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