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.
Wang et al. (Fri,) studied this question.