Let $C$ be a nonsingular projective curve defined over $\mathbb{Q}$, which does **not** admit a map of degree 1 or 2 to $\mathbb{P}^1$ or to an elliptic curve. It is then a consequence of Corollary 3 of [1] that $C$ possesses only finitely many quadratic points; i.e., the set

$$\Gamma_C := \left\{p \in C : [\mathbb{Q}(p) : \mathbb{Q}] = 2\right\}$$

is finite. In particular, there exists a bound $D_C$ such that, if $D$ is a squarefree integer satisfying $|D| > D_C$, then $C(\mathbb{Q}(\sqrt{D})) = C(\mathbb{Q})$.

It is natural to ask if there is an algorithm to effectively compute $D_C$, given a model of $C$. Going through the proof of the Harris-Silverman result does not suggest any such algorithm to me (unless the rank of the Jacobian of $C$ is 0).

Given a model of $C$, is there an algorithm to effectively compute $D_C$?

I would also like to ask if $D_C$ is known when $C$ is the modular curve $X_0(N)$; there are only 55 values of $N$ for which $X_0(N)$ does **not** have finitely many quadratic points (see Theorem 4.9 in [2] for this list). Let $S$ be this set of 55 integers.

Given $N \notin S$, is there an algorithm to effectively compute $D_{X_0(N)}$?

[1]: J. Harris, J.H. Silverman. Bielliptic Curves and Symmetric Products. Proc. Amer. Math. Soc. 112 (1991), 347-356

[2]: F. Bars. On Quadratic Points of Classical Modular Curves. Manuscript, 2012, available from the author's website.