When finding out whether an equation in 2 variables has rational solutions (or, equivalently, whether an algebraic curve has any rational points), many authors recommend checking the local solubility of the equation (that is, solubility in the field ${\mathbb Q}_p$ of $p$-adic numbers) as the first "easy" step. If the equation is not solvable in some ${\mathbb Q}_p$, then it has no rational solutions, otherwise we may start looking for more complicated methods.
The usual justification that this step is "easy" is the fact that for each individual prime the problem is known to be "decidable", together with the fact that it suffices to check this for a finite set of primes only (say, prime $p=\infty$ (which just means checking for real solutions), primes of bad reduction, and primes $p$ such that $p+1-2g\sqrt{p}\leq 0$ where $g$ is the genus).
However, "decidable" does not meet practical. Are you aware of any implementation in computer algebra system (Mathematica, Maple, etc.) where I can check the solvability in ${\mathbb Q}_p$ even for individual $p$? Also, are there an easy methods to compute the set of primes of bad reduction?
As an example, how can I do this for the specific equation $y^3+x y + x^4 + 4 = 0$?
