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Properties

Label 18.12.804...776.2
Degree $18$
Signature $[12, 3]$
Discriminant $-8.042\times 10^{25}$
Root discriminant \(27.49\)
Ramified primes $2,7,53$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_4^2:C_2^2$ (as 18T175)

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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 - 25*x^14 + 3*x^12 + 115*x^10 - 50*x^8 - 136*x^6 + 115*x^4 - 25*x^2 + 1)
 
Copy content gp:K = bnfinit(y^18 - 25*y^14 + 3*y^12 + 115*y^10 - 50*y^8 - 136*y^6 + 115*y^4 - 25*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 25*x^14 + 3*x^12 + 115*x^10 - 50*x^8 - 136*x^6 + 115*x^4 - 25*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 25*x^14 + 3*x^12 + 115*x^10 - 50*x^8 - 136*x^6 + 115*x^4 - 25*x^2 + 1)
 

\( x^{18} - 25x^{14} + 3x^{12} + 115x^{10} - 50x^{8} - 136x^{6} + 115x^{4} - 25x^{2} + 1 \) 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:  $[12, 3]$
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:   \(-80421398252924334225227776\) \(\medspace = -\,2^{18}\cdot 7^{12}\cdot 53^{6}\) 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:  \(27.49\)
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:  $2^{31/16}7^{2/3}53^{1/2}\approx 102.04277129514405$
Ramified primes:   \(2\), \(7\), \(53\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{21668}a^{16}-\frac{412}{5417}a^{14}+\frac{981}{10834}a^{12}+\frac{144}{5417}a^{10}+\frac{4259}{21668}a^{8}+\frac{775}{10834}a^{6}+\frac{7705}{21668}a^{4}+\frac{1285}{5417}a^{2}-\frac{1987}{10834}$, $\frac{1}{21668}a^{17}-\frac{412}{5417}a^{15}+\frac{981}{10834}a^{13}+\frac{144}{5417}a^{11}+\frac{4259}{21668}a^{9}+\frac{775}{10834}a^{7}+\frac{7705}{21668}a^{5}+\frac{1285}{5417}a^{3}-\frac{1987}{10834}a$ 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:  $C_{2}$, which has order $2$ (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:  $14$
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:   $\frac{3563}{21668}a^{16}+\frac{51}{5417}a^{14}-\frac{89413}{21668}a^{12}+\frac{2331}{10834}a^{10}+\frac{418909}{21668}a^{8}-\frac{66349}{10834}a^{6}-\frac{134181}{5417}a^{4}+\frac{76928}{5417}a^{2}-\frac{4741}{21668}$, $\frac{3563}{21668}a^{16}+\frac{51}{5417}a^{14}-\frac{89413}{21668}a^{12}+\frac{2331}{10834}a^{10}+\frac{418909}{21668}a^{8}-\frac{66349}{10834}a^{6}-\frac{134181}{5417}a^{4}+\frac{76928}{5417}a^{2}-\frac{26409}{21668}$, $\frac{4741}{21668}a^{17}+\frac{3563}{21668}a^{15}-\frac{118321}{21668}a^{13}-\frac{37595}{10834}a^{11}+\frac{549877}{21668}a^{9}+\frac{181859}{21668}a^{7}-\frac{388737}{10834}a^{5}+\frac{8491}{21668}a^{3}+\frac{189187}{21668}a$, $\frac{889}{10834}a^{16}+\frac{453}{21668}a^{14}-\frac{10862}{5417}a^{12}-\frac{2551}{10834}a^{10}+\frac{91857}{10834}a^{8}-\frac{55523}{21668}a^{6}-\frac{105683}{10834}a^{4}+\frac{184619}{21668}a^{2}-\frac{5918}{5417}$, $\frac{4194}{5417}a^{17}+\frac{6937}{21668}a^{15}-\frac{416289}{21668}a^{13}-\frac{60063}{10834}a^{11}+\frac{468259}{5417}a^{9}-\frac{101723}{21668}a^{7}-\frac{2281931}{21668}a^{5}+\frac{1078817}{21668}a^{3}-\frac{76643}{21668}a$, $\frac{579}{21668}a^{17}+\frac{4617}{21668}a^{15}-\frac{6203}{10834}a^{13}-\frac{55345}{10834}a^{11}+\frac{28311}{21668}a^{9}+\frac{437005}{21668}a^{7}-\frac{56583}{21668}a^{5}-\frac{409565}{21668}a^{3}+\frac{34176}{5417}a$, $\frac{8959}{21668}a^{16}+\frac{1155}{10834}a^{14}-\frac{55676}{5417}a^{12}-\frac{14551}{10834}a^{10}+\frac{1006595}{21668}a^{8}-\frac{109709}{10834}a^{6}-\frac{1207727}{21668}a^{4}+\frac{196202}{5417}a^{2}-\frac{30429}{5417}$, $\frac{2553}{5417}a^{17}+\frac{1243}{21668}a^{15}-\frac{128069}{10834}a^{13}-\frac{375}{10834}a^{11}+\frac{299243}{5417}a^{9}-\frac{362813}{21668}a^{7}-\frac{749487}{10834}a^{5}+\frac{1011929}{21668}a^{3}-\frac{69583}{10834}a$, $\frac{1893}{21668}a^{17}-\frac{5493}{21668}a^{16}+\frac{132}{5417}a^{15}-\frac{1190}{5417}a^{14}-\frac{11333}{5417}a^{13}+\frac{33143}{5417}a^{12}-\frac{1933}{10834}a^{11}+\frac{48533}{10834}a^{10}+\frac{175135}{21668}a^{9}-\frac{534947}{21668}a^{8}-\frac{27551}{5417}a^{7}-\frac{40284}{5417}a^{6}-\frac{170343}{21668}a^{5}+\frac{600743}{21668}a^{4}+\frac{70693}{5417}a^{3}-\frac{38073}{5417}a^{2}-\frac{19956}{5417}a-\frac{5749}{5417}$, $\frac{1259}{10834}a^{17}+\frac{7297}{21668}a^{16}+\frac{2648}{5417}a^{15}+\frac{5701}{21668}a^{14}-\frac{27079}{10834}a^{13}-\frac{89589}{10834}a^{12}-\frac{125285}{10834}a^{11}-\frac{59847}{10834}a^{10}+\frac{53421}{10834}a^{9}+\frac{775225}{21668}a^{8}+\frac{239013}{5417}a^{7}+\frac{275921}{21668}a^{6}+\frac{14999}{10834}a^{5}-\frac{904297}{21668}a^{4}-\frac{457083}{10834}a^{3}+\frac{37191}{21668}a^{2}+\frac{38940}{5417}a+\frac{1085}{5417}$, $\frac{75}{21668}a^{17}+\frac{4779}{10834}a^{16}-\frac{2213}{10834}a^{15}+\frac{263}{5417}a^{14}-\frac{2263}{10834}a^{13}-\frac{119603}{10834}a^{12}+\frac{27051}{5417}a^{11}+\frac{434}{5417}a^{10}+\frac{59409}{21668}a^{9}+\frac{277313}{5417}a^{8}-\frac{114488}{5417}a^{7}-\frac{170933}{10834}a^{6}-\frac{50497}{21668}a^{5}-\frac{695947}{10834}a^{4}+\frac{263171}{10834}a^{3}+\frac{223788}{5417}a^{2}-\frac{105689}{10834}a-\frac{21540}{5417}$, $\frac{14371}{21668}a^{17}-\frac{15371}{21668}a^{16}-\frac{5701}{21668}a^{15}-\frac{3901}{21668}a^{14}-\frac{181261}{10834}a^{13}+\frac{95786}{5417}a^{12}+\frac{92349}{10834}a^{11}+\frac{12963}{5417}a^{10}+\frac{1716595}{21668}a^{9}-\frac{1750335}{21668}a^{8}-\frac{1359321}{21668}a^{7}+\frac{307685}{21668}a^{6}-\frac{2042551}{21668}a^{5}+\frac{2138031}{21668}a^{4}+\frac{2454629}{21668}a^{3}-\frac{1170067}{21668}a^{2}-\frac{136510}{5417}a+\frac{76969}{10834}$, $\frac{3295}{21668}a^{17}-\frac{7743}{21668}a^{16}-\frac{7743}{21668}a^{15}-\frac{7405}{21668}a^{14}-\frac{22445}{5417}a^{13}+\frac{187097}{21668}a^{12}+\frac{98491}{10834}a^{11}+\frac{38829}{5417}a^{10}+\frac{534241}{21668}a^{9}-\frac{757121}{21668}a^{8}-\frac{921871}{21668}a^{7}-\frac{328015}{21668}a^{6}-\frac{776135}{21668}a^{5}+\frac{202539}{5417}a^{4}+\frac{1189081}{21668}a^{3}-\frac{141163}{21668}a^{2}-\frac{111769}{10834}a-\frac{3295}{21668}$, $\frac{7817}{21668}a^{17}+\frac{3563}{21668}a^{16}+\frac{4627}{21668}a^{15}+\frac{51}{5417}a^{14}-\frac{193585}{21668}a^{13}-\frac{89413}{21668}a^{12}-\frac{22756}{5417}a^{11}+\frac{2331}{10834}a^{10}+\frac{866441}{21668}a^{9}+\frac{418909}{21668}a^{8}+\frac{117695}{21668}a^{7}-\frac{66349}{10834}a^{6}-\frac{268551}{5417}a^{5}-\frac{134181}{5417}a^{4}+\frac{294009}{21668}a^{3}+\frac{76928}{5417}a^{2}+\frac{66741}{21668}a-\frac{4741}{21668}$ 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:  \( 3126708.06364 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{3}\cdot 3126708.06364 \cdot 1}{2\cdot\sqrt{80421398252924334225227776}}\cr\approx \mathstrut & 0.177121418308 \end{aligned}\] (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 - 25*x^14 + 3*x^12 + 115*x^10 - 50*x^8 - 136*x^6 + 115*x^4 - 25*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 - 25*x^14 + 3*x^12 + 115*x^10 - 50*x^8 - 136*x^6 + 115*x^4 - 25*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 - 25*x^14 + 3*x^12 + 115*x^10 - 50*x^8 - 136*x^6 + 115*x^4 - 25*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 - 25*x^14 + 3*x^12 + 115*x^10 - 50*x^8 - 136*x^6 + 115*x^4 - 25*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

$A_4^2:C_2^2$ (as 18T175):

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 576
The 28 conjugacy class representatives for $A_4^2:C_2^2$
Character table for $A_4^2:C_2^2$

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1

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]
 

Sibling fields

Degree 12 sibling: data not computed
Degree 16 sibling: data not computed
Degree 18 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.4.217975320865705984.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ R ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$ R ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.3.2.6a2.2$x^{6} + 4 x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 7$$2$$3$$6$$A_4\times C_2$$$[2, 2, 2]^{3}$$
2.6.2.12a6.1$x^{12} + 4 x^{10} + 4 x^{9} + 3 x^{8} + 8 x^{7} + 7 x^{6} + 4 x^{5} + 10 x^{4} + 6 x^{3} + x^{2} + 4 x + 5$$2$$6$$12$12T87$$[2, 2, 2, 2, 2]^{6}$$
\(7\) Copy content Toggle raw display 7.6.3.12a1.3$x^{18} + 3 x^{16} + 15 x^{15} + 15 x^{14} + 48 x^{13} + 109 x^{12} + 171 x^{11} + 333 x^{10} + 497 x^{9} + 717 x^{8} + 1032 x^{7} + 1216 x^{6} + 1296 x^{5} + 1143 x^{4} + 783 x^{3} + 432 x^{2} + 162 x + 34$$3$$6$$12$$C_6 \times C_3$$$[\ ]_{3}^{6}$$
\(53\) Copy content Toggle raw display 53.3.1.0a1.1$x^{3} + 3 x + 51$$1$$3$$0$$C_3$$$[\ ]^{3}$$
53.3.1.0a1.1$x^{3} + 3 x + 51$$1$$3$$0$$C_3$$$[\ ]^{3}$$
53.3.2.3a1.2$x^{6} + 6 x^{4} + 102 x^{3} + 9 x^{2} + 306 x + 2654$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
53.3.2.3a1.2$x^{6} + 6 x^{4} + 102 x^{3} + 9 x^{2} + 306 x + 2654$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$

Spectrum of ring of integers

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