# This file contains Belyi maps producing 5 non removable 
# singularities above 0,1,infinity with exp. diff 1/6,1/2,0 
# respectively. Let the branching pattern of f be 
# [a1,a2,..,ai],[b1,b2,..,bj],[c1,c2,..ck] above 0,1,infinity
# respectively. Then " #ai's not divisible by 6 + #bj's not 
# divisible by 2 + #ck's = 5".

# Syntax := [Belyi map, 5 point invariant, min poly of 5 point invariant];

# There are 44 such maps (without counting conjugates). 


Belyi_620 := [[-(x-1)*(x^2+x+1), 2048, x-2048], [(2*x-1)*(x+1)^2/((2*x+3)*x^2), 2583940/81, 81*x-2583940], [x^3, 2048, x-2048], [-1/((x-1)*(x^2+x+1)), 2048, x-2048], [-(1/27)*(x-4)^3*x, 9570238/6561, 6561*x-9570238], [(2*x^2-1)^2, 325448, x-325448], [-27/((x-4)^3*x), 9570238/6561, 6561*x-9570238], [1/(2*x^2-1)^2, 325448, x-325448], [-(1/64)*(9*x^2-33*x+64)*(x-1)^3, 1720308119/1244160, 1244160*x-1720308119], [(1/4)*(x^2-5*x+5)^2*x, 601024/5, 5*x-601024], [(-7+24*RootOf(_Z^2+1))*(-x+1+2*RootOf(_Z^2+1))^4*x/(5*x-1+2*RootOf(_Z^2+1))^4, -399392/625-(2125656/625)*RootOf(_Z^2+1), 625*x^2+798784*x+7484683840], [(1/4)*(5*x-1)^4/((2*x+5)^3*x^2), 29261134309681/16402500, 16402500*x-29261134309681], [4*(2*x+5)^3*x^2/(5*x-1)^4, 29261134309681/16402500, 16402500*x-29261134309681], [(4*x+5)^2*x^3/(5*x+4)^2, 99220249/5625, 5625*x-99220249], [-64/((9*x^2-33*x+64)*(x-1)^3), 1720308119/1244160, 1244160*x-1720308119], [4/((x^2-5*x+5)^2*x), 601024/5, 5*x-601024], [-4*(x-1)*(x^2+x+1)*x^3, 2048, x-2048], [-(x-1)*(x+1)*(2*x-1)^2*(2*x+1)^2, 2583940/81, 81*x-2583940], [(1/4)*(x^2+2*x+5)*x^4/(2*x-1), -57968164/15625, 15625*x+57968164], [(27/4)*(x^2+4)/(x^2+3)^3, 9787148, x-9787148], [(1/64)*(x+5)^2*(x+8)*x^3/(3*x-1), 54816732281761/81000000, 81000000*x-54816732281761], [-4*(x-3)*(x-1)^3*x^2/(3*x-1)^2, 2583940/81, 81*x-2583940], [4*(2*x-1)/((x^2+2*x+5)*x^4), -57968164/15625, 15625*x+57968164], [64*(3*x-1)/((x+5)^2*(x+8)*x^3), 54816732281761/81000000, 81000000*x-54816732281761], [(27/4)*(x-1)^2*x^2/(x^2-x+1)^3, 2048, x-2048], [-(1/4)*(3*x-1)^2/((x-3)*(x-1)^3*x^2), 2583940/81, 81*x-2583940], [(4/27)*(x^2+3)^3/(x^2+4), 9787148, x-9787148], [(4/27)*(x^2-x+1)^3/((x-1)^2*x^2), 2048, x-2048], [1/((2*x^2+3)*x^4), 2048, x-2048], [-(1/27)/((20*x^2+12*x+5)*x^4), 109416125286667/48828125000, 48828125000*x-109416125286667], [(1/3125)*(-1196+1968*RootOf(_Z^3-2)+681*RootOf(_Z^3-2)^2)/(x^3*(x-1)^2*(-25*x+4+18*RootOf(_Z^3-2)+6*RootOf(_Z^3-2)^2)), -(229598609256696/6103515625)*RootOf(_Z^3-2)+(615094487550404/18310546875)*RootOf(_Z^3-2)^2-530272598401664/18310546875, 54931640625*x^3+4772453385614976*x^2+554699028250877780928*x+10914643825816191478026496], [1/((4*x^2-3)^2*x^2), 9787148, x-9787148], [(1/2352980)*(293-72*RootOf(_Z^3+3*_Z-1)+42*RootOf(_Z^3+3*_Z-1)^2)*(35*x+23+6*RootOf(_Z^3+3*_Z-1)^2+4*RootOf(_Z^3+3*_Z-1))^6/((x^2+RootOf(_Z^3+3*_Z-1)*x+RootOf(_Z^3+3*_Z-1)^2+2)*(-5*x+25+3*RootOf(_Z^3+3*_Z-1)^2-7*RootOf(_Z^3+3*_Z-1))^5), -(175019114828112848/778656005859375)*RootOf(_Z^3+3*_Z-1)+(15006678959891632/86517333984375)*RootOf(_Z^3+3*_Z-1)^2+168981449140071808/86517333984375, 32328884875774383544921875*x^3-155784391944331002423547500000*x^2+255991253320594564750724652000000*x-145727765186048576999958823030501376], [(-9*x-5-6*RootOf(_Z^4-2*_Z^3+3)^2+2*RootOf(_Z^4-2*_Z^3+3)^3+8*RootOf(_Z^4-2*_Z^3+3))^6*(x-1)/((9*x-11-2*RootOf(_Z^4-2*_Z^3+3)^2+2*RootOf(_Z^4-2*_Z^3+3)^3)^4*(9*x-71+102*RootOf(_Z^4-2*_Z^3+3)-50*RootOf(_Z^4-2*_Z^3+3)^2+8*RootOf(_Z^4-2*_Z^3+3)^3)^2*x), (52922364402746202304/120135498046875)*RootOf(_Z^4-2*_Z^3+3)-(41212884993819730912/120135498046875)*RootOf(_Z^4-2*_Z^3+3)^2+(15904285196914523744/360406494140625)*RootOf(_Z^4-2*_Z^3+3)^3-11692854645806164768/120135498046875, 139628860198736572265625*x^4+73648792903925130899030683776*x^3+91109762230745120311720878766110720*x^2+11272334831020010837938935187558076055552*x+2426181393230105176625532338786080558374977536], [(3/7)*(RootOf(_Z^2+3)+9)*(7*x+2+6*RootOf(_Z^2+3))/((x-1)*(6*x^2+6*x*RootOf(_Z^2+3)-RootOf(_Z^2+3)-3)^3), -(166313960/567)*RootOf(_Z^2+3)-1312664/7, 15309*x^2+5741592336*x+4489817724564736], [(2*x+35)*x^6/(35*x^2-56*x+20)^2, 29650515133348979209/512695312500, 512695312500*x-29650515133348979209], [4*(x^2+2)*x^6/((2*x-1)*(2*x+1)), 14680078/6561, 6561*x-14680078], [-108*(7*x^2-12*x+12)/((x^2+6*x+21)*x^6), 9155180473681/3087580356, 3087580356*x-9155180473681], [-(1/64)*(280*x^2+48*x+9)/((x-2)*(5*x-7)^2*x^5), -8437992764661092702606882/39260614471435546875, 39260614471435546875*x+8437992764661092702606882], [-(9/64)*(-71+39*RootOf(_Z^2+3))*(-2*x^2+3+RootOf(_Z^2+3))*(-6*x+7+RootOf(_Z^2+3))^6/((-21*x+27+4*RootOf(_Z^2+3))^4*(-12*x+15+2*RootOf(_Z^2+3))), (289911237642/117649)*RootOf(_Z^2+3)-135897720094/117649, 117649*x^2+271795440188*x+2300179070456151472], [64*(x-1)*(x+1)/((9*x^2-8)^3*x^2), 341808898, x-341808898], [-(1/4)*(4*x+3)^2/((x^2-2*x+3)*(x+1)^2*x^4), -259058528/59049, 59049*x+259058528], [-64*x^2/((x-1)*(x+1)*(3*x-1)^3*(3*x+1)^3), 12240, x-12240], [(1/16)*(x+4)^2*x^6/((2*x-1)*(x^2+2*x-2)^2), 2605264/3, 3*x-2605264]]: