accessibility Defining equations for X1(N)

The table below lists (links to) defining equations f(x,y)=0 that are birationally equivalent to the "raw" equations F(r,s)=0 for X1(N) listed here. In every case, the map from f(x,y)=0 to F(r,s)=0 is given by

r = (x2yxy + y − 1) / (x2yx),        s = (xyy + 1) / (xy),

and the inverse map from F(r,s)=0 to f(x,y)=0 is

x = (sr) / (rs − 2r + 1),       y = (rs − 2r + 1) / (s2sr + 1).

The polynomials f(x,y) generally have substantially lower degrees than F(r,s). When N=101, for example, F(r,s) has degree 170 in r and 255 in s, while f(x,y) has degree 140 in x and 134 in y (it also has fewer terms and smaller coefficients). In most cases the equations f(x,y)=0 listed below achieve the same bi-degree as the optimized equations listed here, and in several cases they are actually identical.

Update July 24,2020: Mark van Hoeij has implemented a maple program that can be used both to recreate these polynomials and compute them for larger values of N.

Ndegxdegyf(x,y) Ndegxdegyf(x,y) Ndegxdegyf(x,y) Ndegxdegyf(x,y) Ndegxdegyf(x,y) Ndegxdegyf(x,y)
10 1 1f(x,y) 40 15 16f(x,y) 70 47 45f(x,y) 100 98 95f(x,y) 130167158f(x,y) 160252241f(x,y)
11 2 2f(x,y) 41 23 22f(x,y) 71 69 66f(x,y) 101140134f(x,y) 131235225f(x,y) 161347332f(x,y)
12 1 1f(x,y) 42 16 15f(x,y) 72 47 45f(x,y) 102 95 91f(x,y) 132159150f(x,y) 162241228f(x,y)
13 3 2f(x,y) 43 26 24f(x,y) 73 73 70f(x,y) 103145139f(x,y) 133237226f(x,y) 163364348f(x,y)
14 2 2f(x,y) 44 19 19f(x,y) 74 56 54f(x,y) 104110106f(x,y) 134184176f(x,y) 164276264f(x,y)
15 2 2f(x,y) 45 25 23f(x,y) 75 66 63f(x,y) 105126121f(x,y) 135211204f(x,y) 165317303f(x,y)
16 2 3f(x,y) 46 22 21f(x,y) 76 59 56f(x,y) 106115110f(x,y) 136189182f(x,y) 166282271f(x,y)
17 4 4f(x,y) 47 30 29f(x,y) 77 79 75f(x,y) 107157150f(x,y) 137257246f(x,y) 167382365f(x,y)
18 4 3f(x,y) 48 22 19f(x,y) 78 57 52f(x,y) 108107102f(x,y) 138173166f(x,y) 168253240f(x,y)
19 5 5f(x,y) 49 32 31f(x,y) 79 85 82f(x,y) 109163156f(x,y) 139265253f(x,y) 169388372f(x,y)
20 4 3f(x,y) 50 25 23f(x,y) 80 64 60f(x,y) 110118114f(x,y) 140189182f(x,y) 170283272f(x,y)
21 5 5f(x,y) 51 31 30f(x,y) 81 79 77f(x,y) 111149143f(x,y) 141242231f(x,y) 171356340f(x,y)
22 5 4f(x,y) 52 28 26f(x,y) 82 69 66f(x,y) 112127120f(x,y) 142207198f(x,y) 172304290f(x,y)
23 7 7f(x,y) 53 39 37f(x,y) 83 95 90f(x,y) 113175167f(x,y) 143275264f(x,y) 173410392f(x,y)
24 5 6f(x,y) 54 25 26f(x,y) 84 63 61f(x,y) 114118114f(x,y) 144189182f(x,y) 174277264f(x,y)
25 8 8f(x,y) 55 39 37f(x,y) 85 95 90f(x,y) 115174166f(x,y) 145275264f(x,y) 175394377f(x,y)
26 6 7f(x,y) 56 31 31f(x,y) 86 75 73f(x,y) 116138132f(x,y) 146219209f(x,y) 176316301f(x,y)
27 10 8f(x,y) 57 39 37f(x,y) 87 91 88f(x,y) 117167158f(x,y) 147258246f(x,y) 177380364f(x,y)
28 8 7f(x,y) 58 34 33f(x,y) 88 79 76f(x,y) 118143136f(x,y) 148225214f(x,y) 178326311f(x,y)
29 12 11f(x,y) 59 48 46f(x,y) 89108104f(x,y) 119189181f(x,y) 149304291f(x,y) 179439420f(x,y)
30 8 8f(x,y) 60 32 31f(x,y) 90 70 67f(x,y) 120127119f(x,y) 150197189f(x,y) 180283272f(x,y)
31 13 13f(x,y) 61 51 49f(x,y) 91110106f(x,y) 121198190f(x,y) 151312299f(x,y) 181449429f(x,y)
32 11 10f(x,y) 62 40 37f(x,y) 92 87 82f(x,y) 122153146f(x,y) 152236227f(x,y) 182332316f(x,y)
33 12 12f(x,y) 63 48 45f(x,y) 93105100f(x,y) 123184176f(x,y) 153285271f(x,y) 183407389f(x,y)
34 12 11f(x,y) 64 42 40f(x,y) 94 91 87f(x,y) 124157151f(x,y) 154236227f(x,y) 184346333f(x,y)
35 16 15f(x,y) 65 54 53f(x,y) 95118113f(x,y) 125206196f(x,y) 155316301f(x,y) 185449430f(x,y)
36 12 11f(x,y) 66 40 39f(x,y) 96 83 81f(x,y) 126141136f(x,y) 156220212f(x,y) 186315303f(x,y)
37 19 18f(x,y) 67 62 59f(x,y) 97129123f(x,y) 127221211f(x,y) 157338323f(x,y) 187473452f(x,y)
38 15 14f(x,y) 68 47 45f(x,y) 98 97 92f(x,y) 128168161f(x,y) 158257245f(x,y) 188363346f(x,y)
39 17 18f(x,y) 69 58 55f(x,y) 99120113f(x,y) 129201194f(x,y) 159306294f(x,y) 189425408f(x,y)

The table below lists an alternative set of defining equations f(x,y)=0 for X1(N). For these equations, the birational map map from f(x,y)=0 to F(r,s)=0 is given by

r = (x2(x + 1) + x + y) / (−xy(x + 1) + x + y),        s = (x2 + x + y) / (−xy + x + y),

and the map from F(r,s)=0 to f(r,s)=0 is defined by

x = −(rs − 2r+1) / (sr),        y = (rs− 2r+1)(s2sr+1) / ((sr)(rs2 + s2 − 3rs + r)).

Ndegxdegyf(x,y) Ndegxdegyf(x,y) Ndegxdegyf(x,y) Ndegxdegyf(x,y) Ndegxdegyf(x,y) Ndegxdegyf(x,y)
40 17 16f(x,y) 70 49 45f(x,y) 100103 95f(x,y) 130173158f(x,y) 160263241f(x,y)
11 2 2f(x,y) 41 24 22f(x,y) 71 72 66f(x,y) 101146134f(x,y) 131245225f(x,y) 161361332f(x,y)
12 0 1f(x,y) 42 17 15f(x,y) 72 48 45f(x,y) 102100 91f(x,y) 132165150f(x,y) 162250228f(x,y)
13 3 2f(x,y) 43 27 24f(x,y) 73 77 70f(x,y) 103152139f(x,y) 133247226f(x,y) 163380348f(x,y)
14 2 2f(x,y) 44 20 19f(x,y) 74 59 54f(x,y) 104116106f(x,y) 134192176f(x,y) 164288264f(x,y)
15 2 2f(x,y) 45 26 23f(x,y) 75 68 63f(x,y) 105133121f(x,y) 135221204f(x,y) 165330303f(x,y)
16 3 3f(x,y) 46 23 21f(x,y) 76 61 56f(x,y) 106120110f(x,y) 136199182f(x,y) 166295271f(x,y)
17 5 4f(x,y) 47 32 29f(x,y) 77 82 75f(x,y) 107164150f(x,y) 137269246f(x,y) 167399365f(x,y)
18 4 3f(x,y) 48 21 19f(x,y) 78 58 52f(x,y) 108111102f(x,y) 138181166f(x,y) 168261240f(x,y)
19 5 5f(x,y) 49 33 31f(x,y) 79 89 82f(x,y) 109170156f(x,y) 139276253f(x,y) 169405372f(x,y)
20 3 3f(x,y) 50 25 23f(x,y) 80 66 60f(x,y) 110123114f(x,y) 140199182f(x,y) 170296272f(x,y)
21 4 5f(x,y) 51 31 30f(x,y) 81 83 77f(x,y) 111155143f(x,y) 141251231f(x,y) 171371340f(x,y)
22 5 4f(x,y) 52 28 26f(x,y) 82 72 66f(x,y) 112131120f(x,y) 142216198f(x,y) 172316290f(x,y)
23 8 7f(x,y) 53 41 37f(x,y) 83 99 90f(x,y) 113183167f(x,y) 143288264f(x,y) 173428392f(x,y)
24 7 6f(x,y) 54 27 26f(x,y) 84 68 61f(x,y) 114123114f(x,y) 144199182f(x,y) 174288264f(x,y)
25 9 8f(x,y) 55 41 37f(x,y) 85 98 90f(x,y) 115181166f(x,y) 145288264f(x,y) 175411377f(x,y)
26 7 7f(x,y) 56 34 31f(x,y) 86 79 73f(x,y) 116144132f(x,y) 146228209f(x,y) 176329301f(x,y)
27 10 8f(x,y) 57 41 37f(x,y) 87 96 88f(x,y) 117174158f(x,y) 147270246f(x,y) 177397364f(x,y)
28 7 7f(x,y) 58 36 33f(x,y) 88 83 76f(x,y) 118149136f(x,y) 148233214f(x,y) 178340311f(x,y)
29 12 11f(x,y) 59 50 46f(x,y) 89113104f(x,y) 119196181f(x,y) 149317291f(x,y) 179458420f(x,y)
30 8 8f(x,y) 60 35 31f(x,y) 90 73 67f(x,y) 120130119f(x,y) 150207189f(x,y) 180295272f(x,y)
31 14 13f(x,y) 61 53 49f(x,y) 91114106f(x,y) 121207190f(x,y) 151326299f(x,y) 181468429f(x,y)
32 11 10f(x,y) 62 41 37f(x,y) 92 89 82f(x,y) 122160146f(x,y) 152247227f(x,y) 182346316f(x,y)
33 13 12f(x,y) 63 51 45f(x,y) 93109100f(x,y) 123192176f(x,y) 153298271f(x,y) 183425389f(x,y)
34 12 11f(x,y) 64 44 40f(x,y) 94 95 87f(x,y) 124164151f(x,y) 154247227f(x,y) 184363333f(x,y)
35 16 15f(x,y) 65 57 53f(x,y) 95124113f(x,y) 125214196f(x,y) 155329301f(x,y) 185469430f(x,y)
36 13 11f(x,y) 66 42 39f(x,y) 96 88 81f(x,y) 126147136f(x,y) 156232212f(x,y)
37 20 18f(x,y) 67 65 59f(x,y) 97135123f(x,y) 127231211f(x,y) 157353323f(x,y)
38 16 14f(x,y) 68 48 45f(x,y) 98101 92f(x,y) 128175161f(x,y) 158268245f(x,y)
39 18 18f(x,y) 69 59 55f(x,y) 99124113f(x,y) 129210194f(x,y) 159319294f(x,y)