac{1345880381}{6886857178}a^{12}-\frac{26752669633}{6886857178}a^{10}-\frac{11083224631}{6886857178}a^{8}+\frac{132345815755}{6886857178}a^{6}+\frac{19892395669}{6886857178}a^{4}-\frac{131929766089}{6886857178}a^{2}+\frac{439306041}{3443428589}$, $\frac{1012539633}{6886857178}a^{17}+\frac{9178455985}{6886857178}a^{15}-\frac{3704009561}{3443428589}a^{13}-\frac{96473204130}{3443428589}a^{11}-\frac{60533563394}{3443428589}a^{9}+\frac{439220973378}{3443428589}a^{7}+\frac{110324390961}{3443428589}a^{5}-\frac{714842672137}{6886857178}a^{3}+\frac{89633081277}{6886857178}a$, $\frac{4566957}{13718839}a^{17}-\frac{333235869}{3443428589}a^{16}+\frac{83607165}{27437678}a^{15}-\frac{6031929277}{6886857178}a^{14}-\frac{30002221}{13718839}a^{13}+\frac{5062904103}{6886857178}a^{12}-\frac{1751460641}{27437678}a^{11}+\frac{127975064559}{6886857178}a^{10}-\frac{1245944675}{27437678}a^{9}+\frac{39093968175}{3443428589}a^{8}+\frac{3960364678}{13718839}a^{7}-\frac{599494148963}{6886857178}a^{6}+\frac{1402897696}{13718839}a^{5}-\frac{83378612719}{3443428589}a^{4}-\frac{3271703235}{13718839}a^{3}+\frac{270342129438}{3443428589}a^{2}+\frac{54790893}{13718839}a-\frac{671823901}{3443428589}$, $\frac{1219788691}{6886857178}a^{17}+\frac{116416108}{3443428589}a^{16}+\frac{5516576471}{3443428589}a^{15}+\frac{2154269721}{6886857178}a^{14}-\frac{4633858624}{3443428589}a^{13}-\frac{654686201}{3443428589}a^{12}-\frac{116756310662}{3443428589}a^{11}-\frac{44667446977}{6886857178}a^{10}-\frac{70674066830}{3443428589}a^{9}-\frac{35965340011}{6886857178}a^{8}+\frac{1084972036477}{6886857178}a^{7}+\frac{98001974950}{3443428589}a^{6}+\frac{138247374568}{3443428589}a^{5}+\frac{42369522967}{3443428589}a^{4}-\frac{472748118193}{3443428589}a^{3}-\frac{75666052311}{3443428589}a^{2}+\frac{64003696345}{6886857178}a+\frac{687037485}{3443428589}$, $\frac{399620706}{3443428589}a^{17}+\frac{3694946}{3443428589}a^{16}+\frac{7285224109}{6886857178}a^{15}+\frac{978305}{3443428589}a^{14}-\frac{5499180789}{6886857178}a^{13}-\frac{358067385}{3443428589}a^{12}-\frac{76347118736}{3443428589}a^{11}-\frac{1719642225}{6886857178}a^{10}-\frac{51279689490}{3443428589}a^{9}+\frac{5409910744}{3443428589}a^{8}+\frac{345955574906}{3443428589}a^{7}+\frac{14001391408}{3443428589}a^{6}+\frac{200371358227}{6886857178}a^{5}-\frac{24311247951}{6886857178}a^{4}-\frac{292798682914}{3443428589}a^{3}-\frac{43452418173}{6886857178}a^{2}+\frac{28809433607}{3443428589}a+\frac{9039395857}{6886857178}$, $\frac{3495567845}{6886857178}a^{17}+\frac{10824285}{56449649}a^{16}+\frac{32099802119}{6886857178}a^{15}+\frac{198385363}{112899298}a^{14}-\frac{21982904557}{6886857178}a^{13}-\frac{140981847}{112899298}a^{12}-\frac{670524066989}{6886857178}a^{11}-\frac{4163444497}{112899298}a^{10}-\frac{495626669071}{6886857178}a^{9}-\frac{3023410243}{112899298}a^{8}+\frac{3011962661123}{6886857178}a^{7}+\frac{18874749655}{112899298}a^{6}+\frac{1140315848865}{6886857178}a^{5}+\frac{7437085571}{112899298}a^{4}-\frac{1244544001068}{3443428589}a^{3}-\frac{15108780885}{112899298}a^{2}-\frac{9918890164}{3443428589}a-\frac{120747275}{56449649}$ 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:  \( 359830335.235 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{5}\cdot 359830335.235 \cdot 1}{2\cdot\sqrt{4405473434934059943108359950336}}\cr\approx \mathstrut & 0.214887320959 \end{aligned}\] (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^18 + 9*x^16 - 8*x^14 - 191*x^12 - 107*x^10 + 893*x^8 + 181*x^6 - 780*x^4 + 104*x^2 + 1) 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^18 + 9*x^16 - 8*x^14 - 191*x^12 - 107*x^10 + 893*x^8 + 181*x^6 - 780*x^4 + 104*x^2 + 1, 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^18 + 9*x^16 - 8*x^14 - 191*x^12 - 107*x^10 + 893*x^8 + 181*x^6 - 780*x^4 + 104*x^2 + 1); 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^18 + 9*x^16 - 8*x^14 - 191*x^12 - 107*x^10 + 893*x^8 + 181*x^6 - 780*x^4 + 104*x^2 + 1); 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

$C_2^3:A_4^2.S_4$ (as 18T662):

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 27648
The 96 conjugacy class representatives for $C_2^3:A_4^2.S_4$
Character table for $C_2^3:A_4^2.S_4$

Intermediate fields

3.3.32009.2, 9.9.32795655776729.3

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 18 sibling: data not computed
Degree 24 siblings: data not computed
Degree 36 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 ${\href{/padicField/3.6.0.1}{6} }^{3}$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{6}$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(2\) Copy content Toggle raw display 2.3.2.6a2.1$x^{6} + 4 x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 3$$2$$3$$6$$A_4\times C_2$$$[2, 2, 2]^{3}$$
2.3.2.6a3.1$x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 6 x + 5$$2$$3$$6$$A_4$$$[2, 2]^{3}$$
2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(32009\) Copy content Toggle raw display $\Q_{32009}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{32009}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{32009}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{32009}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{32009}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{32009}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$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)