366 nonparametric cases (degree <= 60) A1 - A24 j = 1728 B1 - B34 j in Q, t in Q. C1 - C42 j in Q, t quadratic, t real D1 - D50 j in Q, t quadratic, t not real E1 - E25 j in Q, t not quadratic [Q(t):Q] = 6. F1 - F25 j quadratic, j real G1 - G52 j quadratic, j not real H1 - H53 [Q(j):Q] = 3 I1 - I33 [Q(j):Q] = 4 or 5 J1 - J28 [Q(j):Q] = 6, 7, 8, 9, 10, 11, 13, 14, or 15. 48 parametric cases (degree <= 12) S1 - S48 Parametric cases, from arxiv.org/abs/1204.2730, where they are called H1 .. H48. 3 additional entries (degree 6, 24, 18) S49 - S51 These three are NOT part of our table of all hyperbolic 4-to-3 rational Belyi maps. They are 2F1 -> 2F1 transformations that were added to our files for convenience (in order to list all decompositions of the 366 + 48 members of our table). Remark 1: If L in C(x)[d/dx] is a Heun equation, and if it has no Liouvillian solutions, and if it has a solution of the form: exp(Int(R, x)) * hypergeom([a,b],[c], F) where R and F are rational functions, then subs(x = m, G) appears in tables A-J or in S1 - S48 for some Mobius transformation m, and some G in {F, 1/F, 1-F, 1-1/F, 1/(1-F), 1/(1-1/F) }. So, up to Mobius transformations (and up to conjugacy over Q), the number of such F's is 366 + 48. Remark 2: Not listed here are those Heun equations that have a 2F1-solution where F is an algebraic but not a rational function. Remark 3: Also not listed are those Heun's that have a Liouvillian solution. In the Liouvillian case, to find F one can use the Maple program DEtools[kovacicsols]. However, in the Liouvillian case, F can have arbitrarily high degree (the degree depends on the exponent-differences, there is no bound that depends only on the order) (in fact, that's already true for 2F1 -> 2F1). For example, to see a Belyi map of degree 1001, type the following: ode := 329329/600*y(x) + (64-127*x)*diff(y(x),x) + 6*x*(x-1)*diff(y(x),x,x); sols := DEtools[kovacicsols](ode, y(x)): # takes about five minutes and 3 Gb. F := op(-1, indets(sols,function)[1]); Remark 4: There exist Heun equations that have a solution of the form: exp(Int(R, x)) * ( R1*hypergeom([a1,b1],[c1], F) + R2*hypergeom([a2,b2],[c2], F) ) (with R,R1,R2,F rational functions) but that do not have a solution of the form as in Remark 1. Such Heun's (and corresponding F's) are not part of our table. Our table covers only those F's described in Remark 1. An example of such an equation is: 16*(x^2+3)*((x-37)*y''+y')-9*(x+9)*y = 0.