An Algorithm for Computing the Singularities of the Plane Model of X0(N)

Let ΦN(X,Y) be the N-th classical modular polynomial and let Z0(N)={(X,Y)∈C2∣ΦN(X,Y)=0} be the plane model of the modular curve X0(N). We present an explicit procedure that, for a prime ℓ, enumerates all non-cuspidal singular points of Z0(ℓ) over C and outputs the corresponding pairs of distinct points on X0(ℓ) mapping to each node. The method relies on the arithmetic (CM) classification of self-intersections of the map X0(ℓ)→Z0(ℓ) and on effective computations of proper ideal classes in imaginary quadratic orders. We also provide a complete and self-contained exposition of Kara’s proof of the automorphism-group equality Aut(E)=Aut(E′) in the self-intersection setting, making explicit where Kolyvagin’s conductor lemma is used essentially. Finally, we discuss termination, correctness, and practical complexity issues, and we report computational evidence for larger primes using a parallel implementation; in particular, for ℓ=389, we obtained 151,288 output pairs in 151,017 seconds on a 56-core machine.