This file gives the gonality of X1(N)/Q for N=1..40, and upper bounds for N <= 300. It gives explicit proofs for these upper bounds, in the form of a function of that degree. Other files in this folder http://www.math.fsu.edu/~hoeij/files/X1N/ are: *) The file LowDegreePlaces contains non-cuspidal points of degrees d for which we have no function of degree d (e.g. degrees below the Q-gonality). *) Subfields: enumerates subfields of Q(X1(N)) over Q(j), for N=1..100, to look for functions of low degree in subfields. *) cusp_divisors, cusp_divisors_large, and cusp_divisors_program. These give the divisors of f[2] ... f[floor(N/2)+1], viewed as functions on X1(N). The degrees of the places occurring in these divisors are listed in the file: cusp_degrees. After weighing the entries in a divisor by the corresponding cusp_degrees, the sum of the positive entries will equal the degree of the corresponding function (and the sum of all entries will be zero). *) The folder LowerBoundsGonality contains Magma files for proving lower bounds (equal to the upper bounds) for all N <= 40. The case N=37 is treated in a separate sub-folder since it this computation is divided into numerous subcases. Let f[2] := b/(16*b^2+(1-20*c-8*c^2)*b+c*(c-1)^3); f[3] := b; f[4] := c; f[5] := b-c; f[6] := s-1; f[7] := s-r; f[8] := r*s-2*r+1; f[9] := s^2-s-r+1; f[10] := x-y+1; f[11] := x^2*y-x*y^2+y-1; f[12] := x-y; f[13] := x^3*y-x^2*y^2-x^2*y+x*y^2-y+1; f[14] := x^2*y-x*y^2-x*y+y^2-1; f[15] := x^2*y-x^2-x*y^2+x*y-x+y-1; f[16] := x^2*y^2-2*x^2*y-x*y^3+2*x*y^2-y+1; f[17] := x^4*y-x^3*y^3-x^3*y+x^2*y^4+x^2*y-x^2-x*y^4+x*y^3-x*y^2+x*y+y^3-2*y^2+y; f[18] := x^4*y-3*x^3*y^2+x^3*y+2*x^2*y^3+x^2*y^2-2*x^2*y-2*x*y^3+2*x*y^2+y^2-2*y+1; f[19] := x^5*y^2-3*x^4*y^3+3*x^3*y^4+2*x^3*y^2-2*x^3*y-x^2*y^5-x^2*y^4-x^2*y^3+x^2*y^2+x^2*y+x*y^5-x*y^4+2*x*y^3-2*x*y^2-x*y+x-y^4+2*y^3-y^2; f[20] := x^4*y-2*x^3*y^2-x^3*y+x^2*y^3+x^2*y^2+2*x^2*y-x^2-2*x*y^2+x*y+y-1; f[21] := x^5*y^2-3*x^4*y^3-x^4*y^2+3*x^3*y^4+3*x^3*y^3-x^2*y^5-3*x^2*y^4-2*x^2*y^2+3*x^2*y+x*y^5+3*x*y^3-4*x*y^2-y^4+y^3+y-1; f[22] := x^5*y^2-x^5*y-2*x^4*y^3+2*x^4*y+x^3*y^4+3*x^3*y^3-4*x^3*y^2+x^3-2*x^2*y^4+x^2*y^3+x^2*y^2-x^2*y+x*y^4-x*y^3+x*y^2-x*y-y^3+2*y^2-y; ... where {c = r*s-s, b = r^2*s-r*s}, {r = b/c , s = c^2/(b-c)} {r = (x^2*y-x*y+y-1)/(x^2*y-x), s = (x*y-y+1)/(x*y)}, {x = (s-r)/(r*s-2*r+1), y = (r*s-2*r+1)/(s^2-r-s+1)} and f[10] .. f[189] can be downloaded from the website of Andrew V. Sutherland, specifically, the first table on: http://math.mit.edu/~drew/X1_altcurves.html A modular unit is a function whose support (roots and poles) consists solely of cusps. For N > 3 we can write a minimal degree modular unit for X1(N) as follows: Gon_Q(X1(1)) = 1. For N=1, the j-invariant is a function of degree 1, but there is no modular unit of degree 1 since there is only 1 cusp. Gon_Q(X1(2)) = 1. For N=2, there is a modular unit of degree 1, but we do not write it in terms of b,c because b,c are not defined for N<4. Gon_Q(X1(3)) = 1. For N=3, same comment. Gon_Q(X1(4)) = 1. Function of minimal degree: f[3] Gon_Q(X1(5)) = 1. Function of minimal degree: f[3] Gon_Q(X1(6)) = 1. Function of minimal degree: f[4] Gon_Q(X1(7)) = 1. Function of minimal degree: f[4]/f[3] Gon_Q(X1(8)) = 1. Function of minimal degree: f[5]/f[4] Gon_Q(X1(9)) = 1. Function of minimal degree: f[5]/f[4]^2 Gon_Q(X1(10)) = 1. Function of minimal degree: f[6] Gon_Q(X1(11)) = 2. Function of minimal degree: f[6] Gon_Q(X1(12)) = 1. Function of minimal degree: f[6]/f[7] Gon_Q(X1(13)) = 2. Function of minimal degree: f[4]*f[7]/(f[5]*f[6]) Gon_Q(X1(14)) = 2. Function of minimal degree: f[6] Gon_Q(X1(15)) = 2. Function of minimal degree: f[7]/f[8] Gon_Q(X1(16)) = 2. Function of minimal degree: f[8]/f[9] Gon_Q(X1(17)) = 4. Function of minimal degree: f[7]/f[8] Gon_Q(X1(18)) = 2. Function of minimal degree: f[4]*f[7]^2/(f[5]*f[9]*f[10]) Gon_Q(X1(19)) = 5. Function of minimal degree: f[10] Gon_Q(X1(20)) = 3. Function of minimal degree: f[10] Gon_Q(X1(21)) = 4. Function of minimal degree: f[10] Gon_Q(X1(22)) = 4. Function of minimal degree: f[7]/f[8] Gon_Q(X1(23)) = 7. Function of minimal degree: f[7]/f[8] Gon_Q(X1(24)) = 4. Function of minimal degree: f[11]/f[13] Gon_Q(X1(25)) = 5. Function of minimal degree: f[3]*f[4]*f[6]^3*f[7]^2*f[10]*f[12]^3/(f[5]*f[8]^2*f[9]^2*f[13]^3) Gon_Q(X1(26)) = 6. Function of minimal degree: f[8]/f[9] Gon_Q(X1(27)) = 6. Function of minimal degree: f[13]/f[10] Gon_Q(X1(28)) = 6. Function of minimal degree: f[6]^2*f[14]/(f[9]*f[13]) Gon_Q(X1(29)) = 11. Function of minimal degree: f[7]/f[8] Gon_Q(X1(30)) = 6. Function of minimal degree: f[6]*f[7]*f[14]/(f[9]*f[16]) Gon_Q(X1(31)) = 12. Function of minimal degree: f[11]/f[13] Gon_Q(X1(32)) = 8. Function of minimal degree: f[7]^2*f[16]/(f[8]^2*f[17]) Gon_Q(X1(33)) = 10. Function of minimal degree: f[10]*f[11]*f[12]/f[17] Gon_Q(X1(34)) = 10. Function of minimal degree: f[6]/f[12] Gon_Q(X1(35)) = 12. Function of minimal degree: f[17]/f[18] Gon_Q(X1(36)) = 8. Function of minimal degree: f[14]*f[17]^2/f[19]^2 Gon_Q(X1(37)) = 18. Function of minimal degree: f[7]/f[8] Gon_Q(X1(38)) = 12. Function of minimal degree: f[8]*f[9]*f[10]*f[11]/(f[6]*f[7]^2*f[14]) Gon_Q(X1(39)) = 14. Function of minimal degree: f[12]*f[13]*f[14]/f[19] Gon_Q(X1(40)) = 12. Function of minimal degree: f[11]*f[20]/(f[12]*f[19]) Gon_Q(X1(41)) <= 22. Function of that degree: f[7]/f[8] Gon_Q(X1(42)) <= 12. Function of that degree: f[11]*f[16]*f[22]/(f[10]*f[19]*f[20]) Gon_Q(X1(43)) <= 24. Function of that degree: f[7]/f[8] Gon_Q(X1(44)) <= 15. Function of that degree: f[6]*f[7]*f[13]*f[21]/(f[9]*f[23]) Gon_Q(X1(45)) <= 18. Function of that degree: f[11]*f[22]/f[23] Gon_Q(X1(46)) <= 19. Function of that degree: f[10]*f[11]*f[12]/f[18] Gon_Q(X1(47)) <= 29. Function of that degree: f[7]/f[8] Gon_Q(X1(48)) <= 16. Function of that degree: f[10]*f[23]/f[25] Gon_Q(X1(49)) <= 21. Function of that degree: f[3]*f[4]^2*f[6]^3*f[7]^2*f[12]^3*f[14]*f[24]^3/(f[5]^2*f[8]^2*f[9]*f[25]^3) Gon_Q(X1(50)) <= 15. Function of that degree: f[5]*f[9]*f[13]*f[15]*f[26]/(f[4]^2*f[7]*f[12]*f[20]*f[24]) Gon_Q(X1(51)) <= 24. Function of that degree: f[12]*f[13]*f[24]/(f[14]*f[17]*f[20]) Gon_Q(X1(52)) <= 21. Function of that degree: f[5]*f[6]*f[15]*f[25]/(f[4]*f[9]*f[11]*f[27]) Gon_Q(X1(53)) <= 37. Function of that degree: f[7]/f[8] Gon_Q(X1(54)) <= 18. Function of that degree: f[12]*f[13]*f[26]/(f[16]*f[18]*f[20]) Gon_Q(X1(55)) <= 30. Function of that degree: f[8]*f[27]/(f[7]*f[14]*f[28]) Gon_Q(X1(56)) <= 24. Function of that degree: f[11]*f[27]/f[29] Gon_Q(X1(57)) <= 30. Function of that degree: f[18]*f[19]*f[20]/f[29] Gon_Q(X1(58)) <= 31. Function of that degree: f[16]*f[18]/f[26] Gon_Q(X1(59)) <= 46. Function of that degree: f[7]/f[8] Gon_Q(X1(60)) <= 24. Function of that degree: f[11]*f[29]/f[31] Gon_Q(X1(61)) <= 49. Function of that degree: f[7]/f[8] Gon_Q(X1(62)) <= 36. Function of that degree: f[16]*f[18]/f[26] Gon_Q(X1(63)) <= 36. Function of that degree: f[16]*f[32]/f[31] Gon_Q(X1(64)) <= 32. Function of that degree: f[12]*f[31]/(f[11]*f[33]) Gon_Q(X1(65)) <= 42. Function of that degree: f[16]*f[32]/(f[11]*f[33]) Gon_Q(X1(66)) <= 30. Function of that degree: f[11]*f[20]*f[22]*f[24]/(f[17]*f[34]) Gon_Q(X1(67)) <= 58. Function of that degree: f[15]*f[17]*f[30]/(f[11]*f[33]) Gon_Q(X1(68)) <= 36. Function of that degree: f[6]^2*f[16]*f[18]*f[34]/(f[9]*f[11]*f[19]*f[33]) Gon_Q(X1(69)) <= 44. Function of that degree: f[13]*f[17]*f[34]/(f[22]*f[23]*f[24]) Gon_Q(X1(70)) <= 36. Function of that degree: f[17]*f[34]/(f[18]*f[36]) Gon_Q(X1(71)) <= 66. Function of that degree: f[7]/f[8] Gon_Q(X1(72)) <= 32. Function of that degree: f[9]*f[13]*f[26]*f[32]/(f[6]*f[7]*f[17]*f[20]*f[34]) Gon_Q(X1(73)) <= 70. Function of that degree: f[7]/f[8] Gon_Q(X1(74)) <= 51. Function of that degree: f[10]*f[27]/f[32] Gon_Q(X1(75)) <= 40. Function of that degree: f[7]^2*f[35]*f[37]^3/(f[6]*f[8]*f[10]*f[19]^3*f[20]*f[30]*f[38]^3) Gon_Q(X1(76)) <= 45. Function of that degree: f[17]*f[21]*f[38]/(f[12]*f[18]*f[23]*f[36]) Gon_Q(X1(77)) <= 60. Function of that degree: f[19]*f[38]/(f[13]*f[39]) Gon_Q(X1(78)) <= 42. Function of that degree: f[16]*f[19]*f[38]/(f[13]*f[24]*f[26]*f[28]) Gon_Q(X1(79)) <= 82. Function of that degree: f[7]/f[8] Gon_Q(X1(80)) <= 48. Function of that degree: f[13]^2*f[39]/f[41] Gon_Q(X1(81)) <= 54. Function of that degree: f[13]*f[18]*f[39]/(f[10]*f[24]*f[27]*f[30]) Gon_Q(X1(82)) <= 58. Function of that degree: f[13]*f[26]/(f[10]*f[30]) Gon_Q(X1(83)) <= 90. Function of that degree: f[7]/f[8] Gon_Q(X1(84)) <= 48. Function of that degree: f[13]*f[41]/f[43] Gon_Q(X1(85)) <= 72. Function of that degree: f[7]*f[11]*f[14]*f[21]*f[42]/(f[8]*f[43]) Gon_Q(X1(86)) <= 64. Function of that degree: f[23]*f[24]/f[31] Gon_Q(X1(87)) <= 70. Function of that degree: f[16]*f[43]/(f[10]*f[28]*f[29]*f[30]) Gon_Q(X1(88)) <= 60. Function of that degree: f[8]*f[43]/(f[7]*f[45]) Gon_Q(X1(89)) <= 104. Function of that degree: f[7]/f[8] Gon_Q(X1(90)) <= 48. Function of that degree: f[4]^4*f[6]*f[7]^5*f[11]^5*f[22]^5*f[35]*f[44]^5/(f[5]^2*f[8]^2*f[9]^3*f[20]*f[23]^5*f[46]^5) Gon_Q(X1(91)) <= 84. Function of that degree: f[5]*f[6]*f[10]*f[15]*f[45]/(f[4]*f[9]*f[23]*f[46]) Gon_Q(X1(92)) <= 66. Function of that degree: f[8]*f[19]*f[27]*f[46]/(f[7]*f[14]*f[21]*f[31]*f[42]) Gon_Q(X1(93)) <= 80. Function of that degree: f[13]*f[26]*f[43]/(f[24]*f[31]*f[38]) Gon_Q(X1(94)) <= 83. Function of that degree: f[18]*f[29]/f[38] Gon_Q(X1(95)) <= 90. Function of that degree: f[9]*f[47]/(f[6]*f[7]*f[16]*f[24]*f[48]) Gon_Q(X1(96)) <= 56. Function of that degree: f[9]*f[16]*f[47]/(f[6]*f[7]*f[49]) Gon_Q(X1(97)) <= 123. Function of that degree: f[7]/f[8] Gon_Q(X1(98)) <= 63. Function of that degree: f[10]*f[21]*f[25]*f[28]*f[50]/(f[12]*f[16]*f[24]*f[35]*f[48]) Gon_Q(X1(99)) <= 90. Function of that degree: f[7]*f[49]/(f[8]*f[25]*f[50]) Gon_Q(X1(100)) <= 60. Function of that degree: f[17]*f[35]*f[40]*f[51]/(f[15]*f[30]*f[45]*f[49]) Gon_Q(X1(101)) <= 133. Function of that degree: f[18]*f[36]*f[43]/(f[16]*f[24]*f[29]*f[48]) Gon_Q(X1(102)) <= 72. Function of that degree: f[10]*f[13]*f[39]*f[41]/(f[23]*f[24]*f[27]*f[46]) Gon_Q(X1(103)) <= 139. Function of that degree: f[7]/f[8] Gon_Q(X1(104)) <= 84. Function of that degree: f[17]^2*f[51]/f[53] Gon_Q(X1(105)) <= 96. Function of that degree: f[13]^2*f[26]*f[52]/f[53] Gon_Q(X1(106)) <= 105. Function of that degree: f[26]*f[27]/f[40] Gon_Q(X1(107)) <= 150. Function of that degree: f[7]/f[8] Gon_Q(X1(108)) <= 72. Function of that degree: f[13]*f[24]*f[26]*f[52]/(f[18]*f[32]*f[36]*f[40]) Gon_Q(X1(109)) <= 156. Function of that degree: f[7]/f[8] Gon_Q(X1(110)) <= 90. Function of that degree: f[9]*f[16]*f[18]*f[27]*f[54]/(f[6]*f[7]*f[14]*f[28]*f[56]) Gon_Q(X1(111)) <= 114. Function of that degree: f[18]*f[23]*f[27]*f[54]/(f[34]*f[37]*f[40]) Gon_Q(X1(112)) <= 96. Function of that degree: f[11]*f[18]*f[55]/(f[10]*f[19]*f[57]) Gon_Q(X1(113)) <= 167. Function of that degree: f[7]/f[8] Gon_Q(X1(114)) <= 90. Function of that degree: f[12]*f[19]*f[36]*f[38]*f[40]/(f[16]*f[29]*f[58]) Gon_Q(X1(115)) <= 132. Function of that degree: f[6]*f[7]*f[14]*f[19]*f[57]/(f[9]*f[29]*f[58]) Gon_Q(X1(116)) <= 105. Function of that degree: f[19]*f[31]*f[57]/(f[27]*f[59]) Gon_Q(X1(117)) <= 126. Function of that degree: f[11]*f[29]*f[58]/f[59] Gon_Q(X1(118)) <= 120. Function of that degree: f[19]*f[38]/(f[11]*f[14]*f[42]) Gon_Q(X1(119)) <= 144. Function of that degree: f[4]*f[9]*f[59]/(f[5]*f[6]*f[10]*f[15]*f[20]*f[30]*f[60]) Gon_Q(X1(120)) <= 96. Function of that degree: f[16]*f[59]/f[61] Gon_Q(X1(121)) <= 132. Function of that degree: f[3]*f[6]^5*f[7]*f[8]*f[10]^3*f[12]^3*f[15]^3*f[20]^3*f[22]*f[30]^3*f[60]^3/(f[9]^5*f[61]^3) Gon_Q(X1(122)) <= 139. Function of that degree: f[31]*f[58]/f[59] Gon_Q(X1(123)) <= 140. Function of that degree: f[11]*f[22]*f[61]/(f[19]*f[31]*f[62]) Gon_Q(X1(124)) <= 120. Function of that degree: f[9]*f[11]*f[33]*f[61]/(f[6]*f[7]*f[21]*f[29]*f[63]) Gon_Q(X1(125)) <= 125. Function of that degree: f[4]*f[5]^2*f[8]^18*f[9]^9*f[10]*f[31]^15*f[50]*f[62]^15/(f[3]^5*f[6]^8*f[7]^26*f[21]^15*f[63]^15) Gon_Q(X1(126)) <= 96. Function of that degree: f[5]^3*f[8]^2*f[9]^3*f[31]^7*f[49]*f[62]^7/(f[4]^7*f[6]*f[7]^4*f[16]^7*f[28]*f[32]^7*f[64]^7) Gon_Q(X1(127)) <= 211. Function of that degree: f[7]/f[8] Gon_Q(X1(128)) <= 112. Function of that degree: f[3]*f[4]*f[6]^2*f[7]^4*f[21]^3*f[32]*f[63]^3/(f[5]^2*f[8]*f[9]^2*f[13]^3*f[65]^3) Gon_Q(X1(129)) <= 154. Function of that degree: f[16]*f[23]*f[32]*f[64]/(f[14]*f[42]*f[43]*f[44]) Gon_Q(X1(130)) <= 126. Function of that degree: f[16]*f[18]*f[32]*f[64]/(f[11]*f[22]*f[33]*f[66]) Gon_Q(X1(131)) <= 225. Function of that degree: f[7]/f[8] Gon_Q(X1(132)) <= 120. Function of that degree: f[13]*f[17]*f[65]/f[67] Gon_Q(X1(133)) <= 180. Function of that degree: f[11]*f[13]*f[22]*f[33]*f[66]/f[67] Gon_Q(X1(134)) <= 156. Function of that degree: f[11]*f[16]*f[48]/f[43] Gon_Q(X1(135)) <= 144. Function of that degree: f[13]*f[30]*f[65]/(f[40]*f[45]*f[50]) Gon_Q(X1(136)) <= 144. Function of that degree: f[7]*f[21]*f[67]/(f[8]*f[13]*f[23]*f[69]) Gon_Q(X1(137)) <= 246. Function of that degree: f[7]/f[8] Gon_Q(X1(138)) <= 132. Function of that degree: f[7]*f[21]*f[41]*f[63]/(f[6]*f[12]*f[23]*f[34]*f[46]*f[58]) Gon_Q(X1(139)) <= 253. Function of that degree: f[7]/f[8] Gon_Q(X1(140)) <= 144. Function of that degree: f[49]/f[56] Gon_Q(X1(141)) <= 184. Function of that degree: f[16]*f[46]*f[47]*f[48]/(f[11]*f[22]*f[71]) Gon_Q(X1(142)) <= 189. Function of that degree: f[26]*f[43]/f[56] Gon_Q(X1(143)) <= 210. Function of that degree: f[18]*f[24]*f[36]*f[72]/f[71] Gon_Q(X1(144)) <= 128. Function of that degree: f[6]*f[8]*f[13]*f[26]*f[52]*f[64]/(f[9]*f[17]*f[34]*f[40]*f[68]) Gon_Q(X1(145)) <= 210. Function of that degree: f[17]*f[18]*f[24]*f[36]*f[72]/f[73] Gon_Q(X1(146)) <= 184. Function of that degree: f[22]*f[59]/(f[31]*f[62]) Gon_Q(X1(147)) <= 168. Function of that degree: f[8]*f[35]*f[63]*f[73]^3/(f[6]^2*f[14]*f[28]*f[37]^3*f[42]*f[56]*f[74]^3) Gon_Q(X1(148)) <= 171. Function of that degree: f[23]*f[29]*f[58]*f[69]/(f[21]*f[32]*f[53]*f[74]) Gon_Q(X1(149)) <= 291. Function of that degree: f[7]/f[8] Gon_Q(X1(150)) <= 120. Function of that degree: f[4]^2*f[6]*f[7]^2*f[19]^3*f[38]^3*f[45]*f[76]^3/(f[5]*f[9]^2*f[12]*f[20]*f[37]^3*f[60]*f[74]^3) Gon_Q(X1(151)) <= 299. Function of that degree: f[7]/f[8] Gon_Q(X1(152)) <= 180. Function of that degree: f[9]*f[23]*f[25]*f[75]/(f[6]*f[7]*f[11]*f[77]) Gon_Q(X1(153)) <= 216. Function of that degree: f[8]*f[13]*f[19]*f[38]*f[76]/(f[7]*f[11]*f[77]) Gon_Q(X1(154)) <= 180. Function of that degree: f[18]*f[19]*f[38]*f[76]/(f[13]*f[26]*f[39]*f[78]) Gon_Q(X1(155)) <= 240. Function of that degree: f[7]*f[11]*f[18]*f[77]/(f[8]*f[13]^2*f[26]*f[39]*f[78]) Gon_Q(X1(156)) <= 168. Function of that degree: f[11]*f[19]*f[77]/f[79] Gon_Q(X1(157)) <= 323. Function of that degree: f[7]/f[8] Gon_Q(X1(158)) <= 234. Function of that degree: f[34]*f[47]/f[64] Gon_Q(X1(159)) <= 234. Function of that degree: f[37]*f[43]*f[74]/(f[11]*f[14]*f[42]*f[53]*f[64]) Gon_Q(X1(160)) <= 184. Function of that degree: f[12]*f[24]*f[79]/(f[16]*f[27]*f[81]) Gon_Q(X1(161)) <= 264. Function of that degree: f[5]*f[6]*f[10]*f[15]*f[16]*f[20]*f[40]*f[80]/(f[4]*f[9]*f[27]*f[81]) Gon_Q(X1(162)) <= 162. Function of that degree: f[13]*f[26]*f[36]*f[39]*f[78]/(f[16]*f[20]*f[27]*f[48]*f[54]*f[60]) Gon_Q(X1(163)) <= 348. Function of that degree: f[7]/f[8] Gon_Q(X1(164)) <= 210. Function of that degree: f[27]*f[43]*f[81]/(f[13]*f[39]*f[83]) Gon_Q(X1(165)) <= 240. Function of that degree: f[13]*f[41]*f[82]/f[83] Gon_Q(X1(166)) <= 240. Function of that degree: f[32]*f[61]/f[67] Gon_Q(X1(167)) <= 365. Function of that degree: f[7]/f[8] Gon_Q(X1(168)) <= 192. Function of that degree: f[49]*f[63]/(f[11]*f[77]) Gon_Q(X1(169)) <= 260. Function of that degree: f[3]*f[4]^2*f[6]^4*f[7]^6*f[12]^3*f[14]^3*f[21]^3*f[26]*f[28]^3*f[42]^3*f[84]^3/(f[5]^2*f[8]^5*f[9]^2*f[17]^3*f[85]^3) Gon_Q(X1(170)) <= 216. Function of that degree: f[19]*f[43]*f[57]*f[86]/(f[12]*f[21]*f[23]*f[31]*f[42]*f[62]*f[84]) Gon_Q(X1(171)) <= 270. Function of that degree: f[19]*f[56]*f[57]*f[58]/(f[27]*f[43]*f[86]) Gon_Q(X1(172)) <= 231. Function of that degree: f[13]*f[39]*f[47]*f[86]/(f[17]*f[41]*f[51]*f[82]) Gon_Q(X1(173)) <= 392. Function of that degree: f[7]/f[8] Gon_Q(X1(174)) <= 210. Function of that degree: f[31]*f[62]*f[79]/(f[10]*f[37]*f[50]*f[83]) Gon_Q(X1(175)) <= 240. Function of that degree: f[3]^7*f[5]^8*f[6]^4*f[9]^6*f[10]*f[14]*f[29]^21*f[70]*f[87]^21/(f[4]^23*f[7]*f[8]^7*f[11]^21*f[22]^21*f[44]^21*f[88]^21) Gon_Q(X1(176)) <= 240. Function of that degree: f[19]*f[29]*f[87]/f[89] Gon_Q(X1(177)) <= 290. Function of that degree: f[17]*f[29]*f[34]*f[87]/(f[56]*f[59]*f[62]) Gon_Q(X1(178)) <= 274. Function of that degree: f[37]*f[56]/f[61] Gon_Q(X1(179)) <= 420. Function of that degree: f[7]/f[8] Gon_Q(X1(180)) <= 192. Function of that degree: f[9]*f[13]*f[16]*f[65]*f[80]/(f[6]*f[7]*f[17]*f[50]*f[85]) Gon_Q(X1(181)) <= 429. Function of that degree: f[7]/f[8] Gon_Q(X1(182)) <= 252. Function of that degree: f[23]^2*f[46]^2*f[92]/(f[10]*f[16]*f[18]*f[30]*f[32]*f[59]*f[90]) Gon_Q(X1(183)) <= 310. Function of that degree: f[43]*f[47]*f[86]/(f[10]*f[11]*f[24]*f[50]*f[61]*f[72]) Gon_Q(X1(184)) <= 264. Function of that degree: f[6]*f[7]*f[13]*f[27]*f[91]/(f[9]*f[19]*f[31]*f[93]) Gon_Q(X1(185)) <= 342. Function of that degree: f[6]*f[7]*f[21]*f[23]*f[46]*f[92]/(f[9]*f[16]*f[31]*f[93]) Gon_Q(X1(186)) <= 240. Function of that degree: f[6]*f[8]*f[22]*f[71]*f[73]/(f[9]*f[40]*f[41]*f[53]*f[82]) Gon_Q(X1(187)) <= 360. Function of that degree: f[6]*f[7]^2*f[14]*f[31]*f[93]/(f[8]*f[9]*f[47]*f[94]) Gon_Q(X1(188)) <= 276. Function of that degree: f[29]*f[37]*f[74]*f[87]/(f[27]*f[40]*f[67]*f[94]) Gon_Q(X1(189)) <= 288. Function of that degree: f[13]*f[42]*f[91]/(f[10]*f[56]*f[63]*f[70]) Gon_Q(X1(190)) <= 270. Function of that degree: f[8]*f[9]*f[47]*f[94]/(f[6]*f[7]^2*f[14]*f[16]*f[32]*f[48]*f[96]) Gon_Q(X1(191)) <= 478. Function of that degree: f[7]/f[8] Gon_Q(X1(192)) <= 224. Function of that degree: f[5]*f[19]^2*f[32]*f[95]^2/(f[4]*f[7]*f[97]^2) Gon_Q(X1(193)) <= 488. Function of that degree: f[7]/f[8] Gon_Q(X1(194)) <= 328. Function of that degree: f[31]*f[62]/(f[10]*f[13]*f[14]*f[70]) Gon_Q(X1(195)) <= 336. Function of that degree: f[49]*f[98]/f[97] Gon_Q(X1(196)) <= 252. Function of that degree: f[10]*f[11]^2*f[33]*f[70]*f[77]*f[99]/(f[12]*f[21]*f[42]*f[63]*f[84]*f[97]) Gon_Q(X1(197)) <= 508. Function of that degree: f[7]/f[8] Gon_Q(X1(198)) <= 240. Function of that degree: f[6]^2*f[7]^8*f[9]*f[14]^11*f[49]^11*f[77]*f[98]^11/(f[4]^5*f[5]*f[8]^2*f[10]^11*f[20]^11*f[25]^11*f[44]*f[50]^11*f[100]^11) Gon_Q(X1(199)) <= 519. Function of that degree: f[7]/f[8] Gon_Q(X1(200)) <= 240. Function of that degree: f[3]^4*f[6]^13*f[7]^7*f[11]^12*f[16]*f[33]^12*f[80]*f[99]^12/(f[4]^4*f[8]^2*f[9]^9*f[101]^12) Gon_Q(X1(201)) <= 374. Function of that degree: f[22]*f[66]*f[67]*f[68]/(f[16]*f[32]*f[101]) Gon_Q(X1(202)) <= 382. Function of that degree: f[38]*f[61]/(f[16]*f[80]) Gon_Q(X1(203)) <= 420. Function of that degree: f[18]*f[101]/(f[11]*f[17]*f[34]*f[51]*f[102]) Gon_Q(X1(204)) <= 288. Function of that degree: f[23]*f[101]/(f[11]*f[103]) Gon_Q(X1(205)) <= 420. Function of that degree: f[17]*f[23]*f[34]*f[51]*f[102]/(f[18]*f[103]) Gon_Q(X1(206)) <= 398. Function of that degree: f[31]*f[40]*f[62]/f[82] Gon_Q(X1(207)) <= 396. Function of that degree: f[9]*f[16]*f[103]/(f[6]*f[7]*f[13]*f[26]*f[52]*f[104]) Gon_Q(X1(208)) <= 336. Function of that degree: f[4]*f[9]*f[103]/(f[5]*f[6]*f[15]*f[35]*f[105]) Gon_Q(X1(209)) <= 450. Function of that degree: f[8]*f[13]*f[26]*f[52]*f[104]/(f[7]*f[21]*f[35]*f[105]) Gon_Q(X1(210)) <= 288. Function of that degree: f[9]*f[13]*f[91]/(f[8]*f[35]*f[98]) Gon_Q(X1(211)) <= 583. Function of that degree: f[7]/f[8] Gon_Q(X1(212)) <= 351. Function of that degree: f[11]*f[33]*f[40]*f[73]*f[106]/(f[31]*f[43]*f[86]*f[93]) Gon_Q(X1(213)) <= 420. Function of that degree: f[37]*f[53]*f[106]/(f[17]*f[34]*f[107]) Gon_Q(X1(214)) <= 396. Function of that degree: f[22]*f[47]*f[66]/(f[37]*f[74]) Gon_Q(X1(215)) <= 462. Function of that degree: f[24]*f[107]/(f[18]*f[19]*f[27]*f[36]*f[54]*f[108]) Gon_Q(X1(216)) <= 288. Function of that degree: f[24]*f[107]/f[109] Gon_Q(X1(217)) <= 480. Function of that degree: f[18]*f[19]*f[27]*f[36]*f[54]*f[108]/f[109] Gon_Q(X1(218)) <= 445. Function of that degree: f[14]*f[42]*f[67]/f[88] Gon_Q(X1(219)) <= 444. Function of that degree: f[19]*f[38]*f[109]/(f[24]*f[72]*f[73]*f[74]) Gon_Q(X1(220)) <= 360. Function of that degree: f[27]*f[109]/(f[17]*f[37]*f[111]) Gon_Q(X1(221)) <= 504. Function of that degree: f[5]*f[6]*f[10]*f[11]*f[15]*f[22]*f[55]*f[110]/(f[4]*f[9]*f[37]*f[111]) Gon_Q(X1(222)) <= 342. Function of that degree: f[9]*f[10]*f[79]*f[101]/(f[6]*f[7]*f[47]*f[53]*f[64]*f[106]) Gon_Q(X1(223)) <= 651. Function of that degree: f[7]/f[8] Gon_Q(X1(224)) <= 384. Function of that degree: f[23]*f[37]*f[111]/f[113] Gon_Q(X1(225)) <= 360. Function of that degree: f[4]^20*f[6]^26*f[7]^45*f[10]*f[14]^27*f[16]^27*f[18]*f[28]^27*f[30]*f[56]^27*f[90]*f[112]^27/(f[5]^10*f[8]^23*f[9]^29*f[113]^27) Gon_Q(X1(226)) <= 444. Function of that degree: f[16]*f[23]*f[80]/f[73] Gon_Q(X1(227)) <= 675. Function of that degree: f[7]/f[8] Gon_Q(X1(228)) <= 360. Function of that degree: f[25]*f[113]/(f[13]*f[23]*f[115]) Gon_Q(X1(229)) <= 687. Function of that degree: f[7]/f[8] Gon_Q(X1(230)) <= 396. Function of that degree: f[6]*f[7]^2*f[14]*f[19]*f[28]*f[38]*f[57]*f[114]/(f[8]*f[9]*f[18]*f[29]*f[58]*f[116]) Gon_Q(X1(231)) <= 480. Function of that degree: f[16]*f[23]*f[115]/(f[29]*f[58]*f[116]) Gon_Q(X1(232)) <= 420. Function of that degree: f[18]*f[49]*f[67]*f[116]/(f[19]*f[38]*f[76]*f[107]) Gon_Q(X1(233)) <= 711. Function of that degree: f[7]/f[8] Gon_Q(X1(234)) <= 336. Function of that degree: f[4]^9*f[7]^4*f[9]^2*f[29]^13*f[58]^13*f[91]*f[116]^13/(f[5]^3*f[6]^2*f[8]^9*f[52]*f[59]^13*f[118]^13) Gon_Q(X1(235)) <= 552. Function of that degree: f[9]*f[13]^2*f[26]*f[39]*f[117]/(f[6]*f[7]*f[21]*f[59]*f[118]) Gon_Q(X1(236)) <= 435. Function of that degree: f[24]*f[47]*f[71]*f[118]/(f[53]*f[83]*f[106]) Gon_Q(X1(237)) <= 520. Function of that degree: f[41]*f[59]*f[118]/(f[16]*f[26]*f[78]*f[79]*f[80]) Gon_Q(X1(238)) <= 432. Function of that degree: f[4]*f[9]*f[59]*f[118]/(f[5]*f[6]*f[10]^2*f[15]*f[20]*f[30]*f[40]*f[60]*f[120]) Gon_Q(X1(239)) <= 748. Function of that degree: f[7]/f[8] Gon_Q(X1(240)) <= 384. Function of that degree: f[17]*f[23]*f[119]/(f[11]*f[121]) Gon_Q(X1(241)) <= 761. Function of that degree: f[7]/f[8] Gon_Q(X1(242)) <= 396. Function of that degree: f[9]^3*f[61]^2*f[77]*f[122]^2/(f[3]*f[6]^3*f[7]*f[10]^2*f[12]^2*f[15]^2*f[20]^2*f[24]^2*f[30]^2*f[33]*f[40]^2*f[60]^2*f[88]*f[120]^2) Gon_Q(X1(243)) <= 486. Function of that degree: f[11]*f[18]*f[121]/(f[61]*f[122]) Gon_Q(X1(244)) <= 465. Function of that degree: f[12]*f[36]*f[43]*f[79]*f[122]/(f[13]*f[26]*f[52]*f[97]*f[104]) Gon_Q(X1(245)) <= 504. Function of that degree: f[4]*f[5]^2*f[8]^5*f[9]^3*f[10]*f[14]*f[61]^15*f[70]*f[122]^15/(f[3]^5*f[6]^2*f[7]^7*f[41]^15*f[123]^15) Gon_Q(X1(246)) <= 420. Function of that degree: f[7]*f[11]*f[83]*f[121]/(f[8]*f[16]*f[43]*f[61]*f[80]*f[122]) Gon_Q(X1(247)) <= 630. Function of that degree: f[16]*f[41]*f[123]/(f[31]*f[62]*f[124]) Gon_Q(X1(248)) <= 480. Function of that degree: f[7]*f[35]*f[41]*f[123]/(f[8]*f[25]*f[27]*f[125]) Gon_Q(X1(249)) <= 574. Function of that degree: f[23]*f[41]*f[46]*f[123]/(f[16]*f[80]*f[83]*f[86]) Gon_Q(X1(250)) <= 375. Function of that degree: f[3]^2*f[5]^3*f[6]^8*f[7]^5*f[14]^5*f[15]*f[18]^5*f[21]^5*f[42]^5*f[63]^5*f[75]*f[126]^5/(f[4]^8*f[8]^4*f[9]^3*f[20]*f[31]^5*f[62]^5*f[100]*f[124]^5) Gon_Q(X1(251)) <= 825. Function of that degree: f[7]/f[8] Gon_Q(X1(252)) <= 384. Function of that degree: f[13]*f[16]*f[91]*f[112]/(f[10]*f[17]*f[70]*f[119]) Gon_Q(X1(253)) <= 660. Function of that degree: f[7]*f[14]*f[17]*f[18]*f[21]*f[42]*f[63]*f[126]/(f[8]*f[127]) Gon_Q(X1(254)) <= 605. Function of that degree: f[14]*f[113]/(f[67]*f[106]) Gon_Q(X1(255)) <= 576. Function of that degree: f[32]*f[64]*f[128]/f[127] Gon_Q(X1(256)) <= 448. Function of that degree: f[4]^3*f[8]^5*f[64]*f[127]^6/(f[3]^2*f[5]*f[6]^2*f[9]^2*f[43]^6*f[129]^6) Gon_Q(X1(257)) <= 865. Function of that degree: f[7]/f[8] Gon_Q(X1(258)) <= 462. Function of that degree: f[7]*f[40]*f[53]*f[89]*f[106]/(f[8]*f[46]*f[83]*f[109]) Gon_Q(X1(259)) <= 684. Function of that degree: f[4]*f[9]*f[11]*f[22]*f[43]*f[129]/(f[5]*f[6]*f[10]*f[13]*f[15]*f[26]*f[65]*f[130]) Gon_Q(X1(260)) <= 504. Function of that degree: f[7]*f[91]/(f[8]*f[104]) Gon_Q(X1(261)) <= 630. Function of that degree: f[13]*f[19]*f[26]*f[65]*f[130]/f[131] Gon_Q(X1(262)) <= 596. Function of that degree: f[11]*f[18]*f[121]/(f[61]*f[122]) Gon_Q(X1(263)) <= 906. Function of that degree: f[7]/f[8] Gon_Q(X1(264)) <= 480. Function of that degree: f[6]*f[7]*f[77]*f[99]/(f[9]*f[121]) Gon_Q(X1(265)) <= 702. Function of that degree: f[8]*f[11]*f[22]*f[29]*f[33]*f[44]*f[66]*f[132]/(f[7]*f[19]*f[24]*f[133]) Gon_Q(X1(266)) <= 540. Function of that degree: f[11]*f[13]*f[22]*f[26]*f[33]*f[44]*f[66]*f[132]/(f[67]*f[134]) Gon_Q(X1(267)) <= 660. Function of that degree: f[7]*f[19]*f[23]*f[46]*f[133]/(f[8]*f[43]*f[67]*f[134]) Gon_Q(X1(268)) <= 561. Function of that degree: f[12]*f[61]*f[73]*f[134]/(f[16]*f[32]*f[64]*f[79]*f[128]) Gon_Q(X1(269)) <= 948. Function of that degree: f[7]/f[8] Gon_Q(X1(270)) <= 432. Function of that degree: f[12]*f[13]*f[26]*f[60]*f[65]*f[130]/(f[16]*f[18]*f[45]*f[80]*f[90]*f[100]) Gon_Q(X1(271)) <= 962. Function of that degree: f[7]/f[8] Gon_Q(X1(272)) <= 576. Function of that degree: f[5]*f[6]*f[85]*f[136]/(f[4]*f[7]*f[102]*f[119]) Gon_Q(X1(273)) <= 672. Function of that degree: f[17]^2*f[34]*f[68]*f[136]/f[137] Gon_Q(X1(274)) <= 650. Function of that degree: f[8]*f[26]*f[71]*f[78]/(f[7]*f[14]*f[49]*f[98]) Gon_Q(X1(275)) <= 600. Function of that degree: f[4]^19*f[8]^20*f[9]^20*f[10]*f[22]*f[110]*f[137]^33/(f[3]^11*f[5]^4*f[6]^43*f[7]^21*f[23]^33*f[46]^33*f[69]^33*f[138]^33) Gon_Q(X1(276)) <= 528. Function of that degree: f[29]*f[137]/(f[17]*f[139]) Gon_Q(X1(277)) <= 1005. Function of that degree: f[7]/f[8] Gon_Q(X1(278)) <= 672. Function of that degree: f[6]*f[7]*f[13]*f[91]/(f[9]*f[32]*f[37]*f[96]) Gon_Q(X1(279)) <= 720. Function of that degree: f[8]*f[139]/(f[7]*f[11]*f[14]*f[28]*f[35]*f[70]*f[140]) Gon_Q(X1(280)) <= 576. Function of that degree: f[49]*f[98]/(f[16]*f[112]) Gon_Q(X1(281)) <= 1034. Function of that degree: f[7]/f[8] Gon_Q(X1(282)) <= 552. Function of that degree: f[14]*f[101]*f[127]/(f[16]*f[61]*f[67]*f[80]*f[134]) Gon_Q(X1(283)) <= 1049. Function of that degree: f[7]/f[8] Gon_Q(X1(284)) <= 630. Function of that degree: f[47]*f[73]*f[141]/(f[11]*f[13]*f[23]*f[69]*f[143]) Gon_Q(X1(285)) <= 720. Function of that degree: f[17]*f[71]*f[142]/(f[11]*f[13]*f[143]) Gon_Q(X1(286)) <= 630. Function of that degree: f[71]*f[142]/(f[16]*f[18]*f[36]*f[48]*f[72]*f[144]) Gon_Q(X1(287)) <= 840. Function of that degree: f[11]*f[13]*f[143]/(f[16]*f[17]*f[18]*f[36]*f[48]*f[72]*f[144]) Gon_Q(X1(288)) <= 504. Function of that degree: f[4]^2*f[8]^2*f[9]*f[11]^3*f[13]^3*f[16]*f[48]*f[143]^3/(f[5]^2*f[6]^2*f[7]*f[29]^3*f[145]^3) Gon_Q(X1(289)) <= 765. Function of that degree: f[3]*f[4]^2*f[6]^3*f[7]*f[12]^3*f[16]^3*f[18]^3*f[24]^3*f[34]*f[36]^3*f[48]^3*f[72]^3*f[144]^3/(f[5]^2*f[8]*f[9]*f[29]^3*f[145]^3) Gon_Q(X1(290)) <= 630. Function of that degree: f[16]*f[17]*f[18]*f[34]*f[36]*f[48]*f[72]*f[144]/(f[73]*f[146]) Gon_Q(X1(291)) <= 784. Function of that degree: f[31]*f[47]*f[62]*f[141]/(f[88]*f[97]*f[106]) Gon_Q(X1(292)) <= 666. Function of that degree: f[5]*f[6]*f[17]*f[24]*f[61]*f[85]*f[146]/(f[4]*f[8]*f[67]*f[97]*f[134]) Gon_Q(X1(293)) <= 1124. Function of that degree: f[7]/f[8] Gon_Q(X1(294)) <= 504. Function of that degree: f[5]^6*f[6]^2*f[9]^6*f[12]*f[28]*f[73]^9*f[84]*f[146]^9/(f[4]^10*f[7]^4*f[8]^4*f[37]^9*f[74]^9*f[148]^9) Gon_Q(X1(295)) <= 870. Function of that degree: f[7]*f[16]*f[21]*f[32]*f[49]*f[147]/(f[8]*f[27]*f[37]*f[74]*f[148]) Gon_Q(X1(296)) <= 684. Function of that degree: f[6]*f[7]*f[21]*f[41]*f[49]*f[147]/(f[9]*f[11]*f[33]*f[149]) Gon_Q(X1(297)) <= 720. Function of that degree: f[13]*f[66]*f[143]/(f[10]*f[88]*f[99]*f[110]) Gon_Q(X1(298)) <= 832. Function of that degree: f[18]*f[131]/(f[79]*f[122]) Gon_Q(X1(299)) <= 924. Function of that degree: f[4]*f[9]*f[18]*f[149]/(f[5]*f[6]*f[10]*f[15]*f[25]*f[30]*f[50]*f[75]*f[150]) Gon_Q(X1(300)) <= 480. Function of that degree: f[6]^3*f[7]^8*f[24]*f[40]*f[120]*f[149]^18/(f[8]^5*f[9]^3*f[151]^18)