Atkin Modular Polynomial Database

The Atkin modular function $f(q)$ of prime level N is a modular function on $X_0^+(N)$ of minimal degree, holomorphic on the upper half plane, and whose $q$-expansion has leading coefficient $1$. It is unique up to a constant. The Atkin modular polynomial is a bivariate polynomial $\Phi_N(X,Y)$ such that $$ \Phi_N(f(q),j(q)) = 0, $$ where $j(q) = q^{-1} + 744 + 196884 q + \cdots$ is the modular $j$-invariant. Since $f(q)$ is invariant under the Atkin-Lehner operator, and the image of $j(q)$ is $j_N(q) = j(q^N)$, we also have $\Phi_N(f(q),j_N(q)) = 0$.

Level Degree Number of monomials
[2] 1 8
[3] 1 10
[5] 1 14
[7] 1 18
[11] 1 26
[13] 1 30
[17] 1 38
[19] 1 42
[23] 1 50
[29] 1 62
[31] 1 66
[37] 2 135
[41] 1 86
[43] 2 156
[47] 1 98
[53] 2 191
[59] 1 122
[61] 2 219
[67] 3 342
[71] 1 146
[73] 3 372
[79] 2 282
[83] 2 296
[89] 2 317
[97] 4 639
[101] 2 359
[103] 3 522
[107] 3 542
[109] 3 552
[113] 4 743
[127] 4 834
[131] 2 464
[137] 5 1106
[139] 4 912
[149] 4 977
[151] 3 762
[157] 6 1503
[163] 7 1806
[167] 3 842
[173] 4 1133
[179] 3 902
[181] 6 1731
[191] 3 962
[193] 6 1845
[197] 6 1883
[199] 5 1602
[211] 7 2334
[223] 7 2466
[227] 5 1826
[229] 7 2532
[233] 7 2576
[239] 4 1562
[241] 6 2301
[251] 4 1640
[257] 6 2453
[263] 5 2114
[269] 6 2567
[271] 6 2586
[277] 9 3894
[281] 7 3104
[283] 8 3552
[293] 7 3236
[307] 10 4776
[311] 5 2498
[313] 11 5340
[317] 11 5408
[331] 11 5646
[337] 12 6255
[347] 9 4874
[349] 10 5427
[353] 8 4427
[359] 6 3422
[367] 10 5706
[373] 12 6921
[379] 12 7032
[383] 8 4802
[389] 12 7217
[397] 12 7365
[401] 10 6233
[409] 10 6357
[419] 7 4622
[421] 14 9075
[431] 7 4754
[433] 14 9333
[439] 11 7482
[443] 12 8216
[449] 11 7652
[457] 13 9162
[461] 11 7856
[463] 14 9978
[467] 11 7958
[479] 8 6002
[487] 15 11226
[491] 8 6152
[499] 16 12252
[503] 10 7814
[509] 12 9437
[521] 12 9659
[523] 15 12054
[541] 17 14094
[547] 18 15072
[557] 16 13673
[563] 13 11282
[569] 14 12257
[571] 19 16590
[577] 20 17631
[587] 14 12644
[593] 14 12773
[599] 10 9302
[601] 15 13848
[607] 18 16722
[613] 19 17808
[617] 20 18851
[619] 17 16122
[631] 15 14538
[641] 16 15731
[643] 22 21576
[647] 13 12962
[653] 24 23873
[659] 11 11222
[661] 21 21186
[673] 24 24603
[677] 16 16613
[683] 19 19838
[691] 21 22146
[701] 21 22466
[709] 23 24852
[719] 12 13322
[727] 21 23298
[733] 25 27894
[739] 24 27012
[743] 15 17114
[751] 18 20682
[757] 26 29943
[761] 18 20957
[769] 19 22332
[773] 18 21287
[787] 26 31128
[797] 19 23144
[809] 20 24707
[811] 25 30858
[821] 24 30005
[823] 24 30078
[827] 23 28982
[829] 27 34032
[839] 14 18062
[853] 29 37578
[857] 20 26171
[859] 27 35262
[863] 18 23762
[877] 28 37317
[881] 21 28226
[883] 32 42876
[887] 18 24422
[907] 29 39954
[911] 15 20978
[919] 23 32202
[929] 22 31157
[937] 33 46902
[941] 22 31559
[947] 26 37448
[953] 28 40547
[967] 30 44046
[971] 16 23816
[977] 34 50369
[983] 20 30014
[991] 24 36210
[997] 34 51399
[1009] 25 38382
[1013] 24 37013
[1019] 17 26522
[1021] 34 52635
[1031] 17 26834
[1033] 32 50151
[1039] 26 41082
[1049] 25 39902
[1051] 35 55758
[1061] 34 54695
[1063] 33 53202
[1069] 35 56712
[1087] 34 56034
[1091] 18 30032
[1093] 42 69471
[1097] 26 43373
[1103] 23 38642
[1109] 26 43847
[1117] 38 64287
[1123] 38 64632
[1129] 28 48027
[1151] 19 33410
[1153] 37 64626
[1163] 32 56456
[1171] 39 69150
[1181] 28 50237
[1187] 33 59402
[1193] 28 50747
[1201] 30 54693
[1213] 42 77091
[1217] 29 53594
[1223] 25 46514
[1229] 37 68882
[1231] 30 56058
[1237] 48 89757
[1249] 31 58752
[1259] 21 40322
[1277] 30 58151
[1279] 32 62082
[1283] 35 68054
[1289] 32 62567
[1291] 42 82044
[1297] 46 90213
[1301] 31 61196
[1303] 39 76938
[1307] 31 61478
[1319] 22 44222
[1321] 33 66102
[1327] 41 82338
[1361] 32 66059
[1367] 28 58142
[1373] 47 97556
[1381] 46 96051
[1399] 35 74202
[1409] 35 74732
[1423] 44 94698
[1427] 34 73544
[1429] 47 101532
[1433] 34 73853
[1439] 24 52562
[1447] 45 98466
[1451] 24 53000
[1459] 43 94902
[1471] 36 80226
[1481] 37 82994
[1487] 31 69938
[1489] 37 83442
[1493] 45 101594
[1499] 25 57002
[1511] 25 57458
[1523] 42 96776
[1543] 48 111942
[1553] 37 87026
[1559] 26 61622
[1571] 26 62096
[1583] 33 79202
[1601] 38 92117
[1607] 33 80402
[1609] 40 97407
[1613] 38 92807
[1619] 27 66422
[1637] 39 96644
[1667] 46 115928
[1697] 40 102731
[1721] 41 106766
[1759] 44 117042
[1801] 45 122538
[1811] 30 82448
[1823] 38 104882
[1831] 45 124578
[1847] 38 106262
[1871] 31 87986
[1889] 45 128522
[1907] 45 129746
[1931] 32 93704
[1979] 33 99002
[2039] 34 105062
[2099] 35 111302
[2111] 35 111938