[Joint work: M. van Hoeij and R. Vidunas] Notation: If the 2F1 equation has exponent differences (1/d1, 1/d2, 1/d3) and if the Heun equation has an exponent-difference e = a/b, then this e comes from a d.i for which b | d.i If there is more than one such d.i, and if e comes from a non-minimal d.i, then that will be indicated at the end of the line. Example: Take for example the following 2F1 --> Heun (1/2,1/3,1/8) --> (1/2,1/2,1/8,1/8) These exponent-differences are compatible with two potential branching types: Type #1: (1^2,2^8) = (1,1,2,2,2,2,2,2,2,2) (3^6) = (3,3,3,3,3,3) (1^2,8^2) = (1,1,8,8) Type #2: (2^9) = (2,2,2,2,2,2,2,2,2) (3^6) = (3,3,3,3,3,3) (1^2,4^2,8) = (1,1,4,4,8) Table 2.3.8 lists the following data for (1/2,1/3,1/8) --> (1/2,1/2,1/8,1/8). F4,J1 no covering (4/8,4/8,1/8,1/8) No further indications are given for F4 and J1, which means that the 1/2 in F4,J1 comes from the 1/2 in (1/2,1/3,1/8). So F4 and J1 have Type #1. The (4/8,4/8,1/8,1/8) refers to Type #2, and the table states that there is no covering for this type. In other words, there is no dessin with 9 edges, 6 vertices of degree 3, and faces of degrees 1,1,4,4,8. Remarks regarding completeness: All our BelyiMaps were computed with the same Maple implementation. The question is, how can we verify the resulting table is complete? The Belyi maps mentioned in the tables below are given in the file BelyiMaps, and their dessins in: Dessins_and_ModuliFields. One can verify that the tables are complete by finding all dessins via a combinatorial search, and then verifying that all those dessins appear in the file Dessins_and_ModuliFields. In Maple, our combinatorial dessin-search reached degree 36. Table 2.3.7 goes up to degree 60 (too high for a combinatorial dessin-search in Maple, though it should be feasible with a C implementation). So we verified completeness in a different way: we implemented not one but two completely independent programs to find these Belyi maps. Thus, completeness of our set of dessins depends on the completeness of our set of BelyiMaps (instead of the other way around). In turn, completeness of our set of BelyiMaps depends on having two algorithms to compute them. One algorithm computed all coverings given in the file BelyiMaps, and the other is used for verification of completeness. We did many other verifications as well, in particular, all 125 dessins belonging to the quadrangles from [Felikson] appear in our tables, see the Figure/Entry references in the file Dessins_and_ModuliFields. Our table has 366 Belyi maps, so this checks about one third of our table (including many high-degree examples), with no gaps found. These quadrangles correspond to BelyiMaps that still have real coefficients when their 4 singularities are moved to 0,1,t,infinity. If our table were incomplete due to a bug, then it would be difficult to understand why none of the 125 cases from [Felikson] are missing. Degree 24 Belyi maps were checked by comparing with [Beukers-Montanus]. Remark concerning the S-table: Some Belyi maps are applicable to more than one 2F1 --> Heun. These Belyi Maps are called parametric (meaning that at least one of the 2F1 exponent-differences is allowed to vary). For details concerning the elements of the S-table (such as j-invariants, exponent-differences, etc), see arxiv.org/abs/1204.2730. An implementation to solve Heun equations in terms of 2F1's would have to treat the parametric cases (the S-table) separately from the non-parametric cases (tables A-J). Each of the 366 BelyiMaps in the A-J tables corresponds to only one 2F1 --> Heun. Solving the non-parametric cases is now a straightforward table lookup (compare the j-invariant of the Heun equation with the j's in the file j_invariant_or_its_minpoly). Members of the S-table correspond to multiple 2F1 -> Heun's, and some members of the S-table correspond to more than one j. This is why arxiv.org/abs/1204.2730 lists 61 cases (for 48 BelyiMaps). A parametric solver would need either a table of those 61 cases, or some code that can recover those cases from the 48 Belyi maps (the file BelyiMaps has 48+3 members in the S-table, but those latter 3 produce only 2F1 -> 2F1, and not 2F1 -> Heun, so S49, S50, S51 are not needed for a Heun solver. They were added to the list in order to describe all decompositions). =========== Table 2.3.7. (1/2,1/3,1/7) --> Heun ========== Degree 7: (1/2,1/2,1/2,1/3) : G35 (1/2,1/3,1/3,2/3) : H35 G35: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[1,3,3] 1:[1^3,2,2] infty:[7] Decomposition: none Galois group: PSL(3,2). GAP transitive group ID = 7T5. Order = 168 H35: ModuliField: Q(x^3-x^2+5*x+1) = Q(28^(1/3)). Obstruction: none Branch type: 0:[1,1,2,3] 1:[1,2^3] infty:[7] Decomposition: none Galois group: S7. Order = 7! Degree 8: (1/2,1/2,2/3,1/7) : H47 (1/3,2/3,2/3,1/7) : C29 H47: ModuliField: Q(x^3-x^2+5*x+15). Obstruction: none Branch type: 0:[2,3,3] 1:[1,1,2^3] infty:[1,7] Decomposition: none Galois group: S8. Order = 8! C29: ModuliField: Q. j-invariant = 2^7*5^6*1607^3*3^(-16)*7^(-5) Obstruction: none Branch type: 0:[1,2,2,3] 1:[2^4] infty:[1,7] Decomposition: none Galois group: A8. Order = 8!/2 [Felikson, Figure 18 (2,3,7), Entry #8] Degree 9: (1/2,1/2,1/2,2/7) : E21 (1/2,1/3,2/3,2/7) : I15 E21: ModuliField: Q. j-invariant = -3^1*223^3*2^(-8) Obstruction: none Branch type: 0:[3^3] 1:[1^3,2^3] infty:[2,7] Decomposition: none Galois group: S9. Order = 9! I15: ModuliField: Q(x^4-x^3+16*x-4). Obstruction: none Branch type: 0:[1,2,3,3] 1:[1,2^4] infty:[2,7] Decomposition: none Galois group: S9. Order = 9! [Felikson, Figure 18 (2,3,7), Entries #9, #12] Degree 10: (1/2,1/2,1/3,3/7) : H34 (1/3,1/3,2/3,3/7) : D30 (2/3,2/3,1/7,2/7) : C40 (4/3,1/7,1/7,1/7) : E20 H34: ModuliField: Q(x^3-x^2+2*x-3). Obstruction: none Branch type: 0:[1,3^3] 1:[1,1,2^4] infty:[3,7] Decomposition: none Galois group: A10. Order = 10!/2 D30: ModuliField: Q. j-invariant = -5281^3*3^(-16)*5^(-1) Obstruction: none Branch type: 0:[1,1,2,3,3] 1:[2^5] infty:[3,7] Decomposition: none Galois group: S10. Order = 10! C40: ModuliField: Q. j-invariant = 3^3*27329^3*2^(-14)*5^(-1)*7^(-5) Obstruction: none Branch type: 0:[2,2,3,3] 1:[2^5] infty:[1,2,7] Decomposition: none Galois group: S10. Order = 10! [Felikson, Figure 18 (2,3,7), Entry #14] E20: ModuliField: Q. j-invariant = -2^1*3^3*7^2 Obstruction: none Branch type: 0:[3,3,4] 1:[2^5] infty:[1^3,7] Decomposition: none Galois group: S10. Order = 10! Degree 11: (1/2,1/3,1/3,4/7) : H27 (1/2,2/3,1/7,3/7) : H53 (1/2,2/3,2/7,2/7) : C42 H27: ModuliField: Q(x^3-x^2+x+1). Obstruction: none Branch type: 0:[1,1,3^3] 1:[1,2^5] infty:[4,7] Decomposition: none Galois group: S11. Order = 11! H53: ModuliField: Q(x^3-x^2+12*x-6). Obstruction: none Branch type: 0:[2,3^3] 1:[1,2^5] infty:[1,3,7] Decomposition: none Galois group: S11. Order = 11! [Felikson, Figure 18 (2,3,7), Entry #15] C42: ModuliField: Q. j-invariant = 3^3*289189^3*2^(-18)*5^(-7)*11^(-1) Obstruction: none Branch type: 0:[2,3^3] 1:[1,2^5] infty:[2,2,7] Decomposition: none Galois group: S11. Order = 11! [Felikson, Figure 18 (2,3,7), Entry #10] Degree 12: (1/2,1/2,1/7,4/7) : G38 (1/2,1/2,2/7,3/7) : F22 (1/3,1/3,1/3,5/7) : E14 (1/3,2/3,1/7,4/7) : B25 (1/3,2/3,2/7,3/7) : B20 G38: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[3^4] 1:[1,1,2^5] infty:[1,4,7] Decomposition: none Galois group: S12. Order = 12! F22: ModuliField: Q(sqrt(7)). Obstruction: none Branch type: 0:[3^4] 1:[1,1,2^5] infty:[2,3,7] Decomposition: none Galois group: S12. Order = 12! [Felikson, Figure 18 (2,3,7), Entry #11] E14: ModuliField: Q. j-invariant = -2^8*3^3*61^3*5^(-7) Obstruction: none Branch type: 0:[1^3,3^3] 1:[2^6] infty:[5,7] Decomposition: none Galois group: A12. Order = 12!/2 B25: ModuliField: Q. j-invariant = 2^6*7^3*31^3*271^3*3^(-10)*11^(-4) Obstruction: none Branch type: 0:[1,2,3^3] 1:[2^6] infty:[1,4,7] Decomposition: none Galois group: S12. Order = 12! [Felikson, Figure 18 (2,3,7), Entry #18] B20: ModuliField: Q. j-invariant = 7^3*2287^3*2^(-6)*3^(-2)*5^(-6) Obstruction: none Branch type: 0:[1,2,3^3] 1:[2^6] infty:[2,3,7] Decomposition: none Galois group: S12. Order = 12! [Felikson, Figure 18 (2,3,7), Entry #16] Degree 13: (1/2,1/3,1/7,5/7) : I21 (1/2,1/3,2/7,4/7) : H50 (1/2,1/3,3/7,3/7) : C33 I21: ModuliField: Q(x^4-x^3+9*x^2+5*x+4). Obstruction: none Branch type: 0:[1,3^4] 1:[1,2^6] infty:[1,5,7] Decomposition: none Galois group: A13. Order = 13!/2 H50: ModuliField: Q(x^3+4*x-2). Obstruction: none Branch type: 0:[1,3^4] 1:[1,2^6] infty:[2,4,7] Decomposition: none Galois group: A13. Order = 13!/2 [Felikson, Figure 18 (2,3,7), Entry #19] C33: ModuliField: Q. j-invariant = 112297^3*2^(-4)*3^(-20)*13^(-1) Obstruction: none Branch type: 0:[1,3^4] 1:[1,2^6] infty:[3,3,7] Decomposition: none Galois group: A13. Order = 13!/2 [Felikson, Figure 18 (2,3,7), Entry #20] Degree 14: (1/2,1/2,1/3,1/3) : G39, H12, I3 (1/3,1/3,1/3,2/3) : E19 (1/3,1/3,1/7,6/7) : D26 (1/3,1/3,2/7,5/7) : no covering (1/3,1/3,3/7,4/7) : no covering (2/3,1/7,1/7,5/7) : C27 (2/3,1/7,2/7,4/7) : G31 (2/3,1/7,3/7,3/7) : C34 (2/3,2/7,2/7,3/7) : C26 G39: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[1,1,3^4] 1:[1,1,2^6] infty:[7,7] Decomposition: G35(deg 2) Galois group: PSL(3,2). GAP transitive group ID = 14T10. Order = 168 H12: ModuliField: Q(x^3-x^2-2*x+1) = Q(Re(zeta_7)). Obstruction: Primes: [infinity, infinity], Conic: X^2+Y^2+RootOf(x^3-x^2-2*x+1)*Z^2 = 0 Branch type: 0:[1,1,3^4] 1:[1,1,2^6] infty:[7,7] Decomposition: none Galois group: PSL(2,13). GAP transitive group ID = 14T30. Order = 1092 I3: ModuliField: Q(x^4-x^3+3*x^2-4*x+2) = Q( (-7)^1/4) ). Obstruction: none Branch type: 0:[1,1,3^4] 1:[1,1,2^6] infty:[7,7] Decomposition: G35(deg 2) Galois group: (C2 x C2 x C2) . PSL(3,2). GAP transitive group ID = 14T33. Order = 1344 E19: ModuliField: Q. j-invariant = -2^12*7^1*17^3*23^3*3^(-13) Obstruction: none Branch type: 0:[1^3,2,3^3] 1:[2^7] infty:[7,7] Decomposition: none Galois group: S14. Order = 14! D26: ModuliField: Q. j-invariant = -2^17*3^3*7^3*13^(-4) Obstruction: none Branch type: 0:[1,1,3^4] 1:[2^7] infty:[1,6,7] Decomposition: none Galois group: S14. Order = 14! C27: ModuliField: Q. j-invariant = 2^4*37^3*271^3*3^(-6)*5^(-4) Obstruction: none Branch type: 0:[2,3^4] 1:[2^7] infty:[1,1,5,7] Decomposition: none Galois group: S14. Order = 14! [Felikson, Figure 18 (2,3,7), Entry #23] G31: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[2,3^4] 1:[2^7] infty:[1,2,4,7] Decomposition: none Galois group: S14. Order = 14! C34: ModuliField: Q. j-invariant = 3^3*127^3*5^(-6) Obstruction: none Branch type: 0:[2,3^4] 1:[2^7] infty:[1,3,3,7] Decomposition: none Galois group: S14. Order = 14! [Felikson, Figure 18 (2,3,7), Entry #1] C26: ModuliField: Q. j-invariant = 3^3*5^3*17^3 Obstruction: none Branch type: 0:[2,3^4] 1:[2^7] infty:[2,2,3,7] Decomposition: none Galois group: S14. Order = 14! [Felikson, Figure 18 (2,3,7), Entry #21] Degree 15: (1/2,1/2,1/2,1/7) : H45 (1/2,1/3,2/3,1/7) : J22 (1/2,1/7,1/7,6/7) : G25 (1/2,1/7,2/7,5/7) : H40 (1/2,1/7,3/7,4/7) : B33 (1/2,2/7,2/7,4/7) : no covering (1/2,2/7,3/7,3/7) : C35 H45: ModuliField: Q(x^3-x^2+5*x-13). Obstruction: none Branch type: 0:[3^5] 1:[1^3,2^6] infty:[1,7,7] Decomposition: none Galois group: A15. Order = 15!/2 J22: ModuliField: Q(x^10-3*x^9+14*x^8-39*x^7+110*x^6-217*x^5+381*x^4-478*x^3+518*x^2-370*x+200). Obstruction: none Branch type: 0:[1,2,3^4] 1:[1,2^7] infty:[1,7,7] Decomposition: none Galois group: S15. Order = 15! G25: ModuliField: Q(sqrt(-3)). Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,1,6,7] Decomposition: none Galois group: S15. Order = 15! H40: ModuliField: Q(x^3+2*x-2). Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,2,5,7] Decomposition: none Galois group: S15. Order = 15! [Felikson, Figure 18 (2,3,7), Entry #27] B33: ModuliField: Q. j-invariant = 2^4*106791301^3*3^(-14)*5^(-2)*7^(-8)*11^(-6) Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,3,4,7] Decomposition: none Galois group: S15. Order = 15! [Felikson, Figure 18 (2,3,7), Entry #13] C35: ModuliField: Q. j-invariant = 3^3*1367^3*2^(-4)*5^(-2) Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[2,3,3,7] Decomposition: none Galois group: S15. Order = 15! [Felikson, Figure 18 (2,3,7), Entry #22] Degree 16: (1/2,1/2,1/3,2/7) : I22 (1/3,1/3,2/3,2/7) : A7, H38 (1/3,1/7,2/7,6/7) : G19 (1/3,1/7,3/7,5/7) : B17 (1/3,1/7,4/7,4/7) : no covering (1/3,2/7,2/7,5/7) : D33 (1/3,2/7,3/7,4/7) : F3 (1/3,3/7,3/7,3/7) : no covering (2/3,2/3,1/7,1/7) : A12, H48 I22: ModuliField: Q(x^5-2*x^3-4*x^2-5*x-4). Obstruction: none Branch type: 0:[1,3^5] 1:[1,1,2^7] infty:[2,7,7] Decomposition: none Galois group: S16. Order = 16! A7: ModuliField: Q(sqrt(-3)). j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[1,1,2,3^4] 1:[2^8] infty:[2,7,7] Decomposition: S18(deg 2) Galois group: GAP transitive group ID = 16T1844. Order = 43008 H38: ModuliField: Q(x^3-x^2-2*x-6). Obstruction: none Branch type: 0:[1,1,2,3^4] 1:[2^8] infty:[2,7,7] Decomposition: none Galois group: S16. Order = 16! G19: ModuliField: Q(sqrt(-3)). Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,2,6,7] Decomposition: none Galois group: A16. Order = 16!/2 B17: ModuliField: Q. j-invariant = 2^6*7^3*97^3*3^(-6)*5^(-4) Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,3,5,7] Decomposition: none Galois group: A16. Order = 16!/2 [Felikson, Figure 18 (2,3,7), Entry #17] D33: ModuliField: Q. j-invariant = 2^7*91423^3*3^(-6)*5^(-7)*7^(-8) Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[2,2,5,7] Decomposition: none Galois group: A16. Order = 16!/2 F3: ModuliField: Q(sqrt(2)). Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[2,3,4,7] Decomposition: none Galois group: A16. Order = 16!/2 [Felikson, Figure 18 (2,3,7), Entries #2, #26] A12: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[2,2,3^4] 1:[2^8] infty:[1,1,7,7] Decomposition: S18(deg 2) Galois group: GAP transitive group ID = 16T1803. Order = 21504 H48: ModuliField: Q(x^3-x^2+5*x+15). Obstruction: none Branch type: 0:[2,2,3^4] 1:[2^8] infty:[1,1,7,7] Decomposition: H47(deg 2) Galois group: GAP transitive group ID = 16T1945. Order = 5160960 [Felikson, Figure 18 (2,3,7), Entry #25] Degree 17: (1/2,1/3,1/3,3/7) : H49 (1/2,2/3,1/7,2/7) : I33 H49: ModuliField: Q(x^3-x^2+6*x+5). Obstruction: none Branch type: 0:[1,1,3^5] 1:[1,2^8] infty:[3,7,7] Decomposition: none Galois group: A17. Order = 17!/2 I33: ModuliField: Q(x^5-2*x^4+2*x^2-3*x-6). Obstruction: none Branch type: 0:[2,3^5] 1:[1,2^8] infty:[1,2,7,7] Decomposition: none Galois group: S17. Order = 17! [Felikson, Figure 18 (2,3,7), Entry #28] Degree 18: (1/2,1/2,1/7,3/7) : I27 (1/2,1/2,2/7,2/7) : A3, H37 (1/3,1/3,1/3,4/7) : E6 (1/3,2/3,1/7,3/7) : I26 (1/3,2/3,2/7,2/7) : no covering (1/7,1/7,1/7,8/7) : no covering (1/7,1/7,3/7,6/7) : C4 (1/7,1/7,4/7,5/7) : no covering (1/7,2/7,2/7,6/7) : no covering (1/7,2/7,3/7,5/7) : B31 (1/7,2/7,4/7,4/7) : no covering (1/7,3/7,3/7,4/7) : D29 (2/7,2/7,2/7,5/7) : no covering (2/7,2/7,3/7,4/7) : no covering (2/7,3/7,3/7,3/7) : no covering I27: ModuliField: Q(x^5-2*x^4+2*x^3-3*x^2+3). Obstruction: none Branch type: 0:[3^6] 1:[1,1,2^8] infty:[1,3,7,7] Decomposition: none Galois group: A18. Order = 18!/2 A3: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[3^6] 1:[1,1,2^8] infty:[2,2,7,7] Decomposition: S11(deg 2) Galois group: GAP transitive group ID = 18T802. Order = 129024 H37: ModuliField: Q(x^3-6*x-12). Obstruction: none Branch type: 0:[3^6] 1:[1,1,2^8] infty:[2,2,7,7] Decomposition: E21(deg 2) Galois group: GAP transitive group ID = 18T964. Order = 92897280 [Felikson, Figure 18 (2,3,7), Entry #29] E6: ModuliField: Q. j-invariant = -3^3*5^3*383^3*2^(-7) Obstruction: none Branch type: 0:[1^3,3^5] 1:[2^9] infty:[4,7,7] Decomposition: none Galois group: S18. Order = 18! I26: ModuliField: Q(x^5-x^4-2*x^3+5*x^2-2*x+2). Obstruction: none Branch type: 0:[1,2,3^5] 1:[2^9] infty:[1,3,7,7] Decomposition: none Galois group: S18. Order = 18! [Felikson, Figure 18 (2,3,7), Entry #36] C4: ModuliField: Q. j-invariant = 73^3*601^3*2^(-1)*3^(-4)*7^(-8) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,1,3,6,7] Decomposition: none Galois group: S18. Order = 18! [Felikson, Figure 18 (2,3,7), Entry #24] B31: ModuliField: Q. j-invariant = 73^3*193^3*409^3*2^(-2)*3^(-2)*5^(-4)*7^(-8) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,2,3,5,7] Decomposition: none Galois group: S18. Order = 18! [Felikson, Figure 18 (2,3,7), Entry #31] D29: ModuliField: Q. j-invariant = -23^3*71^3*3^(-1)*7^(-8) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,3,3,4,7] Decomposition: none Galois group: S18. Order = 18! Degree 19: (1/2,1/3,1/7,4/7) : J11 (1/2,1/3,2/7,3/7) : I20 J11: ModuliField: Q(x^6-3*x^5-6*x^4+4*x^3+45*x^2+27*x+4). Obstruction: none Branch type: 0:[1,3^6] 1:[1,2^9] infty:[1,4,7,7] Decomposition: none Galois group: S19. Order = 19! [Felikson, Figure 18 (2,3,7), Entries #44, #45] I20: ModuliField: Q(x^4+3*x^2-7*x+4). Obstruction: none Branch type: 0:[1,3^6] 1:[1,2^9] infty:[2,3,7,7] Decomposition: none Galois group: S19. Order = 19! Degree 20: (1/3,1/3,1/7,5/7) : H42 (1/3,1/3,2/7,4/7) : A11 (1/3,1/3,3/7,3/7) : H33 (2/3,1/7,1/7,4/7) : A14, B32 (2/3,1/7,2/7,3/7) : F23 (2/3,2/7,2/7,2/7) : A15 H42: ModuliField: Q(x^3+2*x-2). Obstruction: none Branch type: 0:[1,1,3^6] 1:[2^10] infty:[1,5,7,7] Decomposition: none Galois group: A20. Order = 20!/2 A11: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[1,1,3^6] 1:[2^10] infty:[2,4,7,7] Decomposition: S8(deg 2) Galois group: GAP transitive group ID = 20T1105. Order = 1857945600 [Felikson, Figure 18 (2,3,7), Entry #4] H33: ModuliField: Q(x^3-x^2+2*x-3). Obstruction: none Branch type: 0:[1,1,3^6] 1:[2^10] infty:[3,3,7,7] Decomposition: H34(deg 2) Galois group: GAP transitive group ID = 20T1100. Order = 928972800 [Felikson, Figure 18 (2,3,7), Entry #5] A14: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[2,3^6] 1:[2^10] infty:[1,1,4,7,7] Decomposition: S8(deg 2) Galois group: GAP transitive group ID = 20T1110. Order = 3715891200 [Felikson, Figure 18 (2,3,7), Entry #6] B32: ModuliField: Q. j-invariant = 49201^3*2^(-8)*3^(-6)*5^(-2)*11^(-4) Obstruction: none Branch type: 0:[2,3^6] 1:[2^10] infty:[1,1,4,7,7] Decomposition: none Galois group: S20. Order = 20! [Felikson, Figure 18 (2,3,7), Entry #30] F23: ModuliField: Q(sqrt(21)). Obstruction: none Branch type: 0:[2,3^6] 1:[2^10] infty:[1,2,3,7,7] Decomposition: none Galois group: S20. Order = 20! [Felikson, Figure 18 (2,3,7), Entries #33, #37] A15: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[2,3^6] 1:[2^10] infty:[2^3,7,7] Decomposition: S8(deg 2) Galois group: GAP transitive group ID = 20T1110. Order = 3715891200 [Felikson, Figure 18 (2,3,7), Entry #3] Degree 21: (1/2,1/3,1/3,1/3) : I9 (1/2,1/7,1/7,5/7) : H44 (1/2,1/7,2/7,4/7) : G30 (1/2,1/7,3/7,3/7) : C36 (1/2,2/7,2/7,3/7) : C39 I9: ModuliField: Q(x^4-x^3+2*x+1) = Q( sqrt(-3), sqrt(3-2*sqrt(-3)) ). Obstruction: none Branch type: 0:[1^3,3^6] 1:[1,2^10] infty:[7^3] Decomposition: none Galois group: A21. Order = 21!/2 H44: ModuliField: Q(x^3+2*x-2). Obstruction: none Branch type: 0:[3^7] 1:[1,2^10] infty:[1,1,5,7,7] Decomposition: none Galois group: A21. Order = 21!/2 [Felikson, Figure 18 (2,3,7), Entry #46] G30: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[3^7] 1:[1,2^10] infty:[1,2,4,7,7] Decomposition: none Galois group: A21. Order = 21!/2 C36: ModuliField: Q. j-invariant = 757^3*11827^3*2^(-4)*3^(-7)*17^(-6) Obstruction: none Branch type: 0:[3^7] 1:[1,2^10] infty:[1,3,3,7,7] Decomposition: none Galois group: A21. Order = 21!/2 [Felikson, Figure 18 (2,3,7), Entry #38] C39: ModuliField: Q. j-invariant = 2^4*3^6*8681^3*5^(-7) Obstruction: none Branch type: 0:[3^7] 1:[1,2^10] infty:[2,2,3,7,7] Decomposition: none Galois group: A21. Order = 21!/2 [Felikson, Figure 18 (2,3,7), Entry #34] Degree 22: (1/2,1/2,1/3,1/7) : J25 (1/3,1/3,2/3,1/7) : J15 (1/3,1/7,1/7,6/7) : H36 (1/3,1/7,2/7,5/7) : B34 (1/3,1/7,3/7,4/7) : H52 (1/3,2/7,2/7,4/7) : G42 (1/3,2/7,3/7,3/7) : no covering J25: ModuliField: Q(x^13-3*x^12+8*x^11-6*x^10+6*x^9-8*x^8+14*x^7+8*x^6+8*x^5-6*x^4+6*x^3+7*x^2+7*x+14). Obstruction: none Branch type: 0:[1,3^7] 1:[1,1,2^10] infty:[1,7^3] Decomposition: none Galois group: A22. Order = 22!/2 J15: ModuliField: Q(x^7-2*x^6-4*x^5-3*x^4+5*x^3-4*x^2+2*x-2). Obstruction: none Branch type: 0:[1,1,2,3^6] 1:[2^11] infty:[1,7^3] Decomposition: none Galois group: S22. Order = 22! H36: ModuliField: Q(x^3-6*x-12). Obstruction: none Branch type: 0:[1,3^7] 1:[2^11] infty:[1,1,6,7,7] Decomposition: none Galois group: S22. Order = 22! [Felikson, Figure 18 (2,3,7), Entry #50] B34: ModuliField: Q. j-invariant = 829^3*30469^3*3^(-6)*5^(-6)*7^(-8)*19^(-4) Obstruction: none Branch type: 0:[1,3^7] 1:[2^11] infty:[1,2,5,7,7] Decomposition: none Galois group: S22. Order = 22! [Felikson, Figure 18 (2,3,7), Entry #32] H52: ModuliField: Q(x^3-x^2+3). Obstruction: none Branch type: 0:[1,3^7] 1:[2^11] infty:[1,3,4,7,7] Decomposition: none Galois group: S22. Order = 22! [Felikson, Figure 18 (2,3,7), Entry #39] G42: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[1,3^7] 1:[2^11] infty:[2,2,4,7,7] Decomposition: none Galois group: S22. Order = 22! Degree 23: (1/2,1/3,1/3,2/7) : J14 (1/2,2/3,1/7,1/7) : J17 J14: ModuliField: Q(x^7-x^6-x^5+5*x^4+5*x^3-4*x^2-6*x-2). Obstruction: none Branch type: 0:[1,1,3^7] 1:[1,2^11] infty:[2,7^3] Decomposition: none Galois group: S23. Order = 23! J17: ModuliField: Q(x^7-3*x^6+2*x^5+6*x^4-6*x^3-4*x^2+4*x+4). Obstruction: none Branch type: 0:[2,3^7] 1:[1,2^11] infty:[1,1,7^3] Decomposition: none Galois group: S23. Order = 23! [Felikson, Figure 18 (2,3,7), Entry #55] Degree 24: (1/2,1/2,1/7,2/7) : I4 (1/3,1/3,1/3,3/7) : D20 (1/3,2/3,1/7,2/7) : B6, I2 (1/7,1/7,2/7,6/7) : no covering (1/7,1/7,3/7,5/7) : F24 (1/7,1/7,4/7,4/7) : G40 (1/7,2/7,2/7,5/7) : no covering (1/7,2/7,3/7,4/7) : no covering (1/7,3/7,3/7,3/7) : no covering (2/7,2/7,2/7,4/7) : no covering (2/7,2/7,3/7,3/7) : F20 I4: ModuliField: Q(x^4-x^3+3*x^2+3*x+2) = Q( sqrt(-7), sqrt(sqrt(-7)-7) ). Obstruction: none Branch type: 0:[3^8] 1:[1,1,2^11] infty:[1,2,7^3] Decomposition: none Galois group: S24. Order = 24! D20: ModuliField: Q(sqrt(-3)). j-invariant = 0 Obstruction: none Branch type: 0:[1^3,3^7] 1:[2^12] infty:[3,7^3] Decomposition: S18(deg 3) Galois group: GAP transitive group ID = 24T21448. Order = 1102248 B6: ModuliField: Q(sqrt(-3)). j-invariant = 2^2*73^3*3^(-4) Obstruction: none Branch type: 0:[1,2,3^7] 1:[2^12] infty:[1,2,7^3] Decomposition: S18(deg 3) Galois group: GAP transitive group ID = 24T24688. Order = 282175488 I2: ModuliField: Q(x^4-x^3+3*x^2-4*x+2) = Q( (-7)^1/4) ). Obstruction: none Branch type: 0:[1,2,3^7] 1:[2^12] infty:[1,2,7^3] Decomposition: none Galois group: S24. Order = 24! F24: ModuliField: Q(sqrt(21)). Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[1,1,3,5,7,7] Decomposition: none Galois group: A24. Order = 24!/2 [Felikson, Figure 18 (2,3,7), Entry #40] G40: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[1,1,4,4,7,7] Decomposition: G38(deg 2) Galois group: GAP transitive group ID = 24T24970. Order = 980995276800 F20: ModuliField: Q(sqrt(7)). Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[2,2,3,3,7,7] Decomposition: F22(deg 2) Galois group: GAP transitive group ID = 24T24970. Order = 980995276800 [Felikson, Figure 18 (2,3,7), Entries #35, #41] Degree 25: (1/2,1/3,1/7,3/7) : J23 (1/2,1/3,2/7,2/7) : H43 J23: ModuliField: Q(x^11-x^10-6*x^9+11*x^8+2*x^7-28*x^6+26*x^5+44*x^4-104*x^3+112*x^2-64*x+32). Obstruction: none Branch type: 0:[1,3^8] 1:[1,2^12] infty:[1,3,7^3] Decomposition: none Galois group: A25. Order = 25!/2 [Felikson, Figure 18 (2,3,7), Entry #52] H43: ModuliField: Q(x^3+2*x-2). Obstruction: none Branch type: 0:[1,3^8] 1:[1,2^12] infty:[2,2,7^3] Decomposition: none Galois group: A25. Order = 25!/2 Degree 26: (1/3,1/3,1/7,4/7) : J12 (1/3,1/3,2/7,3/7) : H51 (2/3,1/7,1/7,3/7) : C12 (2/3,1/7,2/7,2/7) : D47 J12: ModuliField: Q(x^6-2*x^5+3*x^4+x^2+4*x+2). Obstruction: none Branch type: 0:[1,1,3^8] 1:[2^13] infty:[1,4,7^3] Decomposition: none Galois group: S26. Order = 26! H51: ModuliField: Q(x^3+4*x-2). Obstruction: none Branch type: 0:[1,1,3^8] 1:[2^13] infty:[2,3,7^3] Decomposition: none Galois group: S26. Order = 26! C12: ModuliField: Q. j-invariant = 3^3*37^3*192637^3*11^(-6)*17^(-4) Obstruction: none Branch type: 0:[2,3^8] 1:[2^13] infty:[1,1,3,7^3] Decomposition: none Galois group: S26. Order = 26! [Felikson, Figure 18 (2,3,7), Entry #51] D47: ModuliField: Q. j-invariant = -3^3*335089^3*2^(-14)*5^(-7)*23^(-4) Obstruction: none Branch type: 0:[2,3^8] 1:[2^13] infty:[1,2,2,7^3] Decomposition: none Galois group: S26. Order = 26! Degree 27: (1/2,1/7,1/7,4/7) : D46 (1/2,1/7,2/7,3/7) : B2, I25 (1/2,2/7,2/7,2/7) : no covering D46: ModuliField: Q. j-invariant = -11^3*59^3*2^(-12)*3^(-1)*5^(-3) Obstruction: none Branch type: 0:[3^9] 1:[1,2^13] infty:[1,1,4,7^3] Decomposition: none Galois group: S27. Order = 27! B2: ModuliField: Q(sqrt(-7)). j-invariant = 2^4*13^3*3^(-2) Obstruction: none Branch type: 0:[3^9] 1:[1,2^13] infty:[1,2,3,7^3] Decomposition: S11(deg 3) Galois group: GAP transitive group ID = 27T2358. Order = 5079158784 I25: ModuliField: Q(x^5-x^4+2*x^3-2*x^2-x-1). Obstruction: none Branch type: 0:[3^9] 1:[1,2^13] infty:[1,2,3,7^3] Decomposition: none Galois group: S27. Order = 27! [Felikson, Figure 18 (2,3,7), Entry #47] Degree 28: (1/3,1/3,1/3,1/3) : G36, H13 (1/3,1/7,1/7,5/7) : I8 (1/3,1/7,2/7,4/7) : B16 (1/3,1/7,3/7,3/7) : no covering (1/3,2/7,2/7,3/7) : no covering G36: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[1^4,3^8] 1:[2^14] infty:[7^4] Decomposition: I3(deg 2) G39(deg 2) I3(deg 2) G35(deg 4)(see #1,2,3) Galois group: (C2 x C2 x C2) . PSL(3,2). GAP transitive group ID = 28T152. Order = 1344 H13: ModuliField: Q(x^3-x^2-2*x+1) = Q(Re(zeta_7)). Obstruction: Primes: [infinity, infinity], Conic: X^2+Y^2+RootOf(x^3-x^2-2*x+1)*Z^2 = 0 Branch type: 0:[1^4,3^8] 1:[2^14] infty:[7^4] Decomposition: H12(deg 2) Galois group: PSL(2,13). GAP transitive group ID = 28T120. Order = 1092 I8: ModuliField: Q(x^4-x^3+2*x+1) = Q( sqrt(-3), sqrt(3-2*sqrt(-3)) ). Obstruction: none Branch type: 0:[1,3^9] 1:[2^14] infty:[1,1,5,7^3] Decomposition: none Galois group: A28. Order = 28!/2 B16: ModuliField: Q. j-invariant = 2^6*7^3*97^3*3^(-6)*5^(-4) Obstruction: none Branch type: 0:[1,3^9] 1:[2^14] infty:[1,2,4,7^3] Decomposition: none Galois group: A28. Order = 28!/2 [Felikson, Figure 18 (2,3,7), Entry #53] Degree 29: (1/2,1/3,1/3,1/7) : J27 J27: ModuliField: Q(x^14-x^13-4*x^12-3*x^11+8*x^10+41*x^9-30*x^8-88*x^7+78*x^6+8*x^5-8*x^4-36*x^3+64*x^2-4*x+4). Obstruction: none Branch type: 0:[1,1,3^9] 1:[1,2^14] infty:[1,7^4] Decomposition: none Galois group: A29. Order = 29!/2 Degree 30: (1/2,1/2,1/7,1/7) : H10, J19 (1/3,1/3,1/3,2/7) : no covering (1/3,2/3,1/7,1/7) : J6 (1/7,1/7,1/7,6/7) : D22 (1/7,1/7,2/7,5/7) : D5 (1/7,1/7,3/7,4/7) : no covering (1/7,2/7,2/7,4/7) : B13 (1/7,2/7,3/7,3/7) : C18 (2/7,2/7,2/7,3/7) : D23 H10: ModuliField: Q(x^3-x^2-2*x+1) = Q(Re(zeta_7)). Obstruction: Primes: [infinity, infinity], Conic: X^2+Y^2+RootOf(x^3-x^2-2*x+1)*Z^2 = 0 Branch type: 0:[3^10] 1:[1,1,2^14] infty:[1,1,7^4] Decomposition: none Galois group: PSL(2,29). GAP transitive group ID = 30T808. Order = 12180 J19: ModuliField: Q(x^9-2*x^8+13*x^7-30*x^6+81*x^5-118*x^4+171*x^3-114*x^2+70*x-8) = Q(a, x^3+(a*(a-3)/2-3)*(x^2+1)+a*x+2) where a^3-7*a-14 = 0. Obstruction: none Branch type: 0:[3^10] 1:[1,1,2^14] infty:[1,1,7^4] Decomposition: H45(deg 2) Galois group: GAP transitive group ID = 30T5688. Order = 10712468422656000 [Felikson, Figure 18 (2,3,7), Entry #49] J6: ModuliField: Q(x^6-x^5+7*x^4-5*x^3+14*x^2-2*x+6) = Q(sqrt(-8-28^(1/3))). Obstruction: none Branch type: 0:[1,2,3^9] 1:[2^15] infty:[1,1,7^4] Decomposition: none Galois group: S30. Order = 30! D22: ModuliField: Q. j-invariant = 0 Obstruction: none Branch type: 0:[3^10] 1:[2^15] infty:[1^3,6,7^3] Decomposition: S8(deg 3) Galois group: GAP transitive group ID = 30T5405. Order = 214277011200 D5: ModuliField: Q. j-invariant = -2^4*109^3*5^(-6) Obstruction: none Branch type: 0:[3^10] 1:[2^15] infty:[1,1,2,5,7^3] Decomposition: none Galois group: S30. Order = 30! B13: ModuliField: Q. j-invariant = 2^2*73^3*3^(-4) Obstruction: none Branch type: 0:[3^10] 1:[2^15] infty:[1,2,2,4,7^3] Decomposition: S8(deg 3) Galois group: GAP transitive group ID = 30T5655. Order = 109709829734400 [Felikson, Figure 18 (2,3,7), Entry #42] C18: ModuliField: Q. j-invariant = 103681^3*3^(-4)*5^(-1) Obstruction: none Branch type: 0:[3^10] 1:[2^15] infty:[1,2,3,3,7^3] Decomposition: none Galois group: S30. Order = 30! [Felikson, Figure 18 (2,3,7), Entry #54] D23: ModuliField: Q. j-invariant = 0 Obstruction: none Branch type: 0:[3^10] 1:[2^15] infty:[2^3,3,7^3] Decomposition: S8(deg 3) Galois group: GAP transitive group ID = 30T5405. Order = 214277011200 Degree 31: (1/2,1/3,1/7,2/7) : J24 J24: ModuliField: Q(x^13-2*x^12-6*x^11+2*x^10+37*x^9-4*x^8-50*x^7-46*x^6+6*x^5+54*x^4+70*x^3+40*x^2+8*x-2). Obstruction: none Branch type: 0:[1,3^10] 1:[1,2^15] infty:[1,2,7^4] Decomposition: none Galois group: S31. Order = 31! [Felikson, Figure 18 (2,3,7), Entry #48] Degree 32: (1/3,1/3,1/7,3/7) : A10, F21 (1/3,1/3,2/7,2/7) : C6, I23 (2/3,1/7,1/7,2/7) : G32 A10: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[1,1,3^10] 1:[2^16] infty:[1,3,7^4] Decomposition: S18(deg 4) Galois group: Order = 72236924928 F21: ModuliField: Q(sqrt(7)). Obstruction: none Branch type: 0:[1,1,3^10] 1:[2^16] infty:[1,3,7^4] Decomposition: none Galois group: A32. Order = 32!/2 [Felikson, Figure 18 (2,3,7), Entry #58] C6: ModuliField: Q. j-invariant = 2^7*53^3*3^(-3) Obstruction: Primes: [infinity, 2], Conic: X^2+Y^2+Z^2 = 0 Branch type: 0:[1,1,3^10] 1:[2^16] infty:[2,2,7^4] Decomposition: S18(deg 4) Galois group: Order = 11010048 I23: ModuliField: Q(x^5-2*x^3-4*x^2-5*x-4). Obstruction: none Branch type: 0:[1,1,3^10] 1:[2^16] infty:[2,2,7^4] Decomposition: I22(deg 2) Galois group: Order = 685597979049984000 [Felikson, Figure 18 (2,3,7), Entry #43] G32: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[2,3^10] 1:[2^16] infty:[1,1,2,7^4] Decomposition: none Galois group: S32. Order = 32! Degree 33: (1/2,1/7,1/7,3/7) : I32 (1/2,1/7,2/7,2/7) : G41 I32: ModuliField: Q(x^5-2*x^4+2*x^3-4*x^2-2*x-2). Obstruction: none Branch type: 0:[3^11] 1:[1,2^16] infty:[1,1,3,7^4] Decomposition: none Galois group: A33. Order = 33!/2 G41: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[3^11] 1:[1,2^16] infty:[1,2,2,7^4] Decomposition: none Galois group: A33. Order = 33!/2 Degree 34: (1/3,1/7,1/7,4/7) : D17 (1/3,1/7,2/7,3/7) : J10 (1/3,2/7,2/7,2/7) : no covering D17: ModuliField: Q. j-invariant = 7607^3*1753^3*2^(-7)*3^(-20)*5^(-4)*11^(-4) Obstruction: none Branch type: 0:[1,3^11] 1:[2^17] infty:[1,1,4,7^4] Decomposition: none Galois group: S34. Order = 34! J10: ModuliField: Q(x^6+x^4-3*x^3+6*x^2-4*x+2). Obstruction: none Branch type: 0:[1,3^11] 1:[2^17] infty:[1,2,3,7^4] Decomposition: none Galois group: S34. Order = 34! Degree 36: (1/3,1/3,1/3,1/7) : I14 (1/7,1/7,1/7,5/7) : no covering (1/7,1/7,2/7,4/7) : A22, D40 (1/7,1/7,3/7,3/7) : B12, I28 (1/7,2/7,2/7,3/7) : no covering (2/7,2/7,2/7,2/7) : A23, E22 I14: ModuliField: Q(x^4-14*x+21). Obstruction: none Branch type: 0:[1^3,3^11] 1:[2^18] infty:[1,7^5] Decomposition: none Galois group: A36. Order = 36!/2 A22: ModuliField: Q(sqrt(-7)). j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[3^12] 1:[2^18] infty:[1,1,2,4,7^4] Decomposition: S51(deg 2) S11(deg 4)(see #1) Galois group: Order = 33822867456 D40: ModuliField: Q. j-invariant = -5^3*37^3*167^3*2^(-8)*3^(-4)*11^(-4) Obstruction: none Branch type: 0:[3^12] 1:[2^18] infty:[1,1,2,4,7^4] Decomposition: none Galois group: A36. Order = 36!/2 B12: ModuliField: Q. j-invariant = 2^2*73^3*3^(-4) Obstruction: Primes: [infinity, 7], Conic: 7*X^2+Y^2+Z^2 = 0 Branch type: 0:[3^12] 1:[2^18] infty:[1,1,3,3,7^4] Decomposition: S11(deg 4) Galois group: Order = 866843099136 I28: ModuliField: Q(x^5-2*x^4+2*x^3-3*x^2+3). Obstruction: none Branch type: 0:[3^12] 1:[2^18] infty:[1,1,3,3,7^4] Decomposition: I27(deg 2) Galois group: Order = 419585963178590208000 [Felikson, Figure 18 (2,3,7), Entry #57] A23: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[3^12] 1:[2^18] infty:[2^4,7^4] Decomposition: S51(deg 2) A3(deg 2) S51(deg 2) S11(deg 4)(see #1,2,3) Galois group: Order = 66060288 E22: ModuliField: Q. j-invariant = -3^1*223^3*2^(-8) Obstruction: none Branch type: 0:[3^12] 1:[2^18] infty:[2^4,7^4] Decomposition: H37(deg 2) H37(deg 2) H37(deg 2) E21(deg 4)(see #1,2,3) Galois group: Order = 23781703680 Degree 37: (1/2,1/3,1/7,1/7) : J28 J28: ModuliField: Q(x^15-5*x^14+15*x^13-21*x^12+19*x^11+17*x^10-31*x^9-3*x^8+411*x^7-1027*x^6+1661*x^5-1515*x^4+1137*x^3-553*x^2+307*x-29). Obstruction: none Branch type: 0:[1,3^12] 1:[1,2^18] infty:[1,1,7^5] Decomposition: none Galois group: A37. Order = 37!/2 Degree 38: (1/3,1/3,1/7,2/7) : J13 (2/3,1/7,1/7,1/7) : I6 J13: ModuliField: Q(x^7-x^6-2*x^4-x^3+2*x^2+2*x+2). Obstruction: none Branch type: 0:[1,1,3^12] 1:[2^19] infty:[1,2,7^5] Decomposition: none Galois group: S38. Order = 38! I6: ModuliField: Q(x^4-x^3+3*x^2+3*x+2) = Q( sqrt(-7), sqrt(sqrt(-7)-7) ). Obstruction: none Branch type: 0:[2,3^12] 1:[2^19] infty:[1^3,7^5] Decomposition: none Galois group: S38. Order = 38! Degree 39: (1/2,1/7,1/7,2/7) : G43 G43: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[3^13] 1:[1,2^19] infty:[1,1,2,7^5] Decomposition: none Galois group: S39. Order = 39! Degree 40: (1/3,1/7,1/7,3/7) : J16 (1/3,1/7,2/7,2/7) : G34 J16: ModuliField: Q(x^7-x^6+x^5+5*x^4+x^3+6*x^2+9*x+3). Obstruction: none Branch type: 0:[1,3^13] 1:[2^20] infty:[1,1,3,7^5] Decomposition: none Galois group: A40. Order = 40!/2 G34: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[1,3^13] 1:[2^20] infty:[1,2,2,7^5] Decomposition: none Galois group: A40. Order = 40!/2 Degree 42: (1/7,1/7,1/7,4/7) : G33 (1/7,1/7,2/7,3/7) : no covering (1/7,2/7,2/7,2/7) : no covering G33: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[3^14] 1:[2^21] infty:[1^3,4,7^5] Decomposition: none Galois group: S42. Order = 42! Degree 44: (1/3,1/3,1/7,1/7) : H11, J26 H11: ModuliField: Q(x^3-x^2-2*x+1) = Q(Re(zeta_7)). Obstruction: Primes: [infinity, infinity], Conic: X^2+Y^2+RootOf(x^3-x^2-2*x+1)*Z^2 = 0 Branch type: 0:[1,1,3^14] 1:[2^22] infty:[1,1,7^6] Decomposition: none Galois group: PSL(2,43). Order = 39732 J26: ModuliField: Q(x^13-3*x^12+8*x^11-6*x^10+6*x^9-8*x^8+14*x^7+8*x^6+8*x^5-6*x^4+6*x^3+7*x^2+7*x+14). Obstruction: none Branch type: 0:[1,1,3^14] 1:[2^22] infty:[1,1,7^6] Decomposition: J25(deg 2) Galois group: Order = 1178600187130132750663680000 [Felikson, Figure 18 (2,3,7), Entry #56] Degree 45: (1/2,1/7,1/7,1/7) : G44 G44: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[3^15] 1:[1,2^22] infty:[1^3,7^6] Decomposition: none Galois group: A45. Order = 45!/2 Degree 46: (1/3,1/7,1/7,2/7) : J7 J7: ModuliField: Q(x^6-x^5+3*x^4-3*x^3-2*x^2+4*x+2) = Q( sqrt(-7), 2*x^3-x^2+x+1+sqrt(-7)*(x^2-x-1) ). Obstruction: none Branch type: 0:[1,3^15] 1:[2^23] infty:[1,1,2,7^6] Decomposition: none Galois group: S46. Order = 46! Degree 48: (1/7,1/7,1/7,3/7) : no covering (1/7,1/7,2/7,2/7) : A21, I5 A21: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[3^16] 1:[2^24] infty:[1,1,2,2,7^6] Decomposition: S50(deg 2) S18(deg 6)(see #1) Galois group: Order = 11010048 [Felikson, Figure 18 (2,3,7), Entry #7] I5: ModuliField: Q(x^4-x^3+3*x^2+3*x+2) = Q( sqrt(-7), sqrt(sqrt(-7)-7) ). Obstruction: none Branch type: 0:[3^16] 1:[2^24] infty:[1,1,2,2,7^6] Decomposition: I4(deg 2) Galois group: Order = 5204698426366666226930810880000 Degree 52: (1/3,1/7,1/7,1/7) : D35 D35: ModuliField: Q. j-invariant = -11^3*23830621091^3*2^(-8)*3^(-20)*5^(-3)*7^(-8)*43^(-4) Obstruction: none Branch type: 0:[1,3^17] 1:[2^26] infty:[1^3,7^7] Decomposition: none Galois group: A52. Order = 52!/2 Degree 54: (1/7,1/7,1/7,2/7) : D28 D28: ModuliField: Q. j-invariant = -23^3*71^3*3^(-1)*7^(-8) Obstruction: none Branch type: 0:[3^18] 1:[2^27] infty:[1^3,2,7^7] Decomposition: none Galois group: S54. Order = 54! Degree 60: (1/7,1/7,1/7,1/7) : H14, H46 H14: ModuliField: Q(x^3-x^2-2*x+1) = Q(Re(zeta_7)). Obstruction: Primes: [infinity, infinity], Conic: X^2+Y^2+RootOf(x^3-x^2-2*x+1)*Z^2 = 0 Branch type: 0:[3^20] 1:[2^30] infty:[1^4,7^8] Decomposition: H10(deg 2) Galois group: PSL(2,29). Order = 12180 H46: ModuliField: Q(x^3-x^2+5*x-13). Obstruction: none Branch type: 0:[3^20] 1:[2^30] infty:[1^4,7^8] Decomposition: J19(deg 2) J19(deg 2) J19(deg 2) H45(deg 4)(see #1,2,3) Galois group: Order = 175513082636795904000 =========== Table 2.3.8. (1/2,1/3,1/8) --> Heun ========== Degree 8: (1/2,1/2,1/3,1/3) : F5, G13 (1/3,1/3,1/3,2/3) : D13 F5: ModuliField: Q(sqrt(2)). Obstruction: none Branch type: 0:[1,1,3,3] 1:[1,1,2^3] infty:[8] Decomposition: none Galois group: PSL(3,2) : C2. GAP transitive group ID = 8T43. Order = 336 G13: ModuliField: Q(sqrt(-2)). Obstruction: none Branch type: 0:[1,1,3,3] 1:[1,1,2^3] infty:[8] Decomposition: S36(deg 2) Galois group: GL(2,3). GAP transitive group ID = 8T23. Order = 48 D13: ModuliField: Q. j-invariant = -2^6*239^3*3^(-10) Obstruction: none Branch type: 0:[1^3,2,3] 1:[2^4] infty:[8] Decomposition: 1-S32(deg 4) Galois group: (S4 x S4) : C2. GAP transitive group ID = 8T47. Order = 1152 Degree 9: (1/2,1/2,1/2,1/8) : G17 (1/2,1/3,2/3,1/8) : J2 G17: ModuliField: Q(sqrt(-2)). Obstruction: none Branch type: 0:[3^3] 1:[1^3,2^3] infty:[1,8] Decomposition: none Galois group: (((C3 x C3) : Q8) : C3) : C2. GAP transitive group ID = 9T26. Order = 432 J2: ModuliField: Q(x^6-8*x^3+9*x^2+18). Obstruction: none Branch type: 0:[1,2,3,3] 1:[1,2^4] infty:[1,8] Decomposition: none Galois group: S9. Order = 9! Degree 10: (1/2,1/2,1/3,1/4) : H20 (1/3,1/3,2/3,1/4) : F19 (2/3,2/3,1/8,1/8) : C32 H20: ModuliField: Q(x^3-x^2+2*x+2). Obstruction: none Branch type: 0:[1,3^3] 1:[1,1,2^4] infty:[2,8] Decomposition: none Galois group: A10. Order = 10!/2 F19: ModuliField: Q(sqrt(6)). Obstruction: none Branch type: 0:[1,1,2,3,3] 1:[2^5] infty:[2,8] Decomposition: 1-S32(deg 5) Galois group: GAP transitive group ID = 10T43. Order = 10!/126 [Felikson, Figure 17 (2,3,8), Entry #9] C32: ModuliField: Q. j-invariant = 11^3*13^3*23^3*2^(-1)*3^(-12)*5^(-1) Obstruction: none Branch type: 0:[2,2,3,3] 1:[2^5] infty:[1,1,8] Decomposition: S29(deg 2) Galois group: C2 x (((C2 x C2 x C2 x C2) : A5) : C2). GAP transitive group ID = 10T39. Order = 3840 [Felikson, Figure 17 (2,3,8), Entry #8] Degree 11: (1/2,1/3,1/3,3/8) : F25 (1/2,2/3,1/4,1/8) : I10 F25: ModuliField: Q(sqrt(22)). Obstruction: none Branch type: 0:[1,1,3^3] 1:[1,2^5] infty:[3,8] Decomposition: none Galois group: S11. Order = 11! I10: ModuliField: Q(x^4-2*x^2-4*x-1). Obstruction: none Branch type: 0:[2,3^3] 1:[1,2^5] infty:[1,2,8] Decomposition: none Galois group: S11. Order = 11! [Felikson, Figure 17 (2,3,8), Entries #6, #7] Degree 12: (1/2,1/2,1/4,1/4) : C1 (1/2,1/2,1/8,3/8) : D9, G14 (1/2,1/3,1/3,1/3) : E2 (4/8,1/3,1/3,1/3) (1/3,2/3,1/4,1/4) : C21 (1/3,2/3,1/8,3/8) : G10 C1: ModuliField: Q. j-invariant = 2^3*3^3*11^3 Obstruction: none Branch type: 0:[3^4] 1:[1,1,2^5] infty:[2,2,8] Decomposition: S27(deg 2) 1/S34(deg 4)(see #1) Galois group: ((C4 x C4) : C3) : C2. GAP transitive group ID = 12T64. Order = 96 [Felikson, Figure 17 (2,3,8), Entry #5] D9: ModuliField: Q. j-invariant = 2^1*47^3*3^(-8) Obstruction: none Branch type: 0:[3^4] 1:[1,1,2^5] infty:[1,3,8] Decomposition: 1/S34(deg 4) Galois group: GAP transitive group ID = 12T291. Order = 41472 G14: ModuliField: Q(sqrt(-2)). Obstruction: none Branch type: 0:[3^4] 1:[1,1,2^5] infty:[1,3,8] Decomposition: none Galois group: S12. Order = 12! E2: ModuliField: Q. j-invariant = -2^6*3^3*23^3 Obstruction: none Branch type: 0:[1^3,3^3] 1:[2^6] infty:[4,8] Decomposition: 1-S32(deg 6) Galois group: A6 : C2. GAP transitive group ID = 12T182. Order = 720 C21: ModuliField: Q. j-invariant = 2^6*19^3*467^3*3^(-7)*5^(-6) Obstruction: none Branch type: 0:[1,2,3^3] 1:[2^6] infty:[2,2,8] Decomposition: 1-S32(deg 6) Galois group: GAP transitive group ID = 12T299. Order = 12!/462 [Felikson, Figure 17 (2,3,8), Entry #10] G10: ModuliField: Q(sqrt(-2)). Obstruction: none Branch type: 0:[1,2,3^3] 1:[2^6] infty:[1,3,8] Decomposition: none Galois group: S12. Order = 12! Degree 13: (1/2,1/2,1/3,1/8) : I18 (1/2,4/8,1/3,1/8) (1/2,1/3,1/4,3/8) : H29 I18: ModuliField: Q(x^4-3*x^2-2*x+6). Obstruction: none Branch type: 0:[1,3^4] 1:[1,2^6] infty:[1,4,8] Decomposition: none Galois group: A13. Order = 13!/2 H29: ModuliField: Q(x^3-x-2). Obstruction: none Branch type: 0:[1,3^4] 1:[1,2^6] infty:[2,3,8] Decomposition: none Galois group: A13. Order = 13!/2 [Felikson, Figure 17 (2,3,8), Entry #11] Degree 14: (1/2,1/3,1/3,1/4) : F8 (4/8,1/3,1/3,1/4) (1/2,2/3,1/8,1/8) : no covering (4/8,2/3,1/8,1/8) (1/3,1/3,1/8,5/8) : H23 (1/3,1/3,3/8,3/8) : C28 (2/3,1/4,1/4,1/4) : no covering (2/3,1/4,1/8,3/8) : B29 F8: ModuliField: Q(sqrt(2)). Obstruction: none Branch type: 0:[1,1,3^4] 1:[2^7] infty:[2,4,8] Decomposition: 1-S32(deg 7) Galois group: PSL(3,2) : C2. GAP transitive group ID = 14T16. Order = 336 [Felikson, Figure 17 (2,3,8), Entry #15] H23: ModuliField: Q(x^3-x^2+2*x+2). Obstruction: none Branch type: 0:[1,1,3^4] 1:[2^7] infty:[1,5,8] Decomposition: none Galois group: S14. Order = 14! C28: ModuliField: Q. j-invariant = 2^2*11^3*107^3*3^(-12)*7^(-1) Obstruction: none Branch type: 0:[1,1,3^4] 1:[2^7] infty:[3,3,8] Decomposition: S23(deg 2) Galois group: GAP transitive group ID = 14T54. Order = 322560 [Felikson, Figure 17 (2,3,8), Entry #12] B29: ModuliField: Q. j-invariant = 2^4*3^3*7^6*103^3*5^(-6)*11^(-4) Obstruction: none Branch type: 0:[2,3^4] 1:[2^7] infty:[1,2,3,8] Decomposition: none Galois group: S14. Order = 14! [Felikson, Figure 17 (2,3,8), Entry #13] Degree 15: (1/2,1/2,1/4,1/8) : G1 (1/2,4/8,1/4,1/8) (1/2,1/4,1/4,3/8) : B15 (1/2,1/8,1/8,5/8) : G9 (1/2,1/8,3/8,3/8) : C8 G1: ModuliField: Q(sqrt(-1)). Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,2,4,8] Decomposition: 1/S34(deg 5) Galois group: (((C5 x ((C5 x C5) : C4)) : C4) : C3) : C2. GAP transitive group ID = 15T65. Order = 12000 B15: ModuliField: Q. j-invariant = 6481^3*3^(-8)*5^(-2) Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[2,2,3,8] Decomposition: 1/S34(deg 5) Galois group: GAP transitive group ID = 15T100. Order = 5184000 [Felikson, Figure 17 (2,3,8), Entry #14] G9: ModuliField: Q(sqrt(-1)). Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,1,5,8] Decomposition: none Galois group: S15. Order = 15! C8: ModuliField: Q. j-invariant = 3^3*5^3*157^3*2^(-2)*11^(-6) Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,3,3,8] Decomposition: none Galois group: S15. Order = 15! [Felikson, Figure 17 (2,3,8), Entry #17] Degree 16: (1/2,1/3,1/4,1/4) : no covering (4/8,1/3,1/4,1/4) (1/2,1/3,1/8,3/8) : no covering (4/8,1/3,1/8,3/8) (1/3,1/3,1/3,1/3) : D8, F2 (1/3,1/4,1/8,5/8) : B30 (1/3,1/4,3/8,3/8) : no covering (1/3,3/4,1/8,1/8) : G22 D8: ModuliField: Q. j-invariant = -2^5*19^3*3^(-6) Obstruction: none Branch type: 0:[1^4,3^4] 1:[2^8] infty:[8,8] Decomposition: G13(deg 2) G13(deg 2) S20(deg 2) S36(deg 4)(see #1,2,3) 1-S32(deg 8)(see #3) Galois group: GL(2,3). GAP transitive group ID = 16T66. Order = 48 F2: ModuliField: Q(sqrt(2)). Obstruction: none Branch type: 0:[1^4,3^4] 1:[2^8] infty:[8,8] Decomposition: F5(deg 2) 1-S32(deg 8) Galois group: PSL(3,2) : C2. GAP transitive group ID = 16T713. Order = 336 B30: ModuliField: Q. j-invariant = 2^4*181^3*2521^3*3^(-6)*5^(-4)*13^(-4) Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,2,5,8] Decomposition: none Galois group: A16. Order = 16!/2 [Felikson, Figure 17 (2,3,8), Entry #16] G22: ModuliField: Q(sqrt(-3)). Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,1,6,8] Decomposition: none Galois group: A16. Order = 16!/2 Degree 17: (1/2,1/3,1/3,1/8) : J20 J20: ModuliField: Q(x^9-x^8-4*x^6+4*x^4+8*x^3+8*x^2+2*x-2). Obstruction: none Branch type: 0:[1,1,3^5] 1:[1,2^8] infty:[1,8,8] Decomposition: none Galois group: A17. Order = 17!/2 Degree 18: (1/2,1/2,1/8,1/8) : F4, J1 : no covering (4/8,4/8,1/8,1/8) (1/2,1/4,1/4,1/4) : no covering (4/8,1/4,1/4,1/4) (1/2,1/4,1/8,3/8) : B1 (4/8,1/4,1/8,3/8) (1/3,1/3,1/3,1/4) : E5 (1/3,2/3,1/8,1/8) : G12 (1/4,1/4,1/8,5/8) : D1 (1/4,1/4,3/8,3/8) : C5 (1/4,3/4,1/8,1/8) : no covering (1/8,1/8,1/8,7/8) : no covering (1/8,1/8,3/8,5/8) : no covering (1/8,3/8,3/8,3/8) : no covering F4: ModuliField: Q(sqrt(2)). Obstruction: Primes: [infinity, infinity], Conic: X^2+Y^2+Z^2 = 0 Branch type: 0:[3^6] 1:[1,1,2^8] infty:[1,1,8,8] Decomposition: none Galois group: PSL(2,17). GAP transitive group ID = 18T377. Order = 2448 J1: ModuliField: Q(x^6-2*x^5-x^4+2*x^2+4*x+2) = Q( sqrt(-2), x^3-x^2-2*x-sqrt(-2)*(x^2-x-1) ). Obstruction: none Branch type: 0:[3^6] 1:[1,1,2^8] infty:[1,1,8,8] Decomposition: G17(deg 2) Galois group: GAP transitive group ID = 18T801. Order = 110592 B1: ModuliField: Q. j-invariant = 2^4*13^3*3^(-2) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,2,3,4,8] Decomposition: S43(deg 3) 1/S34(deg 6)(see #1) Galois group: GAP transitive group ID = 18T762. Order = 69984 [Felikson, Figure 17 (2,3,8), Entry #1] E5: ModuliField: Q. j-invariant = -3^3*5^3*383^3*2^(-7) Obstruction: none Branch type: 0:[1^3,3^5] 1:[2^9] infty:[2,8,8] Decomposition: 1-S32(deg 9) Galois group: (((C3 x C3 x C3 x C3) : Q8) : C3) : C2. GAP transitive group ID = 18T451. Order = 3888 G12: ModuliField: Q(sqrt(-2)). Obstruction: none Branch type: 0:[1,2,3^5] 1:[2^9] infty:[1,1,8,8] Decomposition: none Galois group: S18. Order = 18! D1: ModuliField: Q. j-invariant = 2^2*3^3*13^3*5^(-4) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,2,2,5,8] Decomposition: 1/S34(deg 6) Galois group: GAP transitive group ID = 18T954. Order = 5184000 C5: ModuliField: Q. j-invariant = 2^2*193^3*3^(-1) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[2,2,3,3,8] Decomposition: S14(deg 2) 1/S34(deg 6)(see #1) Galois group: GAP transitive group ID = 18T721. Order = 41472 [Felikson, Figure 17 (2,3,8), Entry #19] Degree 19: (1/2,1/3,1/4,1/8) : J21 J21: ModuliField: Q(x^10-4*x^9+4*x^8+8*x^7-12*x^6-32*x^5+80*x^4-16*x^3-60*x^2+16*x+16). Obstruction: none Branch type: 0:[1,3^6] 1:[1,2^9] infty:[1,2,8,8] Decomposition: none Galois group: S19. Order = 19! Degree 20: (1/3,1/3,1/4,1/4) : A9, H22 (1/3,1/3,1/8,3/8) : D16 (2/3,1/4,1/8,1/8) : A13, G3 A9: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[1,1,3^6] 1:[2^10] infty:[2,2,8,8] Decomposition: S7(deg 2) 1-S32(deg 10) Galois group: A6 : C2. GAP transitive group ID = 20T146. Order = 720 H22: ModuliField: Q(x^3-x^2+2*x+2). Obstruction: none Branch type: 0:[1,1,3^6] 1:[2^10] infty:[2,2,8,8] Decomposition: H20(deg 2) 1-S32(deg 10) Galois group: GAP transitive group ID = 20T1006. Order = 3628800 [Felikson, Figure 17 (2,3,8), Entry #18] D16: ModuliField: Q. j-invariant = -254977^3*2^(-5)*3^(-12)*19^(-4) Obstruction: none Branch type: 0:[1,1,3^6] 1:[2^10] infty:[1,3,8,8] Decomposition: none Galois group: A20. Order = 20!/2 A13: ModuliField: Q(sqrt(-2)). j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[2,3^6] 1:[2^10] infty:[1,1,2,8,8] Decomposition: S7(deg 2) Galois group: GAP transitive group ID = 20T947. Order = 737280 G3: ModuliField: Q(sqrt(-1)). Obstruction: none Branch type: 0:[2,3^6] 1:[2^10] infty:[1,1,2,8,8] Decomposition: none Galois group: S20. Order = 20! Degree 21: (1/2,1/4,1/4,1/8) : no covering (1/2,1/8,1/8,3/8) : G16 G16: ModuliField: Q(sqrt(-2)). Obstruction: none Branch type: 0:[3^7] 1:[1,2^10] infty:[1,1,3,8,8] Decomposition: none Galois group: A21. Order = 21!/2 Degree 22: (1/2,1/3,1/8,1/8) : no covering (4/8,1/3,1/8,1/8) (1/3,1/4,1/4,1/4) : no covering (1/3,1/4,1/8,3/8) : I16 I16: ModuliField: Q(x^4-2*x^3+x^2+2) = Q( sqrt(-2), sqrt(1+4*sqrt(-2)) ). Obstruction: none Branch type: 0:[1,3^7] 1:[2^11] infty:[1,2,3,8,8] Decomposition: none Galois group: S22. Order = 22! Degree 24: (1/2,1/4,1/8,1/8) : A5 (4/8,1/4,1/8,1/8) (1/4,1/4,1/4,1/4) : A18 (1/4,1/4,1/8,3/8) : no covering (1/8,1/8,1/8,5/8) : no covering (1/8,1/8,3/8,3/8) : B14, G15 A5: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[1,1,2,4,8,8] Decomposition: S2(deg 2) S43(deg 4)(see #1) 1/S34(deg 8)(see #1,2) Galois group: (((C4 x C2) : C4) : C3) : C2. GAP transitive group ID = 24T428. Order = 192 [Felikson, Figure 17 (2,3,8), Entry #4] A18: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[2^4,8,8] Decomposition: S2(deg 2) C1(deg 2) C1(deg 2) S2(deg 2) S5(deg 2) S43(deg 4)(see #1,4,5) S27(deg 4)(see #2,3,5) 1/S49(deg 4)(see #5) 1/S34(deg 8)(see #5,8) 1/S34(deg 8)(see #1,2,3,4,5,6,7,8) 1/S34(deg 8)(see #5,8) 1-S32(deg 12)(see #5,8) Galois group: ((C4 x C4) : C3) : C2. GAP transitive group ID = 24T194. Order = 96 [Felikson, Figure 17 (2,3,8), Entry #2] B14: ModuliField: Q. j-invariant = 2^2*73^3*3^(-4) Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[1,1,3,3,8,8] Decomposition: D9(deg 2) S43(deg 4) 1/S34(deg 8)(see #1,2) Galois group: GAP transitive group ID = 24T14598. Order = 41472 [Felikson, Figure 17 (2,3,8), Entry #3] G15: ModuliField: Q(sqrt(-2)). Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[1,1,3,3,8,8] Decomposition: G14(deg 2) Galois group: GAP transitive group ID = 24T24970. Order = 980995276800 Degree 26: (1/3,1/3,1/8,1/8) : C22, I19 C22: ModuliField: Q. j-invariant = 11^3*1259^3*2^(-1)*3^(-3)*5^(-4) Obstruction: none Branch type: 0:[1,1,3^8] 1:[2^13] infty:[1,1,8^3] Decomposition: none Galois group: PSL(2,25) : C2. GAP transitive group ID = 26T55. Order = 15600 I19: ModuliField: Q(x^4-3*x^2-2*x+6). Obstruction: none Branch type: 0:[1,1,3^8] 1:[2^13] infty:[1,1,8^3] Decomposition: I18(deg 2) Galois group: GAP transitive group ID = 26T89. Order = 25505877196800 Degree 27: (1/2,1/8,1/8,1/8) : G6 G6: ModuliField: Q(sqrt(-1)). Obstruction: none Branch type: 0:[3^9] 1:[1,2^13] infty:[1^3,8^3] Decomposition: none Galois group: S27. Order = 27! Degree 28: (1/3,1/4,1/8,1/8) : G8 G8: ModuliField: Q(sqrt(-1)). Obstruction: none Branch type: 0:[1,3^9] 1:[2^14] infty:[1,1,2,8^3] Decomposition: none Galois group: A28. Order = 28!/2 Degree 30: (1/4,1/4,1/8,1/8) : B10, G5 (1/8,1/8,1/8,3/8) : D24 B10: ModuliField: Q. j-invariant = 2^2*73^3*3^(-4) Obstruction: none Branch type: 0:[3^10] 1:[2^15] infty:[1,1,2,2,8^3] Decomposition: S7(deg 3) 1/S34(deg 10) Galois group: (C3 x A6) : C2. GAP transitive group ID = 30T368. Order = 2160 G5: ModuliField: Q(sqrt(-1)). Obstruction: none Branch type: 0:[3^10] 1:[2^15] infty:[1,1,2,2,8^3] Decomposition: G1(deg 2) 1/S34(deg 10)(see #1) Galois group: GAP transitive group ID = 30T4037. Order = 49152000 D24: ModuliField: Q(sqrt(-2)). j-invariant = 0 Obstruction: none Branch type: 0:[3^10] 1:[2^15] infty:[1^3,3,8^3] Decomposition: S7(deg 3) Galois group: GAP transitive group ID = 30T3837. Order = 42515280 Degree 36: (1/8,1/8,1/8,1/8) : F6, G18 F6: ModuliField: Q(sqrt(2)). Obstruction: Primes: [infinity, infinity], Conic: X^2+Y^2+Z^2 = 0 Branch type: 0:[3^12] 1:[2^18] infty:[1^4,8^4] Decomposition: F4(deg 2) Galois group: C2 x PSL(2,17). Order = 4896 G18: ModuliField: Q(sqrt(-2)). Obstruction: none Branch type: 0:[3^12] 1:[2^18] infty:[1^4,8^4] Decomposition: J1(deg 2) J1(deg 2) J1(deg 2) G17(deg 4)(see #1,2,3) Galois group: Order = 28311552 =========== Table 2.3.9. (1/2,1/3,1/9) --> Heun ========== Degree 9: (1/2,1/3,1/3,1/3) : G26 G26: ModuliField: Q(sqrt(-3)). Obstruction: none Branch type: 0:[1^3,3,3] 1:[1,2^4] infty:[9] Decomposition: none Galois group: PSL(2,8) : C3. GAP transitive group ID = 9T32. Order = 1512 Degree 10: (1/2,1/2,1/3,1/9) : J4 (1/3,1/3,2/3,1/9) : H3 J4: ModuliField: Q(x^6-3*x^5+6*x^4-3*x^3+3*x^2-3*x+2). Obstruction: none Branch type: 0:[1,3^3] 1:[1,1,2^4] infty:[1,9] Decomposition: none Galois group: A10. Order = 10!/2 H3: ModuliField: Q(x^3-3) = Q(3^(1/3)). Obstruction: none Branch type: 0:[1,1,2,3,3] 1:[2^5] infty:[1,9] Decomposition: none Galois group: S10. Order = 10! Degree 11: (1/2,1/3,1/3,2/9) : H32 (1/2,2/3,1/9,1/9) : H31 H32: ModuliField: Q(x^3+6*x-1). Obstruction: none Branch type: 0:[1,1,3^3] 1:[1,2^5] infty:[2,9] Decomposition: none Galois group: S11. Order = 11! H31: ModuliField: Q(x^3-3*x-8). Obstruction: none Branch type: 0:[2,3^3] 1:[1,2^5] infty:[1,1,9] Decomposition: none Galois group: S11. Order = 11! Degree 12: (1/2,1/2,1/9,2/9) : H7 (1/3,1/3,1/3,1/3) : D19 (1/3,1/3,1/3,3/9) (1/3,2/3,1/9,2/9) : B7, H2 H7: ModuliField: Q(x^3-3*x-4). Obstruction: none Branch type: 0:[3^4] 1:[1,1,2^5] infty:[1,2,9] Decomposition: none Galois group: S12. Order = 12! D19: ModuliField: Q. j-invariant = 0 Obstruction: none Branch type: 0:[1^3,3^3] 1:[2^6] infty:[3,9] Decomposition: S47(deg 3) Galois group: ((C3 x ((C3 x C3) : C2)) : C2) : C3. GAP transitive group ID = 12T133. Order = 324 B7: ModuliField: Q. j-invariant = 2^2*73^3*3^(-4) Obstruction: none Branch type: 0:[1,2,3^3] 1:[2^6] infty:[1,2,9] Decomposition: S47(deg 3) Galois group: GAP transitive group ID = 12T280. Order = 15552 [Felikson, Figure 16 (2,3,9), Entry #2] H2: ModuliField: Q(x^3-3) = Q(3^(1/3)). Obstruction: none Branch type: 0:[1,2,3^3] 1:[2^6] infty:[1,2,9] Decomposition: none Galois group: S12. Order = 12! [Felikson, Figure 16 (2,3,9), Entry #1] Degree 13: (1/2,1/3,1/3,1/9) : I17 (1/2,1/3,3/9,1/9) (1/2,1/3,2/9,2/9) : C41 I17: ModuliField: Q(x^4-x^3+3*x^2+x+2). Obstruction: none Branch type: 0:[1,3^4] 1:[1,2^6] infty:[1,3,9] Decomposition: none Galois group: A13. Order = 13!/2 C41: ModuliField: Q. j-invariant = 5^3*349^3*8516873^3*2^(-30)*3^(-3)*7^(-9)*11^(-6)*13^(-1) Obstruction: none Branch type: 0:[1,3^4] 1:[1,2^6] infty:[2,2,9] Decomposition: none Galois group: A13. Order = 13!/2 [Felikson, Figure 16 (2,3,9), Entry #3] Degree 14: (1/3,1/3,1/3,2/9) : D39 (1/3,1/3,3/9,2/9) (1/3,1/3,1/9,4/9) : G21 (1/3,2/3,1/9,1/9) : D2 (3/9,2/3,1/9,1/9) (2/3,1/9,2/9,2/9) : C9 D39: ModuliField: Q. j-invariant = -5^3*1637^3*2^(-18)*7^(-1) Obstruction: none Branch type: 0:[1,1,3^4] 1:[2^7] infty:[2,3,9] Decomposition: none Galois group: S14. Order = 14! G21: ModuliField: Q(sqrt(-3)). Obstruction: none Branch type: 0:[1,1,3^4] 1:[2^7] infty:[1,4,9] Decomposition: none Galois group: S14. Order = 14! D2: ModuliField: Q. j-invariant = -2^4*109^3*5^(-6) Obstruction: none Branch type: 0:[2,3^4] 1:[2^7] infty:[1,1,3,9] Decomposition: none Galois group: S14. Order = 14! C9: ModuliField: Q. j-invariant = 13^3*541^3*3^(-3)*11^(-4) Obstruction: none Branch type: 0:[2,3^4] 1:[2^7] infty:[1,2,2,9] Decomposition: none Galois group: S14. Order = 14! [Felikson, Figure 16 (2,3,9), Entry #5] Degree 15: (1/2,1/3,1/9,2/9) : H4 (1/2,3/9,1/9,2/9) (1/2,1/9,1/9,4/9) : D49 (1/2,2/9,2/9,2/9) : no covering H4: ModuliField: Q(x^3-2) = Q(2^(1/3)). Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,2,3,9] Decomposition: none Galois group: S15. Order = 15! [Felikson, Figure 16 (2,3,9), Entry #6] D49: ModuliField: Q. j-invariant = -17^3*29^3*5197^3*2^(-30)*3^(-3)*5^(-2)*13^(-3) Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,1,4,9] Decomposition: none Galois group: S15. Order = 15! Degree 16: (1/3,1/3,1/3,1/9) : A6 (1/3,3/9,3/9,1/9) (1/3,1/3,2/9,2/9) : C7 (1/3,3/9,2/9,2/9) (1/3,1/9,1/9,5/9) : H25 (1/3,1/9,2/9,4/9) : H6 A6: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,3,3,9] Decomposition: S47(deg 4) Galois group: GAP transitive group ID = 16T1877. Order = 82944 [Felikson, Figure 16 (2,3,9), Entry #8] C7: ModuliField: Q. j-invariant = 2^7*53^3*3^(-3) Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[2,2,3,9] Decomposition: S47(deg 4) Galois group: GAP transitive group ID = 16T1877. Order = 82944 [Felikson, Figure 16 (2,3,9), Entry #4] H25: ModuliField: Q(x^3+3*x-1). Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,1,5,9] Decomposition: none Galois group: A16. Order = 16!/2 H6: ModuliField: Q(x^3+3*x-2). Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,2,4,9] Decomposition: none Galois group: A16. Order = 16!/2 [Felikson, Figure 16 (2,3,9), Entry #7] Degree 18: (1/3,1/3,1/9,2/9) : no covering (3/9,3/9,1/9,2/9) (1/3,1/9,1/9,4/9) : no covering (3/9,1/9,1/9,4/9) (1/3,2/9,2/9,2/9) : no covering (3/9,2/9,2/9,2/9) (2/3,1/9,1/9,1/9) : E3 (6/9,1/9,1/9,1/9) (1/9,1/9,2/9,5/9) : C23 (1/9,2/9,2/9,4/9) : no covering E3: ModuliField: Q. j-invariant = -3^3*17^3*2^(-1) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1^3,6,9] Decomposition: none Galois group: S18. Order = 18! C23: ModuliField: Q. j-invariant = 11^3*1259^3*2^(-1)*3^(-3)*5^(-4) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,1,2,5,9] Decomposition: none Galois group: S18. Order = 18! [Felikson, Figure 16 (2,3,9), Entry #9] Degree 20: (1/3,1/3,1/9,1/9) : H1, J3 H1: ModuliField: Q(x^3-3*x-1) = Q(Re(zeta_9)). Obstruction: Primes: [infinity, infinity], Conic: X^2+Y^2-RootOf(x^3-3*x-1)*Z^2 = 0 Branch type: 0:[1,1,3^6] 1:[2^10] infty:[1,1,9,9] Decomposition: none Galois group: PSL(2,19). GAP transitive group ID = 20T272. Order = 3420 J3: ModuliField: Q(x^6-3*x^5+6*x^4-3*x^3+3*x^2-3*x+2). Obstruction: none Branch type: 0:[1,1,3^6] 1:[2^10] infty:[1,1,9,9] Decomposition: J4(deg 2) Galois group: GAP transitive group ID = 20T1100. Order = 928972800 Degree 21: (1/2,1/9,1/9,1/9) : G52 G52: ModuliField: Q(sqrt(-15)). Obstruction: none Branch type: 0:[3^7] 1:[1,2^10] infty:[1^3,9,9] Decomposition: none Galois group: A21. Order = 21!/2 Degree 22: (1/3,1/9,1/9,2/9) : H9 H9: ModuliField: Q(x^3-3*x-4). Obstruction: none Branch type: 0:[1,3^7] 1:[2^11] infty:[1,1,2,9,9] Decomposition: none Galois group: S22. Order = 22! Degree 24: (1/3,1/9,1/9,1/9) : no covering (3/9,1/9,1/9,1/9) (1/9,1/9,2/9,2/9) : A24, H8 A24: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[1,1,2,2,9,9] Decomposition: S1(deg 2) S47(deg 6)(see #1) Galois group: GAP transitive group ID = 24T16559. Order = 82944 H8: ModuliField: Q(x^3-3*x-4). Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[1,1,2,2,9,9] Decomposition: H7(deg 2) Galois group: GAP transitive group ID = 24T24970. Order = 980995276800 [Felikson, Figure 16 (2,3,9), Entry #10] =========== Table 2.3.10. (1/2,1/3,1/10) --> Heun ========== Degree 10: (1/3,1/3,1/3,1/3) : B19 B19: ModuliField: Q. j-invariant = 7^3*127^3*2^(-2)*3^(-6)*5^(-2) Obstruction: none Branch type: 0:[1^4,3,3] 1:[2^5] infty:[10] Decomposition: 1-S37(deg 2) 1-S32(deg 5) Galois group: C2 x A5. GAP transitive group ID = 10T11. Order = 120 Degree 11: (1/2,1/3,1/3,1/10) : J8 J8: ModuliField: Q(x^6-3*x^5+5*x^4-5*x^3+x+2). Obstruction: none Branch type: 0:[1,1,3^3] 1:[1,2^5] infty:[1,10] Decomposition: none Galois group: S11. Order = 11! Degree 12: (1/2,1/2,1/10,1/10) : F10, G7 (1/3,1/3,1/3,1/5) : E13 (1/3,2/3,1/10,1/10) : H30 F10: ModuliField: Q(sqrt(5)). Obstruction: none Branch type: 0:[3^4] 1:[1,1,2^5] infty:[1,1,10] Decomposition: none Galois group: PSL(2,11) : C2. GAP transitive group ID = 12T218. Order = 1320 G7: ModuliField: Q(sqrt(-1)). Obstruction: none Branch type: 0:[3^4] 1:[1,1,2^5] infty:[1,1,10] Decomposition: S26(deg 2) Galois group: ((C2 x C2 x C2 x C2 x C2) : A5) : C2. GAP transitive group ID = 12T255. Order = 3840 E13: ModuliField: Q. j-invariant = -2^8*3^3*61^3*5^(-7) Obstruction: none Branch type: 0:[1^3,3^3] 1:[2^6] infty:[2,10] Decomposition: 1-S32(deg 6) Galois group: A6 : C2. GAP transitive group ID = 12T182. Order = 720 H30: ModuliField: Q(x^3-x^2-3*x-3) = Q(10^(1/3)). Obstruction: none Branch type: 0:[1,2,3^3] 1:[2^6] infty:[1,1,10] Decomposition: none Galois group: S12. Order = 12! Degree 13: (1/2,1/3,1/5,1/10) : J5 J5: ModuliField: Q(x^6-2*x^5+5*x^2+5). Obstruction: none Branch type: 0:[1,3^4] 1:[1,2^6] infty:[1,2,10] Decomposition: none Galois group: A13. Order = 13!/2 Degree 14: (1/3,1/3,1/5,1/5) : C38 (1/3,1/3,1/10,3/10) : D48 (2/3,1/5,1/10,1/10) : F14 C38: ModuliField: Q. j-invariant = 37^3*5653^3*2^(-2)*3^(-3)*5^(-12)*7^(-1) Obstruction: none Branch type: 0:[1,1,3^4] 1:[2^7] infty:[2,2,10] Decomposition: S22(deg 2) 1-S32(deg 7) Galois group: S7. GAP transitive group ID = 14T46. Order = 5040 [Felikson, Figure 15 (2,3,>=10), Entry #1] D48: ModuliField: Q. j-invariant = 1685104151^3*2^(-6)*3^(-32)*5^(-7)*7^(-1)*13^(-4) Obstruction: none Branch type: 0:[1,1,3^4] 1:[2^7] infty:[1,3,10] Decomposition: none Galois group: S14. Order = 14! F14: ModuliField: Q(sqrt(5)). Obstruction: none Branch type: 0:[2,3^4] 1:[2^7] infty:[1,1,2,10] Decomposition: none Galois group: S14. Order = 14! [Felikson, Figure 15 (2,3,>=10), Entry #2] Degree 15: (1/2,1/5,1/5,1/10) : C15 (1/2,1/10,1/10,3/10) : D44, D50 C15: ModuliField: Q. j-invariant = 2^4*17^3 Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,2,2,10] Decomposition: 1/S34(deg 5) Galois group: ((C5 x C5) : C3) : C2. GAP transitive group ID = 15T14. Order = 150 [Felikson, Figure 15 (2,3,>=10), Entry #3] D44: ModuliField: Q. j-invariant = -269^3*2^(-10)*3^(-5) Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,1,3,10] Decomposition: 1/S34(deg 5) Galois group: GAP transitive group ID = 15T96. Order = 1296000 D50: ModuliField: Q. j-invariant = -17^3*179^3*1223^3*2861^3*2^(-10)*3^(-5)*5^(-12)*11^(-6)*13^(-3) Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,1,3,10] Decomposition: none Galois group: S15. Order = 15! Degree 16: (1/3,1/5,1/5,1/5) : no covering (1/3,1/5,1/10,3/10) : G2 (1/3,2/5,1/10,1/10) : D42 G2: ModuliField: Q(sqrt(-1)). Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,2,3,10] Decomposition: none Galois group: A16. Order = 16!/2 D42: ModuliField: Q. j-invariant = -2^5*199287631^3*3^(-26)*5^(-6)*7^(-3) Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,1,4,10] Decomposition: none Galois group: A16. Order = 16!/2 Degree 18: (1/2,1/10,1/10,1/10) : E7 (5/10,1/10,1/10,1/10) (1/5,1/5,1/5,1/5) : no covering (1/5,1/5,1/10,3/10) : B24 (1/5,2/5,1/10,1/10) : D4 (1/10,1/10,3/10,3/10) : C25 E7: ModuliField: Q. j-invariant = -5^2*241^3*2^(-3) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1^3,5,10] Decomposition: none Galois group: S18. Order = 18! B24: ModuliField: Q. j-invariant = 7^3*2287^3*2^(-6)*3^(-2)*5^(-6) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,2,2,3,10] Decomposition: 1/S34(deg 6) Galois group: GAP transitive group ID = 18T970. Order = 279936000 [Felikson, Figure 15 (2,3,>=10), Entry #5] D4: ModuliField: Q. j-invariant = -2^4*109^3*5^(-6) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,1,2,4,10] Decomposition: 1/S34(deg 6) Galois group: GAP transitive group ID = 18T935. Order = 1296000 C25: ModuliField: Q. j-invariant = 11^3*1979^3*2^(-3)*3^(-1)*5^(-12) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,1,3,3,10] Decomposition: S13(deg 2) Galois group: GAP transitive group ID = 18T966. Order = 92897280 [Felikson, Figure 15 (2,3,>=10), Entry #4] Degree 24: (1/10,1/10,1/10,1/10) : D6, F17 D6: ModuliField: Q. j-invariant = -2^4*109^3*5^(-6) Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[1^4,10,10] Decomposition: G7(deg 2) G7(deg 2) S4(deg 2) S26(deg 4)(see #1,2,3) Galois group: ((C2 x C2 x C2 x C2 x C2) : A5) : C2. GAP transitive group ID = 24T7252. Order = 3840 F17: ModuliField: Q(sqrt(5)). Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[1^4,10,10] Decomposition: F10(deg 2) Galois group: GAP transitive group ID = 24T22755. Order = 2703360 =========== Table 2.3.11. (1/2,1/3,1/11) --> Heun ========== Degree 12: (1/3,1/3,1/3,1/11) : G45 G45: ModuliField: Q(sqrt(-11)). Obstruction: none Branch type: 0:[1^3,3^3] 1:[2^6] infty:[1,11] Decomposition: none Galois group: M12. GAP transitive group ID = 12T295. Order = 95040 Degree 13: (1/2,1/3,1/11,1/11) : J9 J9: ModuliField: Q(x^6-3*x^5+9*x^4-11*x^3+30*x^2-18*x+46). Obstruction: none Branch type: 0:[1,3^4] 1:[1,2^6] infty:[1,1,11] Decomposition: none Galois group: A13. Order = 13!/2 Degree 14: (1/3,1/3,1/11,2/11) : H28 (2/3,1/11,1/11,1/11) : E15 H28: ModuliField: Q(x^3-x^2+x+1). Obstruction: none Branch type: 0:[1,1,3^4] 1:[2^7] infty:[1,2,11] Decomposition: none Galois group: S14. Order = 14! E15: ModuliField: Q. j-invariant = -2^4*11^2*13^3*3^(-6) Obstruction: none Branch type: 0:[2,3^4] 1:[2^7] infty:[1^3,11] Decomposition: none Galois group: S14. Order = 14! Degree 15: (1/2,1/11,1/11,2/11) : H26 H26: ModuliField: Q(x^3-x^2+x+1). Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1,1,2,11] Decomposition: none Galois group: S15. Order = 15! Degree 16: (1/3,1/11,1/11,3/11) : D41 (1/3,1/11,2/11,2/11) : C13 D41: ModuliField: Q. j-invariant = -2^6*5^3*14411^3*3^(-6)*7^(-3)*11^(-10) Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,1,3,11] Decomposition: none Galois group: A16. Order = 16!/2 C13: ModuliField: Q. j-invariant = 2^7*5^3*1301^3*43889^3*3^(-17)*11^(-10)*13^(-4) Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,2,2,11] Decomposition: none Galois group: A16. Order = 16!/2 [Felikson, Figure 15 (2,3,>=10), Entry #6] Degree 18: (1/11,1/11,1/11,4/11) : no covering (1/11,1/11,2/11,3/11) : D15 (1/11,2/11,2/11,2/11) : no covering D15: ModuliField: Q. j-invariant = -482641^3*2^(-1)*3^(-2)*11^(-10) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,1,2,3,11] Decomposition: none Galois group: S18. Order = 18! =========== Table 2.3.12. (1/2,1/3,1/12) --> Heun ========== Degree 14: (1/3,1/3,1/12,1/12) : F9, G23 F9: ModuliField: Q(sqrt(3)). Obstruction: none Branch type: 0:[1,1,3^4] 1:[2^7] infty:[1,1,12] Decomposition: none Galois group: PSL(2,13) : C2. GAP transitive group ID = 14T39. Order = 2184 G23: ModuliField: Q(sqrt(-3)). Obstruction: none Branch type: 0:[1,1,3^4] 1:[2^7] infty:[1,1,12] Decomposition: S21(deg 2) Galois group: (((C2 x C2 x C2 x C2 x C2 x C2) : C7) : C3) : C2. GAP transitive group ID = 14T40. Order = 2688 Degree 15: (1/2,1/12,1/12,1/12) : G27 G27: ModuliField: Q(sqrt(-3)). Obstruction: none Branch type: 0:[3^5] 1:[1,2^7] infty:[1^3,12] Decomposition: none Galois group: S15. Order = 15! Degree 16: (1/3,1/6,1/12,1/12) : D14, G24 D14: ModuliField: Q. j-invariant = -2^6*239^3*3^(-10) Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,1,2,12] Decomposition: S47(deg 4) Galois group: GAP transitive group ID = 16T1790. Order = 18432 G24: ModuliField: Q(sqrt(-3)). Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1,1,2,12] Decomposition: none Galois group: A16. Order = 16!/2 Degree 18: (1/4,1/12,1/12,1/12) : D21 (1/6,1/6,1/12,1/12) : B11 D21: ModuliField: Q. j-invariant = 0 Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1^3,3,12] Decomposition: S43(deg 3) 1/S34(deg 6)(see #1) Galois group: C3 x ((((C3 x ((C3 x C3) : C2)) : C2) : C3) : C2). GAP transitive group ID = 18T346. Order = 1944 B11: ModuliField: Q. j-invariant = 2^2*73^3*3^(-4) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,1,2,2,12] Decomposition: S12(deg 2) S43(deg 3) 1/S34(deg 6)(see #1,2) Galois group: ((C6 x C6) : C3) : C2. GAP transitive group ID = 18T97. Order = 216 [Felikson, Figure 15 (2,3,>=10), Entry #7] =========== Table 2.3.13. (1/2,1/3,1/13) --> Heun ========== Degree 16: (1/3,1/13,1/13,1/13) : G28 G28: ModuliField: Q(sqrt(-3)). Obstruction: none Branch type: 0:[1,3^5] 1:[2^8] infty:[1^3,13] Decomposition: none Galois group: A16. Order = 16!/2 Degree 18: (1/13,1/13,1/13,2/13) : E16 E16: ModuliField: Q. j-invariant = -3^3*41^3*83^3*2^(-1)*13^(-7) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1^3,2,13] Decomposition: none Galois group: S18. Order = 18! =========== Table 2.3.14. (1/2,1/3,1/14) --> Heun ========== Degree 18: (1/14,1/14,1/14,1/14) : C2 C2: ModuliField: Q. j-invariant = 5^3*11^3*31^3*2^(-3)*7^(-6) Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1^4,14] Decomposition: S11(deg 2) Galois group: GAP transitive group ID = 18T855. Order = 258048 =========== Table 2.4.5. (1/2,1/4,1/5) --> Heun ========== Degree 5: (1/2,1/2,1/2,1/4) : E10 E10: ModuliField: Q. j-invariant = 5^1*211^3*2^(-15) Obstruction: none Branch type: 0:[1,4] 1:[1^3,2] infty:[5] Decomposition: none Galois group: S5. Order = 5! Degree 6: (1/2,1/2,1/2,1/5) : G46 (1/2,1/2,2/4,1/5) G46: ModuliField: Q(sqrt(-15)). Obstruction: none Branch type: 0:[2,4] 1:[1,1,2,2] infty:[1,5] Decomposition: none Galois group: A6. Order = 6!/2 Degree 7: (1/2,1/2,1/4,2/5) : H39 (1/2,2/4,1/4,2/5) (1/2,3/4,1/5,1/5) : G29 H39: ModuliField: Q(x^3+2*x-2). Obstruction: none Branch type: 0:[1,2,4] 1:[1,2^3] infty:[2,5] Decomposition: none Galois group: S7. Order = 7! [Felikson, Figure 13 (2,4,5), Entry #2] G29: ModuliField: Q(sqrt(-5)). Obstruction: none Branch type: 0:[3,4] 1:[1,2^3] infty:[1,1,5] Decomposition: none Galois group: S7. Order = 7! Degree 8: (1/2,1/2,1/5,2/5) : H19 : no covering (2/4,2/4,1/5,2/5) (1/2,1/4,1/4,3/5) : C19 (2/4,1/4,1/4,3/5) (1/4,3/4,1/5,2/5) : B26 H19: ModuliField: Q(x^3-x^2+2*x+2). Obstruction: none Branch type: 0:[4,4] 1:[1,1,2^3] infty:[1,2,5] Decomposition: none Galois group: S8. Order = 8! C19: ModuliField: Q. j-invariant = 2^6*971^3*3^(-5) Obstruction: none Branch type: 0:[1,1,2,4] 1:[2^4] infty:[3,5] Decomposition: none Galois group: A8. Order = 8!/2 [Felikson, Figure 13 (2,4,5), Entry #3] B26: ModuliField: Q. j-invariant = 4993^3*2^(-2)*3^(-8)*7^(-4) Obstruction: none Branch type: 0:[1,3,4] 1:[2^4] infty:[1,2,5] Decomposition: none Galois group: S8. Order = 8! [Felikson, Figure 13 (2,4,5), Entry #1] Degree 9: (1/2,1/4,1/5,3/5) : I24 (1/2,1/4,2/5,2/5) : D27 I24: ModuliField: Q(x^5-2*x^3-4*x^2-6*x-4). Obstruction: none Branch type: 0:[1,4,4] 1:[1,2^4] infty:[1,3,5] Decomposition: none Galois group: A9. Order = 9!/2 [Felikson, Figure 13 (2,4,5), Entry #7] D27: ModuliField: Q. j-invariant = -23^3*71^3*3^(-1)*7^(-8) Obstruction: none Branch type: 0:[1,4,4] 1:[1,2^4] infty:[2,2,5] Decomposition: none Galois group: A9. Order = 9!/2 Degree 10: (1/2,1/2,1/4,1/4) : C14, C30, H17 : F13 (2/4,2/4,1/4,1/4) (1/2,1/5,1/5,3/5) : D43 (2/4,1/5,1/5,3/5) (1/2,1/5,2/5,2/5) : C16 (2/4,1/5,2/5,2/5) (1/4,1/4,1/4,3/4) : no covering (1/4,1/4,1/5,4/5) : B8 (1/4,1/4,2/5,3/5) : no covering C14: ModuliField: Q. j-invariant = 2^11 Obstruction: none Branch type: 0:[1,1,4,4] 1:[1,1,2^4] infty:[5,5] Decomposition: S45(deg 2) Galois group: ((C2 x C2 x C2 x C2) : C5) : C2. GAP transitive group ID = 10T15. Order = 160 [Felikson, Figure 13 (2,4,5), Entry #10] C30: ModuliField: Q. j-invariant = 7949^3*2^(-5)*3^(-10) Obstruction: Primes: [infinity, 5], Conic: 5*X^2+2*Y^2+Z^2 = 0 Branch type: 0:[1,1,4,4] 1:[1,1,2^4] infty:[5,5] Decomposition: none Galois group: A6. GAP transitive group ID = 10T26. Order = 360 H17: ModuliField: Q(x^3-x^2+2*x+2). Obstruction: none Branch type: 0:[1,1,4,4] 1:[1,1,2^4] infty:[5,5] Decomposition: E10(deg 2) Galois group: ((C2 x C2 x C2 x C2) : A5) : C2. GAP transitive group ID = 10T37. Order = 1920 [Felikson, Figure 13 (2,4,5), Entry #8] F13: ModuliField: Q(sqrt(5)). Obstruction: none Branch type: 0:[1,1,2,2,4] 1:[2^5] infty:[5,5] Decomposition: S45(deg 2) Galois group: ((C2 x C2 x C2 x C2) : C5) : C2. GAP transitive group ID = 10T16. Order = 160 [Felikson, Figure 13 (2,4,5), Entry #9] D43: ModuliField: Q. j-invariant = -269^3*2^(-10)*3^(-5) Obstruction: none Branch type: 0:[2,4,4] 1:[2^5] infty:[1,1,3,5] Decomposition: 1/(1-S32)(deg 5) Galois group: (A5 x A5) : C2. GAP transitive group ID = 10T40. Order = 7200 C16: ModuliField: Q. j-invariant = 2^4*17^3 Obstruction: none Branch type: 0:[2,4,4] 1:[2^5] infty:[1,2,2,5] Decomposition: 1/(1-S32)(deg 5) Galois group: (D10 x D10) : C2. GAP transitive group ID = 10T21. Order = 200 [Felikson, Figure 13 (2,4,5), Entry #4] B8: ModuliField: Q. j-invariant = 2^2*73^3*3^(-4) Obstruction: none Branch type: 0:[1,1,4,4] 1:[2^5] infty:[1,4,5] Decomposition: none Galois group: S10. Order = 10! [Felikson, Figure 13 (2,4,5), Entry #11] Degree 11: (1/2,1/2,1/4,1/5) : I29 (1/2,2/4,1/4,1/5) I29: ModuliField: Q(x^5-2*x^4+2*x^3-4*x^2+x+8). Obstruction: none Branch type: 0:[1,2,4,4] 1:[1,2^5] infty:[1,5,5] Decomposition: none Galois group: S11. Order = 11! [Felikson, Figure 13 (2,4,5), Entry #13] Degree 12: (1/2,1/2,1/5,1/5) : F16, I11 : A2, G50 (2/4,2/4,1/5,1/5) (1/2,1/4,1/4,2/5) : A4 (2/4,1/4,1/4,2/5) (1/4,3/4,1/5,1/5) : F15 (1/5,1/5,1/5,4/5) : D3 (1/5,1/5,2/5,3/5) : B23 (1/5,2/5,2/5,2/5) : no covering F16: ModuliField: Q(sqrt(5)). Obstruction: none Branch type: 0:[4^3] 1:[1,1,2^5] infty:[1,1,5,5] Decomposition: none Galois group: PSL(2,11) : C2. GAP transitive group ID = 12T218. Order = 1320 I11: ModuliField: Q(x^4-3*x^2+6) = Q( sqrt(-15), sqrt(6+2*sqrt(-15)) ). Obstruction: none Branch type: 0:[4^3] 1:[1,1,2^5] infty:[1,1,5,5] Decomposition: G46(deg 2) Galois group: GAP transitive group ID = 12T286. Order = 23040 A2: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[2,2,4,4] 1:[2^6] infty:[1,1,5,5] Decomposition: S42(deg 2) 1/(1-S32)(deg 6) Galois group: S5. GAP transitive group ID = 12T74. Order = 120 G50: ModuliField: Q(sqrt(-15)). Obstruction: none Branch type: 0:[2,2,4,4] 1:[2^6] infty:[1,1,5,5] Decomposition: G46(deg 2) 1/(1-S32)(deg 6) Galois group: C2 x A6. GAP transitive group ID = 12T180. Order = 720 A4: ModuliField: Q(sqrt(-1)). j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[1,1,2,4,4] 1:[2^6] infty:[2,5,5] Decomposition: S42(deg 2) Galois group: GAP transitive group ID = 12T270. Order = 7680 F15: ModuliField: Q(sqrt(5)). Obstruction: none Branch type: 0:[1,3,4,4] 1:[2^6] infty:[1,1,5,5] Decomposition: none Galois group: A12. Order = 12!/2 [Felikson, Figure 13 (2,4,5), Entry #14] D3: ModuliField: Q. j-invariant = -2^4*109^3*5^(-6) Obstruction: none Branch type: 0:[4^3] 1:[2^6] infty:[1^3,4,5] Decomposition: 1/(1-S32)(deg 6) Galois group: (S5 x S5) : C2. GAP transitive group ID = 12T288. Order = 28800 B23: ModuliField: Q. j-invariant = 7^3*2287^3*2^(-6)*3^(-2)*5^(-6) Obstruction: none Branch type: 0:[4^3] 1:[2^6] infty:[1,1,2,3,5] Decomposition: 1/(1-S32)(deg 6) Galois group: GAP transitive group ID = 12T299. Order = 12!/462 [Felikson, Figure 13 (2,4,5), Entry #5] Degree 13: (1/2,1/4,1/5,2/5) : I31 I31: ModuliField: Q(x^5-x^4-6*x^3+10*x^2-5*x+5). Obstruction: none Branch type: 0:[1,4^3] 1:[1,2^6] infty:[1,2,5,5] Decomposition: none Galois group: S13. Order = 13! [Felikson, Figure 13 (2,4,5), Entry #16] Degree 14: (1/2,1/5,1/5,2/5) : D38 (2/4,1/5,1/5,2/5) (1/4,1/4,1/5,3/5) : C10 (1/4,1/4,2/5,2/5) : H41 D38: ModuliField: Q. j-invariant = -5^3*1637^3*2^(-18)*7^(-1) Obstruction: none Branch type: 0:[2,4^3] 1:[2^7] infty:[1,1,2,5,5] Decomposition: 1/(1-S32)(deg 7) Galois group: GAP transitive group ID = 14T61. Order = 14!/1716 C10: ModuliField: Q. j-invariant = 109^3*9133^3*2^(-4)*3^(-5)*13^(-4) Obstruction: none Branch type: 0:[1,1,4^3] 1:[2^7] infty:[1,3,5,5] Decomposition: none Galois group: S14. Order = 14! [Felikson, Figure 13 (2,4,5), Entry #15] H41: ModuliField: Q(x^3+2*x-2). Obstruction: none Branch type: 0:[1,1,4^3] 1:[2^7] infty:[2,2,5,5] Decomposition: H39(deg 2) Galois group: GAP transitive group ID = 14T57. Order = 645120 [Felikson, Figure 13 (2,4,5), Entry #12] Degree 15: (1/2,1/4,1/4,1/4) : E25 E25: ModuliField: Q. j-invariant = -5^1*3410909^3*2^(-20)*3^(-10)*13^(-5) Obstruction: none Branch type: 0:[1^3,4^3] 1:[1,2^7] infty:[5^3] Decomposition: none Galois group: S15. Order = 15! Degree 16: (1/2,1/4,1/4,1/5) : H18 (2/4,1/4,1/4,1/5) (1/5,1/5,1/5,3/5) : E18 (1/5,1/5,2/5,2/5) : H16 H18: ModuliField: Q(x^3-x^2+2*x+2). Obstruction: none Branch type: 0:[1,1,2,4^3] 1:[2^8] infty:[1,5^3] Decomposition: none Galois group: ((C2 x C2 x C2 x C2) : A5) : C2. GAP transitive group ID = 16T1328. Order = 1920 [Felikson, Figure 13 (2,4,5), Entry #17] E18: ModuliField: Q. j-invariant = -2^9*5^4*11^3*3^(-5) Obstruction: none Branch type: 0:[4^4] 1:[2^8] infty:[1^3,3,5,5] Decomposition: 1/(1-S32)(deg 8) Galois group: GAP transitive group ID = 16T1949. Order = 16!/25740 H16: ModuliField: Q(x^3-x^2+2*x+2). Obstruction: none Branch type: 0:[4^4] 1:[2^8] infty:[1,1,2,2,5,5] Decomposition: H19(deg 2) 1/(1-S32)(deg 8) Galois group: GAP transitive group ID = 16T1873. Order = 80640 [Felikson, Figure 13 (2,4,5), Entry #6] Degree 17: (1/2,1/4,1/5,1/5) : J18 J18: ModuliField: Q(x^8-2*x^7+6*x^6-8*x^5+18*x^4-20*x^3+16*x^2-20*x+10) = Q(I, sqrt(1+2*I), sqrt(3-I-(2+I)*sqrt(1+2*I)) ). Obstruction: none Branch type: 0:[1,4^4] 1:[1,2^8] infty:[1,1,5^3] Decomposition: none Galois group: A17. Order = 17!/2 Degree 18: (1/2,1/5,1/5,1/5) : no covering (2/4,1/5,1/5,1/5) (1/4,1/4,1/5,2/5) : G51 G51: ModuliField: Q(sqrt(-15)). Obstruction: none Branch type: 0:[1,1,4^4] 1:[2^9] infty:[1,2,5^3] Decomposition: none Galois group: S18. Order = 18! Degree 20: (1/4,1/4,1/4,1/4) : C17, D45, E11 (1/5,1/5,1/5,2/5) : E9 C17: ModuliField: Q. j-invariant = 2^4*17^3 Obstruction: none Branch type: 0:[1^4,4^4] 1:[2^10] infty:[5^4] Decomposition: F13(deg 2) C14(deg 2) F13(deg 2) S45(deg 4)(see #1,2,3) Galois group: ((C2 x C2 x C2 x C2) : C5) : C2. GAP transitive group ID = 20T45. Order = 160 [Felikson, Figure 13 (2,4,5), Entry #18] D45: ModuliField: Q. j-invariant = -269^3*2^(-10)*3^(-5) Obstruction: Primes: [infinity, 5], Conic: 5*X^2+2*Y^2+Z^2 = 0 Branch type: 0:[1^4,4^4] 1:[2^10] infty:[5^4] Decomposition: C30(deg 2) Galois group: C2 x A6. GAP transitive group ID = 20T152. Order = 720 E11: ModuliField: Q. j-invariant = 5^1*211^3*2^(-15) Obstruction: none Branch type: 0:[1^4,4^4] 1:[2^10] infty:[5^4] Decomposition: H17(deg 2) H17(deg 2) H17(deg 2) E10(deg 4)(see #1,2,3) Galois group: GAP transitive group ID = 20T567. Order = 30720 E9: ModuliField: Q. j-invariant = -5^2*241^3*2^(-3) Obstruction: none Branch type: 0:[4^5] 1:[2^10] infty:[1^3,2,5^3] Decomposition: 1/(1-S32)(deg 10) Galois group: GAP transitive group ID = 20T1115. Order = 20!/92378 Degree 22: (1/4,1/4,1/5,1/5) : I30 I30: ModuliField: Q(x^5-2*x^4+2*x^3-4*x^2+x+8). Obstruction: none Branch type: 0:[1,1,4^5] 1:[2^11] infty:[1,1,5^4] Decomposition: I29(deg 2) Galois group: GAP transitive group ID = 22T50. Order = 40874803200 [Felikson, Figure 13 (2,4,5), Entry #20] Degree 24: (1/5,1/5,1/5,1/5) : A20, F12, G48 A20: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[4^6] 1:[2^12] infty:[1^4,5^4] Decomposition: A2(deg 2) S42(deg 4)(see #1) 1/(1-S32)(deg 12)(see #1) Galois group: S5. GAP transitive group ID = 24T202. Order = 120 [Felikson, Figure 13 (2,4,5), Entry #19] F12: ModuliField: Q(sqrt(5)). Obstruction: none Branch type: 0:[4^6] 1:[2^12] infty:[1^4,5^4] Decomposition: F16(deg 2) 1/(1-S32)(deg 12) Galois group: PSL(2,11) : C2. GAP transitive group ID = 24T2949. Order = 1320 G48: ModuliField: Q(sqrt(-15)). Obstruction: none Branch type: 0:[4^6] 1:[2^12] infty:[1^4,5^4] Decomposition: I11(deg 2) G50(deg 2) I11(deg 2) G46(deg 4)(see #1,2,3) 1/(1-S32)(deg 12)(see #2) Galois group: GAP transitive group ID = 24T12619. Order = 23040 =========== Table 2.4.6. (1/2,1/4,1/6) --> Heun ========== Degree 6: (1/2,1/2,1/4,1/4) : C24, D25 C24: ModuliField: Q. j-invariant = 11^3*1979^3*2^(-3)*3^(-1)*5^(-12) Obstruction: none Branch type: 0:[1,1,4] 1:[1,1,2,2] infty:[6] Decomposition: none Galois group: S5. GAP transitive group ID = 6T14. Order = 120 D25: ModuliField: Q. j-invariant = 2^11*3^(-1) Obstruction: none Branch type: 0:[1,1,4] 1:[1,1,2,2] infty:[6] Decomposition: S34(deg 2) Galois group: C2 x S4. GAP transitive group ID = 6T11. Order = 48 Degree 7: (1/2,1/2,1/4,1/6) : I12 (1/2,2/4,1/4,1/6) I12: ModuliField: Q(x^4-2*x^3+3). Obstruction: none Branch type: 0:[1,2,4] 1:[1,2^3] infty:[1,6] Decomposition: none Galois group: S7. Order = 7! Degree 8: (1/2,1/2,1/6,1/6) : C3, G11 : no covering (2/4,2/4,1/6,1/6) (1/2,1/3,1/4,1/4) : D12 (2/4,1/3,1/4,1/4) (1/4,3/4,1/6,1/6) : G20 C3: ModuliField: Q. j-invariant = 73^3*601^3*2^(-1)*3^(-4)*7^(-8) Obstruction: none Branch type: 0:[4,4] 1:[1,1,2^3] infty:[1,1,6] Decomposition: none Galois group: PSL(3,2) : C2. GAP transitive group ID = 8T43. Order = 336 G11: ModuliField: Q(sqrt(-2)). Obstruction: none Branch type: 0:[4,4] 1:[1,1,2^3] infty:[1,1,6] Decomposition: 1/S36(deg 2) Galois group: ((((C2 x D8) : C2) : C3) : C2) : C2. GAP transitive group ID = 8T44. Order = 384 D12: ModuliField: Q. j-invariant = -2^6*239^3*3^(-10) Obstruction: none Branch type: 0:[1,1,2,4] 1:[2^4] infty:[2,6] Decomposition: 1-S32(deg 4) Galois group: (((C2 x C2 x C2 x C2) : C3) : C2) : C2. GAP transitive group ID = 8T41. Order = 192 G20: ModuliField: Q(sqrt(-3)). Obstruction: none Branch type: 0:[1,3,4] 1:[2^4] infty:[1,1,6] Decomposition: none Galois group: S8. Order = 8! Degree 9: (1/2,1/3,1/4,1/6) : I1 I1: ModuliField: Q(x^4-2*x^3+6*x^2-4*x+2). Obstruction: none Branch type: 0:[1,4,4] 1:[1,2^4] infty:[1,2,6] Decomposition: none Galois group: A9. Order = 9!/2 Degree 10: (1/2,1/3,1/6,1/6) : no covering (2/4,1/3,1/6,1/6) (1/2,1/4,1/4,1/6) : H5 (3/6,1/4,1/4,1/6) (1/3,1/3,1/4,1/4) : D32 H5: ModuliField: Q(x^3-2) = Q(2^(1/3)). Obstruction: none Branch type: 0:[1,1,4,4] 1:[2^5] infty:[1,3,6] Decomposition: none Galois group: (A6 : C2) : C2. GAP transitive group ID = 10T35. Order = 1440 D32: ModuliField: Q. j-invariant = -5281^3*3^(-16)*5^(-1) Obstruction: none Branch type: 0:[1,1,4,4] 1:[2^5] infty:[2,2,6] Decomposition: 1/S29(deg 2) 1-S32(deg 5) Galois group: C2 x S5. GAP transitive group ID = 10T22. Order = 240 Degree 12: (1/2,1/6,1/6,1/6) : D18 (3/6,1/6,1/6,1/6) (1/3,1/3,1/6,1/6) : B5 (1/4,1/4,1/4,1/4) : B4, B22 D18: ModuliField: Q. j-invariant = 0 Obstruction: none Branch type: 0:[4^3] 1:[2^6] infty:[1^3,3,6] Decomposition: 1/S35(deg 3) 1/(1-S32)(deg 6)(see #1) Galois group: (C3 x S3 x S3) : C2. GAP transitive group ID = 12T116. Order = 216 B5: ModuliField: Q. j-invariant = 2^2*73^3*3^(-4) Obstruction: none Branch type: 0:[4^3] 1:[2^6] infty:[1,1,2,2,6] Decomposition: 1/S25(deg 2) 1/S35(deg 3) 1/(1-S32)(deg 6)(see #1,2) Galois group: (S3 x S3) : C2. GAP transitive group ID = 12T36. Order = 72 [Felikson, Figure 12, Entry #3] B4: ModuliField: Q. j-invariant = 2^4*13^3*3^(-2) Obstruction: none Branch type: 0:[1^4,4,4] 1:[2^6] infty:[6,6] Decomposition: 1-S39(deg 2) 1/S43(deg 2) D25(deg 2) S34(deg 4)(see #1,2,3) 1-S32(deg 6)(see #1) Galois group: C2 x S4. GAP transitive group ID = 12T24. Order = 48 [Felikson, Figure 12, Entry #4] B22: ModuliField: Q. j-invariant = 7^3*2287^3*2^(-6)*3^(-2)*5^(-6) Obstruction: none Branch type: 0:[1^4,4,4] 1:[2^6] infty:[6,6] Decomposition: C24(deg 2) 1-S32(deg 6) Galois group: C2 x S5. GAP transitive group ID = 12T123. Order = 240 Degree 14: (1/4,1/4,1/6,1/6) : C11, I13 C11: ModuliField: Q. j-invariant = 109^3*9133^3*2^(-4)*3^(-5)*13^(-4) Obstruction: none Branch type: 0:[1,1,4^3] 1:[2^7] infty:[1,1,6,6] Decomposition: none Galois group: PSL(2,13) : C2. GAP transitive group ID = 14T39. Order = 2184 I13: ModuliField: Q(x^4-2*x^3+3). Obstruction: none Branch type: 0:[1,1,4^3] 1:[2^7] infty:[1,1,6,6] Decomposition: I12(deg 2) Galois group: GAP transitive group ID = 14T57. Order = 645120 Degree 16: (1/6,1/6,1/6,1/6) : B28, D11 B28: ModuliField: Q. j-invariant = 4993^3*2^(-2)*3^(-8)*7^(-4) Obstruction: none Branch type: 0:[4^4] 1:[2^8] infty:[1^4,6,6] Decomposition: C3(deg 2) 1/(1-S32)(deg 8) Galois group: C2 x (PSL(3,2) : C2). GAP transitive group ID = 16T1035. Order = 672 D11: ModuliField: Q. j-invariant = 2^1*47^3*3^(-8) Obstruction: none Branch type: 0:[4^4] 1:[2^8] infty:[1^4,6,6] Decomposition: G11(deg 2) G11(deg 2) 1/S20(deg 2) 1/S36(deg 4)(see #1,2,3) 1/(1-S32)(deg 8)(see #3) Galois group: ((((C2 x D8) : C2) : C3) : C2) : C2. GAP transitive group ID = 16T752. Order = 384 =========== Table 2.4.7. (1/2,1/4,1/7) --> Heun ========== Degree 7: (1/2,1/4,1/4,1/4) : E24 E24: ModuliField: Q. j-invariant = 3^3*7^1*2099^3*2^(-14)*5^(-7) Obstruction: none Branch type: 0:[1^3,4] 1:[1,2^3] infty:[7] Decomposition: none Galois group: S7. Order = 7! Degree 8: (1/2,1/4,1/4,1/7) : G37 (2/4,1/4,1/4,1/7) G37: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[1,1,2,4] 1:[2^4] infty:[1,7] Decomposition: none Galois group: (C2 x C2 x C2) : PSL(3,2). GAP transitive group ID = 8T48. Order = 8!/30 Degree 9: (1/2,1/4,1/7,1/7) : I7 I7: ModuliField: Q(x^4-2*x^3+2*x^2+2). Obstruction: none Branch type: 0:[1,4,4] 1:[1,2^4] infty:[1,1,7] Decomposition: none Galois group: A9. Order = 9!/2 Degree 10: (1/2,1/7,1/7,1/7) : no covering (2/4,1/7,1/7,1/7) (1/4,1/4,1/7,2/7) : D34 D34: ModuliField: Q. j-invariant = -11^3*88811^3*2^(-6)*3^(-4)*5^(-1)*7^(-12) Obstruction: none Branch type: 0:[1,1,4,4] 1:[2^5] infty:[1,2,7] Decomposition: none Galois group: S10. Order = 10! Degree 12: (1/7,1/7,1/7,2/7) : no covering =========== Table 2.4.8. (1/2,1/4,1/8) --> Heun ========== Degree 8: (1/4,1/4,1/4,1/4) : A16 A16: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[1^4,4] 1:[2^4] infty:[8] Decomposition: S35(deg 2) 1-S32(deg 4)(see #1) Galois group: (C4 x C4) : C2. GAP transitive group ID = 8T17. Order = 32 Degree 10: (1/4,1/4,1/8,1/8) : B9, G4 B9: ModuliField: Q. j-invariant = 2^2*73^3*3^(-4) Obstruction: none Branch type: 0:[1,1,4,4] 1:[2^5] infty:[1,1,8] Decomposition: none Galois group: A6 : C2. GAP transitive group ID = 10T30. Order = 720 G4: ModuliField: Q(sqrt(-1)). Obstruction: none Branch type: 0:[1,1,4,4] 1:[2^5] infty:[1,1,8] Decomposition: S44(deg 2) Galois group: C2 x (((C2 x C2 x C2 x C2) : C5) : C4). GAP transitive group ID = 10T29. Order = 640 Degree 12: (1/8,1/8,1/8,1/8) : no covering =========== Table 2.5.5. (1/2,1/5,1/5) --> Heun ========== Degree 6: (1/2,1/2,1/5,1/5) : A1, G49 A1: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[1,5] 1:[1,1,2,2] infty:[1,5] Decomposition: none Galois group: A5. GAP transitive group ID = 6T12. Order = 60 G49: ModuliField: Q(sqrt(-15)). Obstruction: none Branch type: 0:[1,5] 1:[1,1,2,2] infty:[1,5] Decomposition: none Galois group: A6. Order = 6!/2 Degree 7: (1/2,1/5,1/5,2/5) : D37 D37: ModuliField: Q. j-invariant = -5^3*1637^3*2^(-18)*7^(-1) Obstruction: none Branch type: 0:[1,1,5] 1:[1,2^3] infty:[2,5] Decomposition: none Galois group: S7. Order = 7! Degree 8: (1/5,1/5,1/5,3/5) : E17 (1/5,1/5,2/5,2/5) : H15 E17: ModuliField: Q. j-invariant = -2^9*5^4*11^3*3^(-5) Obstruction: none Branch type: 0:[1^3,5] 1:[2^4] infty:[3,5] Decomposition: none Galois group: A8. Order = 8!/2 H15: ModuliField: Q(x^3-x^2+2*x+2). Obstruction: none Branch type: 0:[1,2,5] 1:[2^4] infty:[1,2,5] Decomposition: none Galois group: S8. Order = 8! [Felikson, Figure 12, Entry #1] Degree 10: (1/5,1/5,1/5,2/5) : E8 E8: ModuliField: Q. j-invariant = -5^2*241^3*2^(-3) Obstruction: none Branch type: 0:[5,5] 1:[2^5] infty:[1^3,2,5] Decomposition: none Galois group: S10. Order = 10! Degree 12: (1/5,1/5,1/5,1/5) : A19, F11, G47 A19: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[1,1,5,5] 1:[2^6] infty:[1,1,5,5] Decomposition: A1(deg 2) Galois group: A5. GAP transitive group ID = 12T33. Order = 60 [Felikson, Figure 12, Entry #2] F11: ModuliField: Q(sqrt(5)). Obstruction: Primes: [infinity, 5], Conic: (2+5^(1/2))*X^2+5^(1/2)*Y^2+Z^2 = 0 Branch type: 0:[1,1,5,5] 1:[2^6] infty:[1,1,5,5] Decomposition: none Galois group: PSL(2,11). GAP transitive group ID = 12T179. Order = 660 G47: ModuliField: Q(sqrt(-15)). Obstruction: none Branch type: 0:[1,1,5,5] 1:[2^6] infty:[1,1,5,5] Decomposition: G49(deg 2) Galois group: (C2 x C2 x C2 x C2 x C2) : A6. GAP transitive group ID = 12T277. Order = 11520 =========== Table 2.5.6. (1/2,1/5,1/6) --> Heun ========== Degree 7: (1/2,1/5,1/5,1/6) : H24 H24: ModuliField: Q(x^3+3*x-1). Obstruction: none Branch type: 0:[1,1,5] 1:[1,2^3] infty:[1,6] Decomposition: none Galois group: S7. Order = 7! Degree 8: (1/3,1/5,1/5,1/5) : no covering (1/5,2/5,1/6,1/6) : D36 D36: ModuliField: Q. j-invariant = -2^3*6359^3*2999^3*3^(-7)*5^(-16)*7^(-4) Obstruction: none Branch type: 0:[1,2,5] 1:[2^4] infty:[1,1,6] Decomposition: none Galois group: S8. Order = 8! Degree 10: (1/6,1/6,1/6,1/6) : C31 C31: ModuliField: Q. j-invariant = 7949^3*2^(-5)*3^(-10) Obstruction: none Branch type: 0:[5,5] 1:[2^5] infty:[1^4,6] Decomposition: 1/(1-S37)(deg 2) Galois group: C2 x ((C2 x C2 x C2 x C2) : A5). GAP transitive group ID = 10T36. Order = 1920 =========== Table 2.5.7. (1/2,1/5,1/7) --> Heun ========== Degree 8: (1/5,1/5,1/5,1/7) : E23 E23: ModuliField: Q. j-invariant = 2^9*3^3*3739^3*5^(-11)*7^(-5) Obstruction: none Branch type: 0:[1^3,5] 1:[2^4] infty:[1,7] Decomposition: none Galois group: A8. Order = 8!/2 =========== Table 2.6.6. (1/2,1/6,1/6) --> Heun ========== Degree 8: (1/6,1/6,1/6,1/6) : B27, D10 B27: ModuliField: Q. j-invariant = 4993^3*2^(-2)*3^(-8)*7^(-4) Obstruction: none Branch type: 0:[1,1,6] 1:[2^4] infty:[1,1,6] Decomposition: none Galois group: PSL(3,2) : C2. GAP transitive group ID = 8T43. Order = 336 D10: ModuliField: Q. j-invariant = 2^1*47^3*3^(-8) Obstruction: none Branch type: 0:[1,1,6] 1:[2^4] infty:[1,1,6] Decomposition: S47(deg 2) Galois group: (((C2 x D8) : C2) : C3) : C2. GAP transitive group ID = 8T38. Order = 192 =========== Table 3.3.4. (1/3,1/3,1/4) --> Heun ========== Degree 5: (1/3,1/3,2/3,1/4) : F18 F18: ModuliField: Q(sqrt(6)). Obstruction: none Branch type: 0:[2,3] 1:[1,1,3] infty:[1,4] Decomposition: none Galois group: S5. Order = 5! [Felikson, Figure 10, Entry #5] Degree 6: (1/2,1/3,1/3,1/3) : E1 (1/3,2/3,1/4,1/4) : C20 E1: ModuliField: Q. j-invariant = -2^6*3^3*23^3 Obstruction: none Branch type: 0:[3,3] 1:[1^3,3] infty:[2,4] Decomposition: none Galois group: A6. Order = 6!/2 C20: ModuliField: Q. j-invariant = 2^6*19^3*467^3*3^(-7)*5^(-6) Obstruction: none Branch type: 0:[1,2,3] 1:[3,3] infty:[1,1,4] Decomposition: none Galois group: S6. Order = 6! [Felikson, Figure 10, Entry #7] Degree 7: (1/2,1/3,1/3,1/4) : F7 F7: ModuliField: Q(x^4-2*x^3-x^2-4*x-2) = Q( sqrt(2), sqrt(3+6*sqrt(2)) ). Note: j in Q(sqrt(2)). Obstruction: none Branch type: 0:[1,3,3] 1:[1,3,3] infty:[1,2,4] Decomposition: none Galois group: PSL(3,2). GAP transitive group ID = 7T5. Order = 168 [Felikson, Figure 10, Entry #9] Degree 8: (1/3,1/3,1/3,1/3) : D7, F1 D7: ModuliField: Q. j-invariant = -2^5*19^3*3^(-6) Obstruction: none Branch type: 0:[1,1,3,3] 1:[1,1,3,3] infty:[4,4] Decomposition: 1/(1-S47)(deg 2) Galois group: SL(2,3). GAP transitive group ID = 8T12. Order = 24 F1: ModuliField: Q(sqrt(2)). Obstruction: Primes: [infinity, 3], Conic: (1-2^(1/2))*X^2+3*Y^2+Z^2 = 0 Branch type: 0:[1,1,3,3] 1:[1,1,3,3] infty:[4,4] Decomposition: none Galois group: PSL(3,2). GAP transitive group ID = 8T37. Order = 168 Degree 9: (1/2,1/4,1/4,1/4) : no covering (1/3,1/3,1/3,1/4) : E4 E4: ModuliField: Q. j-invariant = -3^3*5^3*383^3*2^(-7) Obstruction: none Branch type: 0:[3^3] 1:[1^3,3,3] infty:[1,4,4] Decomposition: none Galois group: ((C3 x C3) : Q8) : C3. GAP transitive group ID = 9T23. Order = 216 Degree 10: (1/3,1/3,1/4,1/4) : A8, H21 A8: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[1,3^3] 1:[1,3^3] infty:[1,1,4,4] Decomposition: none Galois group: A6. GAP transitive group ID = 10T26. Order = 360 H21: ModuliField: Q(x^3-x^2+2*x+2). Obstruction: none Branch type: 0:[1,3^3] 1:[1,3^3] infty:[1,1,4,4] Decomposition: none Galois group: A10. Order = 10!/2 [Felikson, Figure 10, Entry #10] Degree 12: (1/4,1/4,1/4,1/4) : A17 A17: ModuliField: Q. j-invariant = 2^6*3^3 Obstruction: none Branch type: 0:[3^4] 1:[3^4] infty:[1^4,4,4] Decomposition: 1/S28(deg 2) 1/(1-S33)(deg 4)(see #1) Galois group: (C4 x C4) : C3. GAP transitive group ID = 12T31. Order = 48 [Felikson, Figure 10, Entry #11] =========== Table 3.3.5. (1/3,1/3,1/5) --> Heun ========== Degree 5: (1/3,1/3,1/3,1/3) : B18 B18: ModuliField: Q. j-invariant = 7^3*127^3*2^(-2)*3^(-6)*5^(-2) Obstruction: none Branch type: 0:[1,1,3] 1:[1,1,3] infty:[5] Decomposition: none Galois group: A5. Order = 5!/2 Degree 6: (1/3,1/3,1/3,1/5) : E12 E12: ModuliField: Q. j-invariant = -2^8*3^3*61^3*5^(-7) Obstruction: none Branch type: 0:[3,3] 1:[1^3,3] infty:[1,5] Decomposition: none Galois group: A6. Order = 6!/2 Degree 7: (1/3,1/3,1/5,1/5) : C37 C37: ModuliField: Q. j-invariant = 37^3*5653^3*2^(-2)*3^(-3)*5^(-12)*7^(-1) Obstruction: none Branch type: 0:[1,3,3] 1:[1,3,3] infty:[1,1,5] Decomposition: none Galois group: A7. Order = 7!/2 [Felikson, Figure 10, Entry #8] Degree 9: (1/5,1/5,1/5,1/5) : no covering =========== Table 3.4.4. (1/3,1/4,1/4) --> Heun ========== Degree 5: (1/3,1/3,1/4,1/4) : D31 D31: ModuliField: Q. j-invariant = -5281^3*3^(-16)*5^(-1) Obstruction: none Branch type: 0:[1,4] 1:[1,1,3] infty:[1,4] Decomposition: none Galois group: S5. Order = 5! Degree 6: (1/4,1/4,1/4,1/4) : B3, B21 B3: ModuliField: Q. j-invariant = 2^4*13^3*3^(-2) Obstruction: none Branch type: 0:[1,1,4] 1:[3,3] infty:[1,1,4] Decomposition: 1-1/S34(deg 2) Galois group: S4. GAP transitive group ID = 6T8. Order = 24 [Felikson, Figure 10, Entry #6] B21: ModuliField: Q. j-invariant = 7^3*2287^3*2^(-6)*3^(-2)*5^(-6) Obstruction: none Branch type: 0:[1,1,4] 1:[3,3] infty:[1,1,4] Decomposition: none Galois group: S5. GAP transitive group ID = 6T14. Order = 120 ====== Table S. 2F1 --> Heun or 2F1, varying exponent-differences ======= Degree 2: S32 S32: ModuliField: Q. Obstruction: none Branch type: 0:[2] 1:[1,1] infty:[2] Decomposition: none Galois group: C2. Order = 2 Degree 3: S33 S34 S33: ModuliField: Q. Obstruction: none Branch type: 0:[3] 1:[1^3] infty:[3] Decomposition: none Galois group: C3. Order = 3 S34: ModuliField: Q. Obstruction: none Branch type: 0:[1,2] 1:[1,2] infty:[3] Decomposition: none Galois group: S3. Order = 3! Degree 4: S31 S35 S36 S46 S47 S48 S31: ModuliField: Q. Obstruction: none Branch type: 0:[2,2] 1:[2,2] infty:[2,2] Decomposition: 1-S32(deg 2) S32(deg 2) 1/(1-S32)(deg 2) Galois group: C2 x C2. GAP transitive group ID = 4T2. Order = 4 S35: ModuliField: Q. Obstruction: none Branch type: 0:[1,1,2] 1:[2,2] infty:[4] Decomposition: 1-S32(deg 2) Galois group: D8. GAP transitive group ID = 4T3. Order = 8 S36: ModuliField: Q. Obstruction: none Branch type: 0:[1,3] 1:[1,1,2] infty:[4] Decomposition: none Galois group: S4. Order = 4! S46: ModuliField: Q. Obstruction: none Branch type: 0:[1,3] 1:[1,3] infty:[1,3] Decomposition: none Galois group: A4. Order = 4!/2 S47: ModuliField: Q. Obstruction: none Branch type: 0:[1,3] 1:[2,2] infty:[1,3] Decomposition: none Galois group: A4. Order = 4!/2 S48: ModuliField: Q. Obstruction: none Branch type: 0:[4] 1:[1^4] infty:[4] Decomposition: S32(deg 2) Galois group: C4. GAP transitive group ID = 4T1. Order = 4 Degree 5: S29 S30 S37 S44 S45 S29: ModuliField: Q. Obstruction: none Branch type: 0:[2,3] 1:[1,2,2] infty:[1,4] Decomposition: none Galois group: S5. Order = 5! S30: ModuliField: Q. Obstruction: none Branch type: 0:[2,3] 1:[1,2,2] infty:[2,3] Decomposition: none Galois group: S5. Order = 5! S37: ModuliField: Q. Obstruction: none Branch type: 0:[1,2,2] 1:[1,1,3] infty:[5] Decomposition: none Galois group: A5. Order = 5!/2 S44: ModuliField: Q(sqrt(-1)). Obstruction: none Branch type: 0:[1,4] 1:[1,2,2] infty:[1,4] Decomposition: none Galois group: C5 : C4. GAP transitive group ID = 5T3. Order = 20 S45: ModuliField: Q. Obstruction: none Branch type: 0:[1,2,2] 1:[1,2,2] infty:[5] Decomposition: none Galois group: D10. GAP transitive group ID = 5T2. Order = 10 Degree 6: S24 S25 S26 S27 S28 S38 S39 S42 S43 S49 S24: ModuliField: Q. Obstruction: none Branch type: 0:[1,2,3] 1:[2^3] infty:[1,5] Decomposition: none Galois group: S6. Order = 6! S25: ModuliField: Q. Obstruction: none Branch type: 0:[1,2,3] 1:[2^3] infty:[2,4] Decomposition: 1-S32(deg 3) Galois group: (S3 x S3) : C2. GAP transitive group ID = 6T13. Order = 72 S26: ModuliField: Q. Obstruction: none Branch type: 0:[3,3] 1:[1,1,2,2] infty:[1,5] Decomposition: none Galois group: A5. GAP transitive group ID = 6T12. Order = 60 S27: ModuliField: Q. Obstruction: none Branch type: 0:[3,3] 1:[1,1,2,2] infty:[2,4] Decomposition: 1/S34(deg 2) Galois group: S4. GAP transitive group ID = 6T7. Order = 24 S28: ModuliField: Q. Obstruction: none Branch type: 0:[1,1,2,2] 1:[3,3] infty:[3,3] Decomposition: 1-S33(deg 2) Galois group: A4. GAP transitive group ID = 6T4. Order = 12 S38: ModuliField: Q. Obstruction: none Branch type: 0:[1^3,3] 1:[2^3] infty:[6] Decomposition: 1-S32(deg 3) Galois group: C3 x S3. GAP transitive group ID = 6T5. Order = 18 S39: ModuliField: Q. Obstruction: none Branch type: 0:[2^3] 1:[1,1,2,2] infty:[6] Decomposition: S34(deg 2) S32(deg 3) Galois group: D12. GAP transitive group ID = 6T3. Order = 12 S42: ModuliField: Q. Obstruction: none Branch type: 0:[1,1,4] 1:[2^3] infty:[1,5] Decomposition: none Galois group: S5. GAP transitive group ID = 6T14. Order = 120 S43: ModuliField: Q. Obstruction: none Branch type: 0:[3,3] 1:[2^3] infty:[1,1,4] Decomposition: 1/S34(deg 2) Galois group: S4. GAP transitive group ID = 6T8. Order = 24 S49: ModuliField: Q. Obstruction: none Branch type: 0:[2^3] 1:[2^3] infty:[3,3] Decomposition: S34(deg 2) S34(deg 2) S34(deg 2) 1/(1-S32)(deg 3) Galois group: S3. GAP transitive group ID = 6T2. Order = 6 Degree 7: S21 S22 S23 S21: ModuliField: Q(sqrt(-3)). Obstruction: none Branch type: 0:[1,3,3] 1:[1,2^3] infty:[1,6] Decomposition: none Galois group: (C7 : C3) : C2. GAP transitive group ID = 7T4. Order = 42 S22: ModuliField: Q. Obstruction: none Branch type: 0:[1,3,3] 1:[1,2^3] infty:[2,5] Decomposition: none Galois group: S7. Order = 7! S23: ModuliField: Q. Obstruction: none Branch type: 0:[1,3,3] 1:[1,2^3] infty:[3,4] Decomposition: none Galois group: S7. Order = 7! Degree 8: S15 S16 S17 S18 S19 S20 S40 S41 S15: ModuliField: Q. Obstruction: none Branch type: 0:[2,3,3] 1:[2^4] infty:[1,1,6] Decomposition: S47(deg 2) Galois group: (((C2 x D8) : C2) : C3) : C2. GAP transitive group ID = 8T38. Order = 192 S16: ModuliField: Q. Obstruction: none Branch type: 0:[2,3,3] 1:[2^4] infty:[1,2,5] Decomposition: none Galois group: S8. Order = 8! S17: ModuliField: Q. Obstruction: none Branch type: 0:[2,3,3] 1:[2^4] infty:[2,3,3] Decomposition: S47(deg 2) Galois group: (((C2 x D8) : C2) : C3) : C2. GAP transitive group ID = 8T38. Order = 192 S18: ModuliField: Q. Obstruction: none Branch type: 0:[1,1,3,3] 1:[2^4] infty:[1,7] Decomposition: none Galois group: PSL(3,2). GAP transitive group ID = 8T37. Order = 168 S19: ModuliField: Q. Obstruction: none Branch type: 0:[1,1,3,3] 1:[2^4] infty:[2,6] Decomposition: S47(deg 2) 1-S32(deg 4) Galois group: C2 x A4. GAP transitive group ID = 8T13. Order = 24 S20: ModuliField: Q. Obstruction: none Branch type: 0:[1,1,3,3] 1:[2^4] infty:[4,4] Decomposition: S36(deg 2) 1-S32(deg 4) Galois group: S4. GAP transitive group ID = 8T14. Order = 24 S40: ModuliField: Q. Obstruction: none Branch type: 0:[1,1,2,4] 1:[2^4] infty:[4,4] Decomposition: S35(deg 2) 1-S32(deg 4)(see #1) Galois group: ((C4 x C2) : C2) : C2. GAP transitive group ID = 8T19. Order = 32 S41: ModuliField: Q. Obstruction: none Branch type: 0:[2^4] 1:[2^4] infty:[4,4] Decomposition: S35(deg 2) 1-S35(deg 2) 1-S35(deg 2) S35(deg 2) S31(deg 2) 1-S32(deg 4)(see #1,4,5) S32(deg 4)(see #2,3,5) 1/(1-S32)(deg 4)(see #5) Galois group: D8. GAP transitive group ID = 8T4. Order = 8 Degree 9: S11 S12 S13 S14 S11: ModuliField: Q. Obstruction: none Branch type: 0:[3^3] 1:[1,2^4] infty:[1,1,7] Decomposition: none Galois group: PSL(2,8). GAP transitive group ID = 9T27. Order = 504 S12: ModuliField: Q. Obstruction: none Branch type: 0:[3^3] 1:[1,2^4] infty:[1,2,6] Decomposition: 1/S34(deg 3) Galois group: ((C3 x C3) : C3) : C2. GAP transitive group ID = 9T11. Order = 54 S13: ModuliField: Q. Obstruction: none Branch type: 0:[3^3] 1:[1,2^4] infty:[1,3,5] Decomposition: none Galois group: A9. Order = 9!/2 S14: ModuliField: Q. Obstruction: none Branch type: 0:[3^3] 1:[1,2^4] infty:[2,3,4] Decomposition: 1/S34(deg 3) Galois group: (((C3 x ((C3 x C3) : C2)) : C2) : C3) : C2. GAP transitive group ID = 9T30. Order = 648 Degree 10: S7 S8 S9 S10 S7: ModuliField: Q. Obstruction: none Branch type: 0:[1,3^3] 1:[2^5] infty:[1,1,8] Decomposition: none Galois group: A6 : C2. GAP transitive group ID = 10T30. Order = 720 S8: ModuliField: Q. Obstruction: none Branch type: 0:[1,3^3] 1:[2^5] infty:[1,2,7] Decomposition: none Galois group: S10. Order = 10! S9: ModuliField: Q. Obstruction: none Branch type: 0:[1,3^3] 1:[2^5] infty:[1,4,5] Decomposition: none Galois group: S10. Order = 10! S10: ModuliField: Q. Obstruction: none Branch type: 0:[1,3^3] 1:[2^5] infty:[2,3,5] Decomposition: none Galois group: S10. Order = 10! Degree 12: S1 S2 S3 S4 S5 S6 S1: ModuliField: Q. Obstruction: none Branch type: 0:[3^4] 1:[2^6] infty:[1^3,9] Decomposition: S47(deg 3) Galois group: ((C3 x ((C3 x C3) : C2)) : C2) : C3. GAP transitive group ID = 12T132. Order = 324 S2: ModuliField: Q. Obstruction: none Branch type: 0:[3^4] 1:[2^6] infty:[1,1,2,8] Decomposition: S43(deg 2) 1/S34(deg 4)(see #1) Galois group: ((C4 x C4) : C3) : C2. GAP transitive group ID = 12T65. Order = 96 S3: ModuliField: Q. Obstruction: none Branch type: 0:[3^4] 1:[2^6] infty:[1,2,3,6] Decomposition: S47(deg 3) 1/S34(deg 4) Galois group: A4 x S3. GAP transitive group ID = 12T43. Order = 72 S4: ModuliField: Q. Obstruction: none Branch type: 0:[3^4] 1:[2^6] infty:[1,1,5,5] Decomposition: S26(deg 2) Galois group: A5. GAP transitive group ID = 12T33. Order = 60 S5: ModuliField: Q. Obstruction: none Branch type: 0:[3^4] 1:[2^6] infty:[2,2,4,4] Decomposition: S43(deg 2) S27(deg 2) 1/S49(deg 2) 1/S34(deg 4)(see #3) 1/S34(deg 4)(see #1,2,3) 1/S34(deg 4)(see #3) 1-S32(deg 6)(see #3) Galois group: S4. GAP transitive group ID = 12T9. Order = 24 S6: ModuliField: Q. Obstruction: none Branch type: 0:[3^4] 1:[2^6] infty:[3^4] Decomposition: 1-S28(deg 2) 1-S28(deg 2) 1-S28(deg 2) S47(deg 3) S47(deg 3) S47(deg 3) S47(deg 3) S33(deg 4)(see #1,2,3) Galois group: A4. GAP transitive group ID = 12T4. Order = 12 Degree 18: S51 S51: ModuliField: Q(sqrt(-7)). Obstruction: none Branch type: 0:[3^6] 1:[2^9] infty:[1,1,2,7,7] Decomposition: S11(deg 2) Galois group: GAP transitive group ID = 18T855. Order = 258048 Degree 24: S50 S50: ModuliField: Q. Obstruction: none Branch type: 0:[3^8] 1:[2^12] infty:[1^3,7^3] Decomposition: S18(deg 3) Galois group: PSL(3,2). GAP transitive group ID = 24T284. Order = 168