# Let H420 be the Gauss Hypergeometric Differential Equation with exponent difference (1/4,1/2,0).
# Let H420(f) be the differential operator obtained after applying the "change of variables" f on H420.
# This table contains ALL (20) Belyi minus 1 maps f such that H420(f) has 5 non removable singularities.
# The second element on each entry gives the list of exponent differences of H420(f). 
# The table omits 1 branching pattern from the paper which have no Belyi minus 1 maps.


Belyi_one_420 := [[s*x*(x-1), [0, 1/4, 1/4, 1/2, 1/2]], [s/(x*(x-1)), [0, 0, 1/2, 1/2, 1/2]], [-s*x^3+2*s*x^2-s*x+1, [0, 1/4, 1/4, 1/4, 1/2]],
 [(s-x)^2*x/((s*x-3*x+2)*(s-1)), [0, 0, 1/4, 1/2, 1/2]], [-1/(s*x^3-2*s*x^2+s*x-1), [0, 0, 0, 1/2, 3/4]], [-s/(x^2*(2*s*x+x^2-3*s-2*x+1)),
 [0, 0, 0, 1/2, 1/2]], [-s*x^4+2*s*x^3-s*x^2+1, [0, 1/4, 1/4, 1/4, 1/4]], [-(1/2)*(-8*s*x^2+s^2+6*s*x-3*x^2+2*x)/((2*x-1)^3*x), [0, 0, 1/4, 1/4, 1/2]],
 [-4*s*(s*x^2-x-1)*x^2/(x+1)^2, [0, 0, 1/4, 1/4, 1/2]], [-2*(2*x-1)^3*x/(-8*s*x^2+s^2+6*s*x-3*x^2+2*x), [0, 0, 0, 1/4, 3/4]], 
[-(1/4)*(x+1)^2/(s*(s*x^2-x-1)*x^2), [0, 0, 0, 1/2, 1/2]], [(1/4)*(3*s*x-x+1)/((s^2*x^2+2*s^2*x+s*x^2+s^2+s*x+1)*x^3*s), [0, 0, 0, 1/4, 1/2]],
 [(2*s^2+2*s*x-2*s-x+1)/((x+1)*(2*s*x+2*x^2-s-2*x)^2), [0, 0, 0, 1/4, 1/2]], [-(s*x^2-2*s*x+s-2)*x^4*s/(2*s*x^2+x^2+2*x+1), [0, 0, 0, 1/4, 1/4]],
 [4*(2*s^2*x+2*s*x^2-s^2+2*s*x-x^2)/((x+4)*x^3*(x+2+s)^2), [0, 0, 0, 1/4, 1/4]], [-(1/4)*s^2*(x+1)^2/(x^3*(x^3+s*x+s)), [0, 0, 0, 0, 1/2]],
 [-s*(s-2)^3*x^2/((2*s*x+2*x^2+s)^2*(2*s*x+x^2-2*x+1)), [0, 0, 0, 0, 1/2]], [-(1/4)*s^2*(2*x-1)^4/(x^4*(x^4+4*s*x^2-4*s*x+s)), [0, 0, 0, 0, 0]],
 [-(1/4)*(s*x+x^2-s)^4/(s^2*(s^2+2*s*x+x^2-2*s)*(2*x-1)*x^2), [0, 0, 0, 0, 0]], [(1/16)*(x^2-1)^4/((x^2-s-1)*(x^2-2*s-1)^2*s), [0, 0, 0, 0, 0]]]: