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
[337] 12 6255
[373] 12 6921
[379] 12 7032
[389] 12 7217
[397] 12 7365
[443] 12 8216
[509] 12 9437
[521] 12 9659
[719] 12 13322