# Let H320 be the Gauss Hypergeometric Differential Equation with exponent difference (1/3,1/2,0).
# Let H320(f) be the differential operator obtained after applying the "change of variables" f on H320.
# This table contains ALL (65) Belyi minus 1 maps f such that H320(f) has 5 non removable singularities.
# The second element on each entry gives the list of exponent differences of H320(f). 
# The table omits 6 branching patterns from the paper which have no Belyi minus 1 maps.



Belyi_one_320 := [[s*x*(x-1), [0, 1/3, 1/3, 1/2, 1/2]], [s/(x*(x-1)), [0, 0, 1/2, 1/2, 2/3]], [s/(x*(x-1)^2), [0, 0, 1/2, 1/2, 1/2]],
 [-s*x^3+2*s*x^2-s*x+1, [0, 1/3, 1/3, 1/3, 1/2]], [(s-x)^2*x/((s*x-3*x+2)*(s-1)), [0, 0, 1/3, 1/2, 2/3]], [(3*x-4+s)*x^3/(3*s*x-2*s-1), [0, 0, 1/3, 1/2, 1/2]],
 [2*(2*x-1)^3*x/(-4*x^2+s+3*x)^2, [0, 0, 1/3, 1/2, 1/2]], [-s*x^4+2*s*x^3-s*x^2+1, [0, 1/3, 1/3, 1/3, 1/3]], [-(1/2)*(-8*s*x^2+s^2+6*s*x-3*x^2+2*x)/((2*x-1)^3*x),
 [0, 0, 1/3, 1/3, 2/3]], [-4*s*(s*x^2-x-1)*x^2/(x+1)^2, [0, 0, 1/3, 1/3, 2/3]], [-(1/4)*(x+1)^2/(s*(s*x^2-x-1)*x^2), [0, 0, 0, 2/3, 2/3]],
 [-1/(s*x^4-2*s*x^3+s*x^2-1), [0, 0, 0, 0, 4/3]], [4*(s^2*x^2+2*s^2*x+s*x^2+s^2+s*x+1)*s*x^3/(3*s*x-x+1), [0, 0, 1/3, 1/3, 1/2]],
 [(4/27)*(s^2*x^2-11*s*x^2+4*s*x+10*x^2+5*x+4)/((s*x-s-x)^2*x^3), [0, 0, 1/3, 1/3, 1/2]], [(27/4)*(s*x-s-x)^2*x^3/(s^2*x^2-11*s*x^2+4*s*x+10*x^2+5*x+4),
 [0, 0, 0, 1/2, 2/3]], [4*s*(16*s^2*x+8*s*x+5*s+x+1)^2*x^3/(4*s*x^2+5*s*x+x^2+2*x+1)^2, [0, 0, 0, 1/2, 2/3]], [(s*x-1)^3*x^3/(3*s*x^2-3*x+2), 
[0, 0, 0, 1/2, 1/2]], [-(2/27)*s^3*(2*s*x^2+1)^3/(2*s^3*x^2+5*s^2*x^2+2*s^2*x+s^2+6*s*x+4*s+5), [0, 0, 0, 1/2, 1/2]],
 [-(1/27)*(s^3*x^2+2*s^2*x^2-6*s^2*x+s*x^2+4*s-2)^3/(s*(s-1)^2*(x-1)^2*(2*s*x-1)), [0, 0, 0, 1/2, 1/2]], [-(s*x-2)^3*x^3/(3*s*x^2-6*x+4)^2, [0, 0, 0, 1/2, 1/2]],
 [-(1/4)*(4*s^2*x^3+s^2*x^2+4*s*x^3-4*s*x^2-2*s*x+1)/(s^2*(x-2)*x^5), [0, 0, 1/3, 1/3, 1/3]], [-s*(-4*x^3+6*x^2+s)/((2*x-3)^2*x^4), [0, 0, 1/3, 1/3, 1/3]],
 [-(1/4)*(8*s^2*x^3-3*s*x^2+12*s*x+4*s-4)/(s*x^2+1)^3, [0, 0, 1/3, 1/3, 1/3]], [-4*(x-2)*(s*x-s-x)^2*x^3/(s^2*x^2-6*s*x^2+2*s*x+5*x^2+2*x+1), [0, 0, 0, 1/3, 2/3]],
 [(1/4)*(-x+12*s+9)*(x-1)^3*x^2/((4*s^2+5*s*x+3*s+4*x)^2*(s^2+2*s*x+x)), [0, 0, 0, 1/3, 2/3]], [8*s*(2*x-3)*x^2/(4*s*x^3-6*s*x^2+1)^2, [0, 0, 0, 1/3, 2/3]],
 [4*(3*s^2*x^2+2*s-x+1)^3*(x+1)/(36*s^4*x^2+36*s^3*x^2-4*s^3*x+12*s^2*x^2+32*s^3-15*s^2*x+45*s^2-12*s*x+24*s-4*x+5), [0, 0, 0, 1/3, 1/2]],
 [-(1/1728)*(1125*s^4+800*s^3-9*s^2*x+210*s^2-2*s*x+24*s-x+1)*(11520*s^6+4864*s^5+45*s^4*x+1792*s^4+88*s^3*x-9*s^2*x^2+256*s^3+42*s^2*x-2*s*x^2-x^2+x)^3/
(s^3*(21*s^2+6*s+1)^6*(63*s^3+5*s^2+5*s+x-1)*x^2), [0, 0, 0, 1/3, 1/2]], [(x+1)*(3*s^4-8*s^3*x-6*s^2*x-x^2)^3/(4*s^4*x+3*s^4+4*s^3*x+6*s^2*x-4*s*x^2-x^2)^3,
 [0, 0, 0, 1/3, 1/2]], [(1/2)*(x^2-2*s+7)^3*(x-s+21)/((-28+3*s)*x^2+8*s*x-27*s+60+2*s^2)^2, [0, 0, 0, 1/3, 1/2]], [-(1/64)*(x^2+8*s-9)*(x+1)^3*(x-1)^3/
((s-1)^3*(-x^2+s)), [0, 0, 0, 1/3, 1/3]], [(1/4)*(s^2-2*s*x+x^2+6*s-6*x+21)*(-x^2+4*s)^3/((s+3)^3*(-3*s*x^2+16*s^2+4*s*x+7*x^2-12*s+12*x+12)), [0, 0, 0, 1/3, 1/3]],
 [(1/64)*(280*s^2+48*s*x+9*x^2-81*s)*(-x^2+s)^3/(s^3*(5*s^2-7*s*x-s+3*x)^2*(-2*x+s+1)), [0, 0, 0, 1/3, 1/3]], [(1/16)*(-2*s^2*x^2+4*s^3-12*s^2*x+12*s*x^2-36*s^2
+62*s*x-18*x^2+105*s-78*x-98)*(2*s^2*x^2+4*s^2*x-12*s*x^2-22*s*x+18*x^2-s+30*x+2)^3/((s*x+2)^3*(s-2)^3*(s-3)^4*x), [0, 0, 0, 1/3, 1/3]],
 [(1/16)*(x^2+s)^3*(x^2+9*s-8*x+16)/((2*s*x-3*x^2+s)^2*s), [0, 0, 0, 1/3, 1/3]], [-(1/64)*(9*x^2+8*s-9*x)*(x-1)^3*x^3/(s*(x^2+s-x)^3), [0, 0, 0, 1/3, 1/3]],
 [-16*(x^2+s)^3*(2*x-1)^2/(s^2*(8*s*x^2+24*x^3+9*s^2+28*s*x-12*x^2-16*s)), [0, 0, 0, 0, 2/3]], [(1/1728)*(-x^2+24*s-16*x-16)^3*x^2/((s-x)^2*(-s*x^2+27*s^2
-18*s*x-x^2-16*x)), [0, 0, 0, 0, 2/3]], [-64*(s*x+x^2-s)^3*x^2/((x-1)^3*s^3*(9*s*x+8*x^2-9*s)), [0, 0, 0, 0, 2/3]], [(2*x+8*s-1)^2*(-x^2+s)^3/(s*(16*s^3
-16*s^2*x-8*s*x^2-4*s^2+8*s*x+3*x^2+s)^2*(-2*x+s+1)), [0, 0, 0, 0, 2/3]], [4*(s*(x-1)*x^2+1)^3/(3*s*(x-1)*x^2+4), [0, 0, 0, 0, 1/2]],
 [-4*(x^3+2*x^2+(2*s+1)*x+s)^3/(s^2*((4*s-3)*x^3+6*(s-1)*x^2+3*(3*s^2-2*s-1)*x-4*s)), [0, 0, 0, 0, 1/2]], [(4/27)*(s^2*x-2*s*x^2+x^3-2*s*x-4*x^2+3*s
+10*x-6)^3/((4*s^3-8*s^2*x+4*s*x^2-24*s^2-4*s*x-4*x^2+48*s+13*x-32)*(s*x-1)^2), [0, 0, 0, 0, 1/2]], [-(4/27)*(s^2*x-4*s*x^2+x^3-2*s*x+4*x^2+10*x+6)^3/
((4*s^2*x-s*x^2-20*s*x+4*x^2+13*x+32)*(s*x-1)^3), [0, 0, 0, 0, 1/2]], [(1/27)*(8*s^3*x^3+4*s^2*x^3-12*s^2*x^2-8*s*x^2+x+1)^3/(s*(2*s^2*x+s*x-3*s-2)*(x+1)^2*x^2),
 [0, 0, 0, 0, 1/2]], [-(1/432)*(8*(1+3*s)^2*x^3-12*s^2*x^2+(-576*s^3-330*s^2-72*s-6)*x-6*s+15*s^2-1+96*s^3)^3/((6*(1+3*s)^2*x+1-2*s-27*s^2)*(4*s+1)^5*(x^2+s)^3),
 [0, 0, 0, 0, 1/2]], [(1/27)*(s*x^3-2*s*x^2+s*x+3)^3/(s*x^3-2*s*x^2+s*x-1)^2, [0, 0, 0, 0, 1/2]], [(8/27)*(2*s*x^2+4*x^3+6*x-3)^3*(2*x+s)/(4*s^2*x^2+8*s*x^3
+12*s*x^2+24*x^3+12*s*x-3*x^2-8*s+38*x-27), [0, 0, 0, 0, 1/3]], [8*(s^4+2*s^3*x-12*s^2*x-36*s*x^2-18*x^3+6*s^2+18*s*x+9)^3*(x-2)/((s+3)^8*(s^4-12*s^2*x-8*s*x^2
+6*s^2+16*s*x+12*x^2+12*x+9)*x^2), [0, 0, 0, 0, 1/3]], [-(1/64)*(x-1)*(s^4*x-9*s^4+6*s^2*x^2+42*s^2*x-48*s*x^2+9*x^3-16*s*x+15*x^2)^3/((s-1)^8*(s^4+8*s^3
+6*s^2*x-40*s*x+9*x^2+16*x)*x^3), [0, 0, 0, 0, 1/3]], [(4/27)*(-5*s^2*x^2+12*s*x^3+6*s^3-18*s^2*x+6*s*x^2-12*x^3+9*s^2-8*s*x+15*x^2-6*x+1)^3/((s-1)^3*(-27*s*x^2
+32*s^2-18*s*x+27*x^2-3*s-14*x+3)*(-x^2+s)^4), [0, 0, 0, 0, 1/3]], [-(1/64)*(x-9)*(s^2*x+6*s*x^2+9*x^3-s^2-22*s*x-57*x^2+64*x)^3/((s^3+6*s^2*x+9*s*x^2-9*s^2
-6*s*x+63*x^2-192*x)^2*x), [0, 0, 0, 0, 1/3]], [-(s^2*x-8*s*x^2+4*x^3+44*s*x-48*x^2-48*s+168*x-168)^3*x/((s*x+6)^3*(6*s^2-3*s*x+36*s-8*x+54)*(3*x-8)^2), 
[0, 0, 0, 0, 1/3]], [(1/4)*s*(-5*s*x^2-4*x^3+4*s^2+14*s*x+8*x^2-s)^3/((-s*x^2+4*s^2+6*s*x+3*s+4*x)^2*(x^2+s)^3), [0, 0, 0, 0, 1/3]],
 [(4/27)*(s*x^4-2*s*x^3+s*x^2-3)^3/(s*x^4-2*s*x^3+s*x^2-4), [0, 0, 0, 0, 0]], [(1/32)*s^2*(4*s^2*x^4-4*s*x^2+16*x+1)^3/(12*s^3*x^4-16*s^2*x^3-12*s^2*x^2
+72*s*x+3*s-72), [0, 0, 0, 0, 0]], [-(4/27)*(x^4-4*s*x^2+s^2+12*s*x)^3/((-s*x^2+8*x^3+4*s^2-36*s*x+108*s)*s^3*(x-2)^2), [0, 0, 0, 0, 0]],
 [64*(-x^3+6*x^2+9*s)^3*s/((-x^3+6*x^2+8*s)*(x-6)^3*x^6), [0, 0, 0, 0, 0]], [(1/1728)*(s^4-12*s^3*x+14*s^2*x^2+12*s*x^3+x^4+240*s^2*x-360*s*x^2-120*x^3
+3600*x^2)^3/((s-15)^3*s^2*(s^2*x-11*s*x^2-x^3-15*s^2+180*s*x+120*x^2-3600*x)*x^4), [0, 0, 0, 0, 0]], [(4/27)*(s^2*x^4+2*s^2*x^3+s^2*x^2-4*s*x^2-s*x+1)^3/
((s*x^2+2*s*x+s-4)*s*(s*x-1)^2*x^2), [0, 0, 0, 0, 0]], [-(1/1728)*(s^4+12*s^3*x+14*s^2*x^2-12*s*x^3+x^4+32*s^2*x+48*s*x^2-16*x^3+24*s*x+40*x^2)^3/
((s+1)^4*(s^3+11*s^2*x-s*x^2+36*s*x-2*x^2+27*x)*(x-1)^2*x^3), [0, 0, 0, 0, 0]], [(1/27)*(s^2*x^2+8*s*x^3+4*x^4-2*s^2*x-8*s*x^2+s^2)^3/
(s^4*(2*s*x+x^2-2*s)*(x-1)^4*x^2), [0, 0, 0, 0, 0]], [-(4/27)*(s^2*x^4+2*s^2*x^3+s^2*x^2-s*x^2-4*s*x+1)^3/(s^2*(4*s*x-1)*(s*x^2+2*s*x+s-1)^2*x^4),
 [0, 0, 0, 0, 0]], [4*(2*s*x^3-2*s*x^2-1)^3/((x-1)*s*(s*x^3-s*x^2+4)^3*x^2), [0, 0, 0, 0, 0]], [-(1/27)*(s*x^4-2*s*x^3+s*x^2-3)^3/(s*x^4-2*s*x^3+s*x^2+1)^2,
 [0, 0, 0, 0, 0]]]: